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WULF ALBERS • WERNER GUTH 

PETER HAMMERSTEIN • BENNY MOLDOVANU 

ERIC VAN DAMME (Editors) 



Understanding 

Strategic 

Interaction 

Essays in Honor 
of Reinhard Selten 




Springer 





Understanding Strategic Interaction 




springer 

Berlin 

Heidelberg 

New York 

Barcelona 

Budapest 

Hong Kong 

London 

Milan 

Paris 

Santa Clara 

Singapore 

Tokyo 




Wulf Albers • Werner Giith 

Peter Hammerstein • Benny Moldovanu 

Eric van Damme (Eds.) 

With the help of Martin Strobel 

Understanding 
Strategic Interaction 

Essays in Honor 
of Reinhard Selten 



With 86 Figures 




Springer 




Prof. Dr. Wulf Albers 

Universitat Bielefeld, FB Wirtschaftswissenschaften, 

Institut fur Mathematische Wirtschaftsforschung, Universitatsstrafie, 

33501 Bielefeld, Germany 

Prof. Dr. Werner Giith 

Humboldt-Universitat zu Berlin, Wirtschaftswissenschaftliche Fakultat, 
Institut fur Wirtschaftstheorie, Spandauer StraCe 1, 10178 Berlin, Germany 

Priv.-Doz. Dr. Peter Hammerstein 

Max-Planck-Institut fur Verhaltensphysiologie, Abteilung Wickler, 

82319 Seewiesen, Germany 

Prof. Dr. Benny Moldovanu 

Universitat Mannheim, Lehrstuhl fiir VWL, insb. Wirtschaftstheorie, 
Seminargebaude A5, 68131 Mannheim, Germany 

Prof. Dr. Eric van Damme 

CentER, Center for Economic Research, Warandelaan 2, 

NL-5037 AB Tilburg, The Netherlands 



Catologing-in-Publication Data applied for 

Die Deutsche Bibliothek - CIP-Einheitsaufhahme 

Understanding strategic interaction: essays in honor of Reinhard SeltenAVulf Albers ... (ed.) with 
the help of Martin Strobel. - Berlin; Heidelberg; New York; Barcelona; Budapest; Hong Kong; 
London; Milan; Paris; Santa Clara; Singapore; Tokyo: Springer, 1996 

ISBN-13: 978-3-642-64430-6 e-ISBN-13: 978-3-642-60495-9 
DOI: 10.1007/978-3-642-60495-9 

NE: Albers; Wulf [Hrsg.]; Selten, Reinhard: Festschrift 

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© Springer- Verlag Berlin • Heidelberg 1997 
Soflcover reprint of the hardcover 1st edition 1997 

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SPIN 10543783 42/2202-5 43210 - Printed on acid-free paper 




Preface 



On the 5th of October, 1995, Reinhard Selten celebrated his 65th birthday. To 
honor his scientific contributions, his infiuence and encouragement of other 
researchers the editors have asked prominent scholars for contributions. We 
are rather proud of the result, but the reader may judge this for her- or 
himself. Also on behalf of my co-editors I thank all authors who helped us to 
honor Reinhard Selten by contributing to this volume. 

The main burden (most of the correspondence and all technical work) of 
editing the volume has rested on the shoulders of Martin Strobel who has been 
responsible for all technical aspects as well as for part of the correspondence 
involved. I gratefully acknowledge his kind help and fruitful cooperation, also 
as an inspiring co-author and as a co-interviewer. 

Among my co-editors Eric van Damme has been most helpful by refereeing 
the papers like the others, but also by approaching authors and especially by 
interviewing Bob Aumann on the occasion of the Jerusalem Conference in 
1995. Without the help and assistance of my co-editors the book would not 
be as interesting as it is - in my view - now. 

Given the prominence of the laureate there was a considerable risk of over- 
burdening the volume. To avoid this it was decided not to ask his present 
and former colleagues at the universities of Bonn, Bielefeld, Berlin and Prank- 
furt/M., nor his present and former assistants for contributions. So what the 
volume tries to accomplish is mainly a recollection of the scientific disputes, 
in which Reinhard Selten was actively and partly dominantly involved, as 
well as the methodologal approaches of a core group of researchers who con- 
tinuously enjoyed the close cooperation with Reinhard Selten. 

Initially our attempt was to have separate sections for game theory, ap- 
plications of game theory, experimental economics and evolutionary game 
theory. Instead the contributions now are ordered more loosely in this way. 
The introductory essay offers a finer partitioning and provides some overview 
and guidance for the more selective readers. 

Berlin, 1996 



Werner Giith 




Contents 



Interview with Elisabeth and Reinhard Selten 1 

On the State of the Art in Game Theory: 

An Interview with Robert Aumann 8 

Working with Reinhard Selten 

Some Recollections on Our Joint Work 1965-88 35 

John C. Harsanyi 

Introduction and Survey 39 

Werner Giith 

Conceptual Ideas in Game Theory 

A Note on Imperfect Recall 51 

Ken Binmore 

Futures Market Contracting 

When You Don’t Know Who the Optimists Are 63 

Ronald M. Harstad, Louis Phlips 

Games of Incomplete Information: The Inconsistent Case 79 

Michael Maschler 

Admissibility and Stability 85 

Robert Wilson 

Equilibrium Selection in Team Games 100 

Eric van Damme 

Sustainable Equilibria in Culturally Familiar Games Ill 

Roger B. Myerson 

Evolutionary Game Theory 

Evolutionary Conflict and the Design of Life 122 

Peter Hammerstein 

Evolutionary Selection Dynamics and Irrational Survivors 128 

Jonas Bjomerstedt, Martin Dufwenberg, Peter Norman, 

Jorgen W. Weibull 

Strict and Symmetric Correlated Equilibria are the Distributions 

of the ESS’s of Biological Conflicts with Asymmetric Roles 149 

Avi Shmida, Bezalel Peleg 




VIII 



Applications of Non-Cooperative Game Theory 

Recurring Bullies, Trembling and Learning 171 

Matthew Jackson, Ehud Kalai 

Dumb Bugs vs. Bright Noncooperative Players: A Comparison 185 

Thomas Quint, Martin Shubik, Dicky Van 

Communication Effort in Teams and in Games 198 

Eric J. Friedman, Thomas Marschak 

Endogenous Agendas in Committees 217 

Eyal Winter 

The Organization of Social Cooperation: 

A Noncooperative Approach 228 

Akira Okada 

Reinhard Selten Meets the Classics 243 

Werner Giith, Hartmut Kliemt 

Equlibrium Selection in Linguistic Games: 

Kial Ni (Ne) Parolas Esperanton? 257 

Werner Giith, Martin Strobel, Bengt-Ame Wickstrom 

Relating Cooperative and Non-Cooperative Game Theory 

Are Stable Demands Vectors in the Core of Two-Sided Markets? 

Some Graph-Theoretical Considerations 270 

Benny Moldovanu 

The Consistent Solution for Non- Atomic Games 279 

Guillermo Owen 

Finite Convergence of the Core in a Piecewise Linear Market Game 286 

Joachim Rosenmiiller 

Credible Threats of Secession, Partnership, and Commonwealths 305 

Philip J. Reny, Myma Holtz Wooders 

Principles in Behavioral Economics 

Rules for Experimenting in Psychology and Economics, 

and Why They Differ 313 

Colin Camerer 



Reciprocity: The Behavioral Foundations of Socio-Economic Games 328 

Elizabeth Hoffman, Kevin McCabe, Vernon Smith 




IX 



Adaption of Aspiration Levels - Theory and Experiment 345 

Reinhardt Tietz 

A Model of Boundedly Rational Experienced Bargaining 

in Characteristic Function Games 365 

Wulf Albers 

Theory, Field, and Laboratory: The Continuing Dialogue 386 

Elinor Ostrom, Roy Gardner, James Walker 



Experimental Studies 



Naive Strategies in Competitive Games 394 

Ariel Rubinstein, Amos Tversky, Dana Heller 

Induction vs. Deterrence in the Chain Store Game: 

How Many Potential Entrants are Needed to Deter Entry 403 

James A. Sundali, Amnon Rapoport 

Cooperation in Intergroup and Single-Group 

Prisoner’s Dilemma Games 418 

Gary Bomstein, Eyal Winter, Harel Goren 

On Styles of Relating to Bargaining Partners 430 

Axel Ostmann, Ulrike Leopold- Wildburger 

What Makes Markets Predict Well? 

Evidence from the Iowa Electronic Markets 444 

Joyce Berg, Robert Forsythe, Thomas Rietz 

Sequencing and the Size of the Budget: Experimental Evidence 464 

Roy J. Gardner, Jurgen von Hagen 

Intertemporal Speculation under Uncertain 

Future Demand: Experimental Results 475 

Charles R. Plott, Theodore L. Turocy III 

Endowment Effect for Risky Assets 494 

Graham Loomes, Martin Weber 



List of Contributors 



513 




Interview with Elisabeth and Reinhard Selten* 



Q: You, Mrs Elisabeth Selten, and your husband married in 1959. Do you 
have any advice for those who also would like to marry a future Nobel- 
laureate? 

E: No! - Do not overestimate it. Often he is physically with you, but not 
mentally. 

Q: You and your husband are both active members of the Esperanto- 
movement. Actually I think you met for the first time at an Esperanto- 
meeting. 

E: This is true, yes. I was in Munich at that time and my sister was already 
in Frankfurt. I visited her and we went to the Esperanto-Club. There I met 
my husband who was studying in Frankfurt. 

Q: Are you disappointed about the success of the Esperanto-idea, e.g. since 
it is not used as the official language of the United Nations or the European 
Community? 

E: No, I wasn’t disappointed because I don’t expect it to become accepted 
that soon. You should not forget that people have to learn the language. It’s 
easy to learn, but still you have to learn. 

Q: Your husband contributed to many sciences, most notably to economics, 
politics, philosophy, psychology and biology. Is there a special field which you 
personally found the most interesting one? 

E: I can only say which was the least interesting one. That was philosophy. 
Q: Do you discuss his theoretical ideas with him? 

E: To a degree not anymore. Some in biology. 

Q: You have translated the collection of papers ’’Game Theory and Related 
Approaches to Social Behavior” (1964) edited by Martin Shubik^ into Ger- 
man. Didn’t you also help your husband to perform experiments? 

E: Yes, but some time ago. He didn’t have any assistance, so I helped him 
both as an assistant and as a secretary. 

Q: You once said that the honorarium per page of translating this book 
edited by Martin Shubik was close to nothing. 



* The interview took place at the home of Elisabeth and Reinhard Selten on 
November 27th in 1995. After being typed it was slightly extended and revised 
by Reinhard Selten. The interviewers were Werner Giith and Martin Strobel. 

^ Shubik, Martin (ed.) (1965): Spieltheorie und Sozialwissenschaften, Hamburg, 
S. Fischer Verlag 




2 



E: It was better than what others got, it was the highest tariff I think. But 
if you would have to live from it ...! Well, actually it was really work. 

Q: Many colleagues often enjoyed your hospitality and your skillful cooking. 
This must have been quite a burden for you. Didn’t you sometimes complain? 
It also must have been more fun when you personally liked your visitors. 
Didn’t you, since and then, ask to have meetings in the office rather than at 
your home? 

E: No, I never did that. (To Reinhardt Selten) Or do you remember anything? 
Sorry, you must ask Reinhard! The people whom I did not like never stayed 
here for longer. So, this answers the question. 

Q: Thankyou, Mrs Selten! 

Q: We expect the readers of this volume to be pretty well informed about 
Reinhard Selten’s major contributions. What we would like to illuminate 
are the driving forces behind your work and how your research interests 
have developed over time. Let us start with high school. You visited the 
Gymnasium in Melsungen. Whoever knows you does not expect you to have 
been a crack in sports. You undoubtedly were more of the reading type. 
Which kind of literature were you reading mainly? 

R: I was reading much world literature, belletristic, but also books about 
special subjects, nonfiction. And when I was in my second last year of high 
school I also had a full time job as a librarian in the America-house at Mel- 
sungen. At this time I got used to read English. In the library they had 
many books on different subjects, psychology, economics. - I cannot remem- 
ber which topic was the most exciting one, I read a lot of books, among others 
a history of economic thought by Heimann, my first exposure to economic 
theory. 

Q: After high school you have studied mathematics at the university of 
Prankfurt/M. You rarely showed up for the early morning lectures. Since 
the later Reinhard Selten hardly ever drops a session of his courses, e.g. by 
attending conferences, one wonders whether you have changed or whether 
you think teaching is more fun than being taught? 

R: No, actually I don’t think teaching is fun. The reason why I don’t drop 
lectures, although I must do it occasionly, is that I think it is my duty to 
teach. It is not a question whether it’s fun or not. This question never occured 
to me. It is my duty. I’m obliged to do it. This is what I’m paid for. As a 
student it was not really a duty to go to lectures, students are free to go there 
or not to go. But as a teacher you can’t stay away. 

Q: You were not only attending mathematics courses but also those of other 
disciplines. What were your main interests in addition to mathematics and 
what do you consider to be the main influences of this time on your later 
work? 




3 



R; I Wets interested in many subjects, especially psychology. I also took part 
in experiments both as a subject and as student helper. 

Q: Did you sometimes think to change subjects and stop with mathematics? 

R: There was a point where I thought about leaving the university. This was 
before the Vordiplom. I had the impression that I would not pass it. Therefore 
I went to the SAS. It was my idea to become a navigator. But they were not 
interested in me as a navigator, they were interested in me as a steward or 
something. (Laughter) No, no, I mean they didn’t say what. Maybe also in 
their administration, I don’t know. Maybe because I spoke a little bit Danish. 

Q: Is it correct that Ewald Burger, who wrote a remarkable introduction 
to game theory^ and who was your thesis advisor, has also been your main 
teacher in mathematics? Were your conceptual ideas already shaped by Ewald 
Burger or was he conceptually more neutral, e.g. by viewing cooperative 
and noncooperative game theory as related, but different fields of applied 
mathematics? 

R: I was very much influenced by Ewald Burger, because he was a very good 
mathematican with a vast knowledge of the literature and he spent a lot of 
time talking to me. A that time I was still very young and not yet well trained 
in reading mathematical texts. It was very difficult for me to understand 
Shapley’s paper on his value^ and we went over this together. Burger was 
an excellent teacher, a very, very knowledgeable man. But he did not like 
conceptual discussions. He did not like to talk about interpretations and 
did this only where it was absolutely necessary. He did not feel secure about 
anything not mathematically precise. Later he concentrated on mathematical 
logic where he also had the same aversion against interpretations. Though I 
always had the impression that he is interested in game theory, he did not like 
to talk about the non-mathematical aspects of the field. Non-cooperative and 
cooperative game theory were not yet distinguished. It was all game theory. 
Nash introduced the notion of a non-cooperative game but in von Neumann’s 
and Morgenstern’s book^ there is no distinction between cooperative games 
and non-cooperative games. There is a distinction between 2-person zero-sum 
games and all other games. It was the idea that 2-person zero-sum games have 
to be handled distinctly. In all other games there is scope for cooperation. 
What people largely believed in was not always expressed. It was some kind of 
’’Coase conjecture”: If there is scope for cooperation, cooperation will occur. 
This was the general feeling at this time, at least this is my impression. 



^ Burger, Ewald (1966): Einfiihrung in die Theorie der Spiele, 2. Aufl., Berlin 
^ Shapley, Lloyd S. (1953): A Value for n-Person Games, in H. W. Kuhn and A. 
W. Tucker (eds.), Contributions to the Theory of Games, Vol. II, Ann. Math. 
Stud. 28, Princeton, N.J., Princeton University Press, pp. 307 - 317 
^ von Neumann, John and Oscar Morgenstern (1944): Theory of Games and Eco- 
nomic Behaviour, Princeton University Press 




4 



Q: Although your doctoral degree (Doktor phil. nat.) is still in mathematics, 
you then changed to economics by joining the team of Heinz Sauermann at the 
economics department in Prankfurt/M. where you initiated the experimental 
tradition, known as the German School of Experimental Economics. Did you 
always plan to supplement normative game theory by a behavioral theory of 
game playing, e.g. due to your experiences with the experimental tradition 
in (social) psychology? 

R: No, when I began to do experiments I was in the process of turning away 
from ” naive rationalism” , but I had not yet left this position completely be- 
hind me. The ” naive rationalist” thinks that people must act rationally and 
it is only necessary to find out what is the right rational solution concept. 
At least in cooperative game theory as well as in oligopoly theory there was 
no agreement about the right rational theory. There were many cooperative 
theories. The problem of deciding between competing cooperative solution 
concepts may have induced Kalisch, Milnor, Nash, and Nering^to do their 
experiments on n-person coalition games. I was much impressed by their pa- 
per. It was already clear from their results that a purely rational theory is not 
a good theory in view of what happened in their experiments. Shortly after I 
had began to run oligopoly experiments I read Herbert Simon and I was im- 
mediately convinced of the necessity to build a theory of bounded rationality. 
At first I still thought it might be possible to do this in an axiomatic way, just 
by pure thinking, as you build up a theory of full rationality. But over the 
years I more and more came to the conclusion that the axiomatic approach is 
premature. Behavior cannot be invented in the armchair, it must be observed 
in experiments. You first have to know what really happens before forming 
an axiomatic theory. 

Q: Two milestones of your academic career seem to be your contribution to 
’’Advances in Game Theory”®, and the famous 1965-article^ where you in- 
troduce the concept of subgame perfect equilibria. Do these two papers also 
reveal a change in your conceptual views, e.g. from cooperative to noncoop- 
erative ways of modelling and solving strategic conflicts? 

R: Yes and No. When I wrote the paper in 1965, 1 was really much motivated 
by my experimental work. This was a paper about a dynamic oligopoly game 
with demand inertia. We had played a somewhat more complicated game 
in the laboratory. I was interested in finding a normative solution, at least 

® Kalisch, G. K., J. W. Milnor, J. Nash, E. D. Nering (1954): Some experimental 
n-person games, in: Thrall, R. M., Coombs, C. H. and Davis, R. L. (eds.). 
Decision Processes, New York, Wiley 

® Selten, Reinhard (1964): Valuation of n-person games. In: Advances in Game 
Theory, Annals of Mathematics Studies no. 52, Princeton, N.J., 565 - 578 

^ Selten, Reinhard (1965): Spieltheoretische Behandlung eines Oligopolmodells 
mit Nachfragetragheit, Teil I: Bestimmung des dynamischen Preisgleichge- 
wichts, Teil II: Eigenschaften des dynamischen Preisgleichgewichts, JITE 
(Zeitschrift fiir die gesamte Staatswissenschaft) 121, 301 - 324, 667 - 689 




5 



for a simplified version. The reason why I took a non-cooperative approach 
goes back to the early oligopoly experiments by Sauermann and myself®. In 
the fifties many oligopoly theorists believed in tacit cooperation. The idea 
was that oligopolists would send signals to each other by their actions - they 
would speak a language, someone called ”oligopolese” - and thereby arrive at 
a Pareto-optimal constellation. But this did not happen in our experiments. 
What we observed had much more similarity to Cournot theory. This was 
confirmed by my second paper on oligopoly experiments. However, in the 
dynamic game with demand inertia, it was not clear, what is the counterpart 
of the Cournot-solution. It took me a long time to answer this question. 
Finally I found an equilibrium solution by backward induction. However while 
I already was in the process of writing the paper, I became aware of the fact 
that the game has many other equilibrium points. In order to make precise 
what is the distinguishing feature of the natural solution I had derived, I 
introduced subgame perfectness as a general game theoretic concept. This 
was only a small part of the paper but, as it turned out later, its most 
important one. 

Q: You have met John C. Harsanyi for the first time in 1961. Although there 
exist only two publications by the two of you^’^®, there has been a continuous 
exchange of ideas and mutual inspiration. Would you have developed differ- 
ently, e.g. by concentrating more on your experimental research, without the 
influence of John C. Harsanyi? 

R: It is very hard to say what would have happened. Probably I would have 
formed a relationship to somebody else and this could have driven me in 
another direction. What we first did, John Harsanyi and I, was dealing with 2- 
person bargaining under incomplete information. Then, after we had finished 
the paper for the fixed threat case^^ , it originally was our intention to go on 
to the variable threat case. However, I proposed that we should first develop 
a general theory of equilibrium selection in games. This theory should then 
be applied to the variable threat case. He agreed to proceed in this way and 
the problem of equilibrium selection kept us busy for decades. Of course, I 
am much influenced by John Harsanyi. He is older, ten years older than I am. 
He was more mature when we met and he was a very systematic thinker. He 
has a clear view of what he wants to achieve. I felt motivated to subordinate 
my own inclinations to his aims as a system builder. Without his influence I 

® Sauermann, Heinz and Reinhard Selten (1959): Ein Oligopolexperiment, 
Zeitschrift fur die Gesamte Staatswissenschaft, 115, pp. 417 - 471 
® Harsanyi, John C. and Reinhard Selten (1972): A generalized Nash-solution for 
2-person bargaining games with incomplete information. Management Science 
18, 80 - 160 

Harsanyi, John C. and Reinhard Selten (1988): A general theory of equilibrium 
selection in games, M.I.T. Press, Cambridge Mass 

Harsanyi, John C. and Reinhard Selten (1972): A generalized Nash-solution for 
2-person bargaining games with incomplete information, Management Science 
18, 80 - 160 




6 



might have worked less on full rationality and more on bounded rationality. 
As you probably know I am a methodological dualist and I think that both 
descriptive and normative theory are important. I am intringued by problems 
in both areas. 

Q: You have impressed many scholars and shaped their way of studying 
social interaction, regardless whether they rely on the game theoretic or the 
behavioral approach or even on evolutionary stability. Who has impressed 
the Nobel-laureate Reinhard Selten? 

R: Of course, I was impressed by my teacher Ewald Burger. And then as 
game theorists von Neumann and Morgenstern were very important. I knew 
Morgenstern personally. He visited Frankfurt sometimes. We had some dis- 
cussions because he always wanted to talk to me when he came to Frankfurt. 
I didn’t know von Neumann. Among the game theorists I was always very 
much impressed by Shapley and also by Aumann. Shapley was a leading, as 
far as I remember, he was the leading game theorist after von Neumann and 
Morgenstern. Nash also, but Nash was not available at this time. Nash was not 
visible. Of course, he was very important for me, both for my work on coop- 
erative and my work on noncooperative theory, but personally it was Shapley 
and Aumann. There were also many other people, I already mentioned Her- 
bert Simon. But I don’t really know him personally, only superficially. You 
can also say that I was impressed by my students. 

Q: You are approaching official retirement although you remain to be the 
director of the computer laboratory at the university of Bonn. If time allows, 
what do you still want to accomplish and what do you encourage younger 
researchers to study most of all? 

R: I hope that during my lifetime I will have the opportunity to see the 
emergence of a strong body of behavioral economic theory which begins to 
replace neoclassical thinking. In my view experimental and empirically based 
theoretical work on bounded rationality is an important and fruitful research 
direction I can recommend to younger colleagues. Of course I myself will also 
try to achieve further progress in this area. 

Q: Let us conclude by asking you what inspired your interests in evolutionary 
game theory to which you contributed some pioneering and pathbreaking 
studies. 

R: I was lucky enough to come into contact with biological game theory 
very early in its development. I was fascinated by the subject matter and it 
became clear to me that there are important open theoretical problems of 
evolutionary stability in extensive games. I think that I succeeded to con- 
tribute something worthwhile in this area. More recently the task of writing 




7 



a survey paper^^induced Peter Hammerstein and myself to look at the pop- 
ulation genetic foundations of evolutionary stability. But I also would like 
to mention that my interests in biological game theory are not entirely con- 
centrated on abstract problems. I enjoy my cooperation with Avi Shmida, a 
botanist at the Hebrew University of Jerusalem with whom I like to discuss 
the interaction of flowers and bees. 

Q: Thank you for the interview! 



Hammerstein, Peter and Reinhard Selten (1994): Game Theory and Evolution- 
ary Biology, in R. I. Aumann and S. Hart (eds.) Handbook of Game Theory, 
Amsterdam, Elsevier, pp. 929 - 993 




On the State of the Art in Game Theory: 
An Interview with Robert Aumann* 



Q: We could take your talk yesterday on “Relationships” as a starting point. 
That talk is a follow-up to your paper “What is Game Theory Trying to 
Accomplish”^, which you presented in Finland some years ago. Selten acted 
as a discussant for that paper. He agreed with some of the points you made, 
but disagreed with others. It would be nice to make an inventory of what has 
happened since that time. Have your opinions gotten closer together or have 
they drifted further apart? Do you agree that this is a good starting point 
for the interview? 

A: Fine. 

Q: Let us recall your position. Yesterday you said that science is not a quest 
for truth, but a quest for understanding. The way you said this made it clear 
that you have to convince people of this point; not everybody agrees with 
your point of view, maybe even the majority does not agree. Is that true? 

A: You are entirely right. The usual, naive, view of science is that it is a quest 
for truth, that there is some objective truth out there and that we are looking 
for it. We haven’t necessarily found it, but it is there, it is independent of hu- 
man beings. If there were no human beings, there would still be some kind of 
truth. Now I agree that without human beings there would still be a universe, 
but not the way we think of it in science. What we do in science is that we 
organize things, we relate them to each other. The title of my talk yesterday 
was “Relationships” . These relationships exist in the minds of human beings, 
in the mind of the observer, in the mind of the scientist. The world without 
the observer is chaos, it is just a bunch of particles flying around, it is “Tohu 
Vavohu” — the biblical description of the situation before the creation^. It 
is impossible to say that it is “true” that there is a gravitational force: the 

* This interview took place one day after the close of the “Games ’95” Conference, 
which was held in Jerusalem between June 25 and 29, 1995. There are several 
references in the interview to events occurring at the conference, in particular to 
Paul Milgrom’s lecture on the spectrum auction on June 29, to Aumann’s lecture 
on “Relationships” on June 29, and to Aumaun’s discussion of the Nobel prize 
winners’ contributions on Sunday evening, June 26. The interviewer was Eric 
van Damme, who combined his own questions with written questions submitted 
by Werner Giith. Most of the editing of the interview was done at the State 
University of New York at Stony Brook, with which Aumann has a part-time 
association. Editing was completed on June 19, 1996. 

^ In Frontiers of Economics ^ K. Arrow and S. Honkapohja, eds., Basil Blackwell, 
Oxford, 1985, 28-76. 

^ Genesis 1,2. “Tohu” is sometimes translated as “formless.” It seems to me that 
the idea of “form” is in the mind of the observer; there can be no form without 
an observer. 




9 



gravitational force is only an abstraction, it is not something that is really 
out there. One cannot even say that energy is really out there, that is also 
an abstraction. Even the idea of “particle” is in some sense an abstraction. 
There certainly are individual particles, but the idea that there is something 
that can be described as a particle — that there is a clciss of objects that 
are all particles — that is already an abstraction, that is already in the mind 
of the human being. When we say that the earth is round, roundness is in 
our minds, it does not exist out there, so to speak. In biology, the idea of 
“species” is certainly an abstraction, and so is that of an individual. So all 
of science really is in our minds, it is in the observer’s mind. Science makes 
sense of what we see, but it is not what is “really” there. 

Let me quote from my talk yesterday: Science is a sort of giant jigsaw 
puzzle. To get the picture you have to fit the pieces together. One might 
almost say that the picture is the fitting together, it is not the pieces. The 
fitting together is when the picture emerges. 

Certainly, that is not the usual view. Most people do not see the world 
that way. Maybe yesterday’s talk gave people something to think about. The 
paper, “What is Game Theory Trying to Accomplish?” had a focus that is 
related, but a little different. 

Q: What I recall from that paper is that you also discuss “applied” science. 
You argue that from understanding follows prediction and that understanding 
might imply control, but I don’t recall whether you use this latter word. 
Perhaps you use the word engineering. Your talk yesterday was confined to 
understanding: what about prediction and engineering? 

A: We heard a wonderful example of that yesterday: Paul Milgrom talking 
about the work of himself and Bob Wilson and many other game theorists in 
the United States — like John Riley, Peter Cramton, Bob Weber, and John 
Macmillan — who were asked by the FCC (Federal Communications Com- 
mission) or one of the communications corporations to consult on the big 
spectrum auction that took place in the US last year, and that netted close 
to 9 billion dollars^. This is something big and it illustrates very well how one 
goes from understanding to prediction and from prediction to engineering. 
Let me elaborate on that. These people have been studying auctions both 
theoretically and empirically for many years. (Bob Wilson wrote a very nice 
survey of auctions for the Handbook of Game Theory^.) They are theoreti- 
cians and they have also consulted on highly practical auction matters. For 
example, Wilson has consulted for oil companies bidding on off-shore oil tracts 
worth upwards of 100 million dollars each. These people have been studying 
auctions very seriously, and they are also excellent theoreticians; they use 
concepts from game theory to predict how these auctions might come out, 

^ See Milgrom, P. (1996): Auction Theory for Privatization^ Oxford: Oxford Uni- 
versity Press (forthcoming). 

Wilson, R. (1992): “Strategic Analysis of Auctions,” in Handbook of Game The- 
ory, Vol. I, R. Aumann and S. Hart, eds., Amsterdam: Elsevier, 227-279. 




10 



and they have gotten a good feel for the relation between theory and what 
happens in a real auction. Milgrom and Weber^ once did a theoretical study, 
looking at what Nash equilibrium would predict on a qualitative basis for how 
oil lease auctions would work out, and then Hendricks and Porter® did an 
empirical study and they found that a very impressive portion of Milgrom’s 
predictions (something like 7 out of 8) were actually true; moreover these 
were not trivial, obvious predictions. So these people make predictions, they 
have a good feel for theory and they have a good feel for how theory works 
out in the real world. And then they do the engineering, like in the design of 
the spectrum auction. So one goes from theory, to prediction, to engineering. 
And as we heard yesterday, this was a tremendously successful venture. 

Another example of how understanding leads to prediction and to en- 
gineering comes from cooperative game theory; it is the work of Roth and 
associates on expert labour markets. This is something that started theo- 
retically with the Gale-Shapley'^ algorithm, and then Roth found that this 
algorithm had actually been implemented by the American medical profes- 
sion in assigning interns to hospitals®. That had happened before Gale and 
Shapley published their paper. It had happened as a result of 50 years of 
development. So there was an empirical development, something that hap- 
pened out there in the real world, and it took 50 years for these things to 
converge; but in the end, it did converge to the game theoretic solution, in 
this case the core. Now this is amazing and beautiful, it is empirical game 
theory at its best. We’re looking at something that it took smart, highly 
motivated people 50 years to evolve. That is something very different from 
taking a few students and putting them in a room and saying: OK, you have 
a few minutes to think about what to decide. These things are sometimes 
quite complex and take a long time to evolve; people have to learn from their 
mistakes, they have to try things out. 

This was the beginning of a very intensive study by Roth and collabora- 
tors, Sotomayor and others^, into the actual working of expert labor markets. 



^ “The Value of Information in a Sealed Bid Auction,” Journal of Mathematical 
Economics^ 10 (1982), 105-114; see also Engelbrecht- Wiggins, R., P. Milgrom, 
and R. Weber: “Competitive Bidding and Proprietary Information,” Journal of 
Mathematical Economics 11 (1983), 161-169. 

® “An Empirical Study of an Auction with Asymmetric Information,” American 
Economic Review 78 (1988), 865-883. 

^ Gale, D. and L. Shapley (1962): “College Admissions and the Stability of Mar- 
riage,” American Mathematical Monthly 69, 9-15. 

® Roth, A. (1984): “The Evolution of the Labor Market for Medical Interns and 
Residents: A Case Study in Game Theory,” Journal of Political Economy 92, 
991-1016. 

^ See Roth, A. and M. Sotomayor (1990): Two-Sided Matching: A Study in Game- 
Theoretic Modeling and Analysis, Econometric Society Monograph Series, Cam- 
bridge: Cambridge University Press; Roth, A. (1991): “A Natural Experiment 
in the Organization of Entry-Level Labor Markets: Regional Markets for New 
Physicians and Surgeons in the United Kingdom,” American Economic Review 




11 



And now they are beginning to consult with people and say: Listen, you guys 
could do this and that better. So again we have a progression from theory to 
empirical observation to prediction and then to engineering. 

Q: From these examples, can one draw some lessons about the type of situ- 
ations in which one can expect game theory to work in applications? 

A: What one needs for game theory to work, in the sense of making verifi- 
able (and falsifiable!) predictions, is that the situation be structured. Both 
the auctions and the markets that Roth studies are highly structured. Some- 
times when people interview me for the newspapers in Israel, they ask ques- 
tions like, can game theory predict whether the Oslo agreement will work 
or whether Saddam Hussein will stay in power. I always say: Those situa- 
tions are not sufficiently structured for me to give a useful answer. They are 
too amorphous. The issues are too unclear, there are too many variables. 
To say something useful we need a structured situation. Besides the above 
examples, another example is the formation of governments. For years I have 
been predicting the government that will form in Israel once the composi- 
tion of the Israeli parliament is known after an election. That is a structured 
situation, with set rules. The parliament has 120 seats. Each party gets a 
number of seats in proportion to the votes it got in the election. To form a 
government, a coalition of 61 members of parliament is required. The presi- 
dent starts by choosing someone to initiate the coalition formation process. 
(Usually, but not necessarily, this “leader” is the representative of the largest 
party in parliament.) The important point is that the process of government 
formation is a structured situation, to which you can apply a theory. That is 
true also of auctions and of labor markets like those discussed above. They 
work with fixed rules, and this is ideal for application. Now, if you don’t have 
a structured situation, that doesn’t necessarily mean that you cannot say 
anything, but usually you can only say something qualitative. For example, 
one prediction of game theory is the formation of prices; that comes from 
the equivalence theorem^^. That kind of general prediction doesn’t require a 
clear structure. On the other hand, the Roth example and the spectrum auc- 
tion are examples of structured situations, and they are beautiful examples 
of applications. 

Q: Could you explain what exactly you mean by a structured situation? Is it 
that there are players with well-defined objectives and well-defined strategy 

81, 415-440; and Roth, A. and X. Xing (1994): “Jumping the Gun: Imperfec- 
tions and Institutions Related to the Timing of Market Interactions,” American 
Economic Review 84, 992-1004. 

See, e.g., Debreu, G. and H. Scarf (1963): “A Limit Theorem on the Core of a 
Market,” International Economic Review 4, 235-246; Aumann, R. (1964): “Mar- 
kets with a Continuum of Traders,” Econometrica 32, 39-50; Aumann, R. and L. 
Shapley (1974): Values of Non- Atomic Games, Princeton: Princeton University 
Press; and Champsaur, P. (1975): “Cooperation versus Competition,” Journal 
of Economic Theory 11, 394-417. 




12 



sets, that there is perhaps even a timing of the moves? In short, is a structured 
situation one in which the extensive or strategic form of the game is given? 

A: No, that is not what is meant by “structured”. It means something more 
general. A structured situation is one that is formally characterized by a lim- 
ited number of parameters in some definite, clear, totally defined way. That 
implies that you can have a theory that makes predictions on the basis of this 
formal structure, and you can check how often that theory works out, and you 
can design a system based on those parameters. That holds for auctions, and 
for Roth’s labor markets, and for the formation of a governmental majority in 
a parliamentary democracy. In each of those cases, important aspects of the 
situation are described by a formal structure, a structure with clear, distinct 
regularities. Of course, there are also aspects that are not described by this 
formal structure, but at least some of the important aspects are described by 
this formal structure. 

That is not so for the Oslo agreement and it is not so for Saddam Hussein. 
Those situations are in no sense repeatable, there is no regularity to them, 
there is nothing on which to base a prediction, and there is no way to check 
the prediction if you make it. Of course the prediction could be right or wrong, 
but there is no way to say how often it’s right, because each prediction is a 
one-time affair, there’s no way to generalize. 

For instance, in the governmental majority matter, one can set up a par- 
liament as a simple game in the sense of Von Neumann and Morgenstern’s 
cooperative game theory, where we model the party as a player; we get a 
coalitional worth function that attributes to a coalition the worth 1 when it 
has a majority and the worth 0 otherwise. And then one can work out what 
the Shapley values are; the structure is there, it is clear, and one can make 
predictions. Now there are all kinds of things that are ignored by this kind of 
procedure, but one can go out and make predictions. Then, if the predictions 
turn out correct, you know that you were right to ignore what you ignored. 

No matter what you do you are going to be ignoring things. This is true 
not only in game theory, it is true in the physical sciences also; there are all 
kinds of things that you are ignoring all the time. Whenever you do something 
out there in the real world, or observe something in the real world, you are 
ignoring, you are sweeping away all kinds of things and you are trying to say: 
Well, what is really important over here is this and that and that; let’s see 
whether that’s significant, let’s see whether that comes out. 

Q: In this example of coalition formation, you make predictions using an 
algorithm that involves the Shapley value. Suppose you show me the data 
and your prediction comes out correct. I might respond by saying that I don’t 
understand what is going on. Why does it work? Why is the Shapley value 
related to coalition formation? Is it by accident, or is it your intuition, or is 
it suggested by theory? 




13 



A: There are two answers to that. First, certainly, this is an intuition that 
arises from understanding the theory. The idea that the Shapley value does 
represent power comes from the theory. Second, for good science it is not 
important that you understand it right off the bat. What is important, in 
the first instance, is that it is correct, that it works. If it works, then that in 
itself tells us that the Shapley value is relevant. 

Let me explain this a little more precisely. The theory that I am testing is 
very simple, almost naive. It is that the leader — the one with the initiative 
— tries to maximize the influence of his party within the government. So, 
one takes each possible government that he can form and one looks at the 
Shapley value of his party within that government; the intuition is that this is 
a measure of the power of his party within the government. This maximization 
is a nontrivial exercise. If you make the government too small, let’s say you 
put together a coalition government that is just a bare majority with 61 
members of parliament — a minimal winning coalition — then it is clear 
that any party in the coalition can bring down the government by leaving. 
Therefore, all the parties in the government have the same Shapley value. 
So the hypothesis is that a wise leader won’t do that. That is also quite 
intuitive, that such a government is unstable, and it finds its expression in 
a low Shapley value for the leader. On the other hand, too large a coalition 
is also not good, since then the leader doesn’t have sufficient punch in the 
government; that also finds its expression in his Shapley value. Consequently, 
the hypothesis that the leader aims to maximize his Shapley value seems a 
reasonable hypothesis to test, and it works not badly. It works not badly, 
but by no means a hundred percent. For example, the current (June 95) 
government of Israel is very far off from that, it is basically a minimal winning 
coalition. In fact, they don’t even have a majority in parliament, but there 
are some parties outside the government that support it; though it is really 
very unstable, somehow it has managed to maintain itself over the past 3 
years. But I have been looking at these things since 1977, and on the whole, 
the predictions based on the Shapley value have done quite well. I think there 
is something there that is significant. It is not something that works 100% 
of the time, but you should know that nothing in science works 100% of the 
time. In physics also not. In physics they are glad if things work more than 
half the time. 

Q: Would you like to see more extensive empirical testing of this theory? 

A: Absolutely. We have tried it in one election in the Netherlands, where it 
seems to work not badly; but we haven’t examined that situation too closely. 
The idea of using the Shapley value is not just an accident, the Shapley value 
ha^ an intuitive content and this hypothesis matches that intuitive content. 

Q: Are game theoretical applications to unstructured situations doomed to 
fail? 




14 



A: No, we already discussed the example of competitive equilibrium, the idea 
of price formation. It is successful although it is not structured. Even to diplo- 
matic negotiations, which we discussed above, one can make contributions, in 
that one can point out certain things to be aware of. Let’s consider an exam- 
ple. We have these negotiations with Syria. Now, the president of Syria, Mr. 
Assad, ha^ stated publicly again and again that he is not sure that he is at all 
interested in reaching any kind of accommodation with Israel. The reason, he 
says, is that the status quo is not too bad for Syria; it is OK. And he is right, 
because we are not talking about a population in the Golan Heights, there 
are almost no Syrian people there. The few Syrian villages that were there 
and that were displaced in the 1967 war were resettled in Syria; no problems 
there, it was a small population. Most of the current inhabitants of the Golan 
Heights are Israeli Jews, all except for two or three villages right near the 
Syrian border, and there are no problems. These people are essentially sat- 
isfied. There is nothing that is burning under Assad’s pants over there, and 
really under anybody’s pants. So, Assad quite openly says: The status quo 
is not bad for me. Now, from the game theoretic point of view we have to 
ask ourselves, is the status quo in the set of feasible agreements? It could be 
that the status quo is outside the convex set of all possible pairs of utilities 
to agreements. It might be beyond the North-East boundary of that set. In 
that case there is no sense in pursuing a possible accommodation. Assad is 
quite happy the way it is; and while we would like to have a peace treaty with 
Syria, obviously that should not be at any cost. So the question is: Is there a 
possible peace treaty that is worthwhile for Assad, and that simultaneously 
is acceptable to us? I don’t think that anybody in our government has an- 
swered that question, or even asked it. Professional diplomats don’t think in 
those terms. They think in terms that are totally different; they don’t have 
to do with payoffs, with outcomes. A diplomat will think in terms of vaguely 
defined objectives, like building trust between leaders, or peace, or signing a 
document, or making a statement to the press. This kind of thing is a totally 
different world. In this case, while game theory cannot make a prediction, it 
can say let us think in certain terms, in certain directions. 

By the way, here again the discussion centers on the cooperative theory. 
It is the idea of a set of possible accommodations, and a status quo point, and 
where is that status quo point in relation to the set of possible agreements. 
That is cooperative game theory. 

That brings to mind another point. I don’t remember who it was, perhaps 
Kissinger, somebody in the early fifties said that a major contribution of 
game theory is just the idea of a strategy (or payoff) matrix. Just have the 
matrix in front of you. Just say: “look, we can do something, and they can do 
something; let’s write down everything that we can do, and everything that 
they can do, and then let’s look it over from that point of view”. Whereas 
before game theory, people had only been thinking about what we can do. 
The example of Syria and Israel is similar in that respect, because it says: 




15 



Can we reach an accommodation that we can buy and they can also buy? Or, 
among all the accommodations that will be acceptable to them, will there 
be one that is acceptable to us? So we are putting ourselves in both shoes 
at the same time. That is already a giant step forward. So in this kind of 
unstructured situation, game theory does have something to contribute. But 
it is conceptual: ways of thinking, approaches - not specific predictions. 

Q: I think that one of the persons who wrote in the fifties that game theory’s 
most important contribution was the introduction of the payoff matrix was 
Thomas Schelling, and perhaps, he provides a nice, natural way to move to 
the next topic. Specifically, I would like to move on to disagreements between 
you and Reinhard Selten, differences of viewpoint. Yesterday you stressed the 
importance of relationships, of establishing links between the complex and the 
simple. Now one might argue that perhaps, relationships are the more easy to 
establish the more abstract the setting in which one is working. If I interpret 
Selten ’s discussion of your earlier paper correctly, then he fears somewhat that 
by moving to a more abstract level one is going to lose the connection with 
the real world. The reason I thought this relates to Schelling is that Schelling 
has argued in the past that game theory should stand on two strong legs, 
one is a strong empirical leg and the other is a deductive, formal leg. You 
are stressing very much this second leg, of establishing formal relationships, 
whereas Schelling and Selten would argue that we need to develop more the 
empirical leg. Do you see a conflict there? 

A: No, that is a misunderstanding. The term “relationship” does not at all 
imply that we are talking about theory only. Empirics are vital, and I agree 
entirely with the view that you just attributed to Schelling. Up to now in this 
interview we have discussed only empirics, so the importance that I attribute 
to that side should be clear. 

Let me point out that at least three major areas of my talk yesterday 
touched on the importance of empirics. The very first item in my talk was an 
empirical item. The Naval electronics problem^^ was my entrance into deci- 
sion theory and from there into game theory. The naval electronics problem 
is a highly empirical problem with both legs in the real world; we are talking 
about real pieces of Naval equipment worth hundreds of thousands of 1955 
dollars each. We are talking altogether about billions of 1995 dollars, and we 
went in there and we used methodology from decision theory, and from linear 
programming, and some methodology that Joe Kruskal and I developed for 
this very purpose ourselves, which later turned out to be connected with util- 
ity theory. We went in there with these tools, which are fairly sophisticated, 
to solve this real empirical problem; we did an engineering job, and I think 
it was a useful one. Yesterday I stressed the relationships within this engi- 



Aumann, R. and J. Kruskal (1959): “Assigning Quantitative Values to Qual- 
itative Factors in the Naval Electronics Problem,” Naval Research Logistics 
Quarterly 6, 1-16. 




16 



neering job, to go from simple hypothetical assignment problems to complex 
real ones. So that is the first empirical item. 

The second one was my analysis of the concept of Nash equilibrium. You 
will recall that I used three slides. The black slide on the bottom, which had 
just the payoff matrix and the word “equilibrium” written on it; and two red 
slides, which I superimposed on the black slide, one after the other. The first 
red slide explained the payoff matrix in terms of conscious decision making, 
and the other one explained it in terms of evolution. In one case the equilib- 
rium was strategic, and in the other case it was a population equilibrium: two 
completely different concepts with the same mathematical structure. Now the 
evolutionary interpretation of Nash equilibrium is certainly closely related to 
empirics. Let me mention in this connection the astoundingly beautiful work 
of Selten’s associate Peter Hammerstein^^, who did this study of spiders in 
the American Southwest: a game theoretic study of the behavior of spiders in 
which mixed strategies turned up in precisely (or almost precisely) the right 
proportion when fitnesses were measured in terms of egg-mass; a mindblowing 
study! This is real empirics and that is empirical game theory. 

The third item from yesterday’s talk that is really empirically rooted is 
the study of the Talmud^^, using the idea of consistency and using the idea 
of the nucleolus. There we take something that happened 2000 years ago and 
happened in the real world; law is part of the real world. We are talking about 
a document that was written by people who had no idea of the discipline of 
game theory. They had something in mind, for 2000 years nobody understood 
it, and now it is understood. That is empirics. They were talking about a real 
bankruptcy problem, they had a real decision problem. This is a relationship; 
but not a theoretical relationship, it is something that is really happening out 
there. So that is the third area. 

We’re doing pretty well, when you take into account also all the other 
examples that we have mentioned, like the work about auctions, and about 
matching people to jobs, and the emergence of prices and many items we have 
not mentioned, like signalling^^, and labor relations^^, and other matters. The 
connection with the real world is indeed very important, and we have it. We 
have it and we are developing it further and it is closely associated with the 



Hammerstein, P. and S. Riechert (1988): “Payoffs and Strategies in Territorial 
Contests: ESS Analysis of Two Ecotypes of the Spider Agelenopsis Aperta,^' 
Evolutionary Ecology 2, 115-138. 

Aumann, R. and M. Maschler (1985): “Game Theoretic Analysis of a 
Bankruptcy Problem from the Talmud,” Journal of Economic Theory 36, pp. 
195-213. 

See, e.g., Kreps, D. and J. Sobel (1994): “Signalling,” in Handbook of Game 
Theory j Vol. II, R. Aumann and S. Hart, eds., Amsterdam: Elsevier, 849-867. 
See, e.g., Kennan, J., and R. Wilson (1993): “Bargaining with Private Informa- 
tion,” Journal of Economic Literature 31, 45-104, and Wilson, R. (1994): Negoti- 
ation with Private Information: Litigation and Strikes, Nancy L. Schwartz Lec- 
ture, J.L. Kellogg School of Management, Evanston: Northwestern University. 




17 



theory; far from rejecting that, I embrace it. Many of the relations I discussed 
yesterday were real world relations. 

Having said that, I’ll say something else also. At some level the theory 
has to be several stages in advance of empirics. So we could be doing very 
complicated theory, let us say equilibrium refinement a la Mertens^® with 
cohomology and all that, and at the same time be doing empirical studies and 
empirical engineering like Milgrom was discussing (the spectrum auction). It 
is very important that we have those two levels. Because it is not possible 
to do the things that Milgrom W3;S discussing without having the Mertens 
cohomology at the same time. This is a somewhat subtle idea, so let me 
explain it. There has to be a body of knowledge out there that has been 
internalized by a body of people. Perhaps Milgrom, as an individual, does 
not necessarily have to understand Mertens ’s cohomology-based refinement 
theory, but the body of science has to develop in such a way that there 
can exist a Milgrom. (As it happens, Milgrom and Wilson are both very 
strong theorists; with all his involvement in the spectrum auction, Wilson has 
just written a highly theoretical piece about Mertens refinements^.). In some 
sense it wouldn’t be possible for him to work out these very practical rules 
without knowing something like Mertens’s cohomology refinement; perhaps 
not necessarily that, but something of similar depth. You have to swim in 
it. You are not going to be able to swim from the shore to an island that 
is a kilometer away if it’s the first time you are in the water. It has to be 
a part of your background. As I said yesterday, these things are all hitched 
together; they are part of a milieu. You have to have a lot of experience, and 
this experience has to be both practical and theoretical. You can’t keep your 
nose too close to the ground. There hcis to be some “spiel” , some room, and 
there has to be some background, and the theory always has to be far in 
advance. 

Q: The second point of difference of view might perhaps be the following. 
You seem to be saying that every relationship is good, the more relations 
- the tighter the web - the better. But some of these relationships might 
actually be misleading. For example, take the rationalistic and evolutionary 
interpretations of Nash equilibrium. Maybe one interpretation applies in one 
context, while the other applies in a completely different context. Insights 
relevant in one context need not necessarily be relevant in the other context. 
Nevertheless, the relation might be used to import insights from one context 

Mertens, J-F. (1989): “Stable Equilibria - A Reformulation, Part I: Definition 
and Basic Properties,” Mathematics of Operations Research^ 14, 575-625, and 
Mertens, J-F. (1989): “Stable Equilibria - A Reformulation, Part II: Discussion 
of the Definition and Further Results,” Mathematics of Operations Research, 
16, 694-753. 

Govindan, S. and R. Wilson (1996): “A Sufficient Condition for the Invariance 
of Essential Components,” Duke Journal of Mathematics, to appear; see also 
Wilson, R. (1992): “Computing Simply Stable Equilibria,” Econometrica, 60, 
1039-1070. 




18 



to the other. One shouldn’t mix these things, one should keep them separate. 
Selten advocates a sharp distinction between descriptive game theory and 
rationalistic game theory, where the former is based on less strong rationality 
assumptions. Hence, this is related to the issue of bounded rationality in 
general. Could you comment on these points? 

A: You say “one shouldn’t mix these things... Selten advocates a sharp dis- 
tinction” . Well, I disagree. When there is a formal relationship between two 
interpretations, like the rationalistic (conscious maximization) and evolution- 
ary interpretations of Nash equilibrium^®, then one can say, “Oh, this is an 
accident, these things really have nothing to do with each other”. Or, one 
can look for a deeper meaning. In my view, it’s a mistake to write off the 
relationship as an accident. 

In fact, the evolutionary interpretation should be considered the funda- 
mental one. Rationality - Homo Sapiens, if you will - is a product of evolution. 
We have become used to thinking of the “fundamental” interpretation as the 
cognitive rationalistic one, and of evolution as an “as if” story, but that’s 
probably less insightful. 

Of course, the evolutionary interpretation is historically subsequent to the 
rationalistic one. But that doesn’t mean that the rationalistic interpretation 
is the more basic. On the contrary, in science we usually progress, in time, 
from the superficial to the fundamental. The historically subsequent theory 
is quite likely to encompass the earlier ones. 

The bottom line is that relationships like this are often trying to tell us 
something, and that we ignore them at our peril. 

Having said this, let me add that game theorists argue too much about 
interpretations. Sure, your starting point is some interpretation of Nash equi- 
librium; but when one is doing science, you have a model and what you must 
say is: What do we ohservel Do we observe Nash equilibrium? Or don’t we 
observe it? Now, why we observe it, that’s a problem that is important for 
philosophers, but less so for scientists. It’s like the old joke of the town where 
they wanted to build a bridge over the river. They started arguing in the 
town council. The merchants wanted the bridge because it would be good for 
commerce. People from each side wanted it because it would be easier to go 
to the other side. The religious people wanted the bridge because then they 
could reach the synagogue more easily. The lovers wanted it because they 
could make romance on a full moon night. They started arguing and fighting 
with each other over why to build the bridge, and in the end they couldn’t 
agree and they did not build the bridge. 

So, the story of why we have Nash equilibrium, it’s an important philo- 
sophical problem, but as science it is not that important. You are quite right, 

Aumann, R. (1987): “Game Theory,” in The New Palgrave, Vol. //, J. Eatwell, 
M. Milgate, and P. Newman, eds., London: Macmillan, 460-482. See specifically 
subsection ii of the section entitled 1970-1986, on page 477, near the bottom of 
the left column. 




19 



when we had a discussion this week, I was a little surprised when Reinhard 
said, “Yes, we must distinguish...”. I don’t quite understand why. 

As far as descriptive Game Theory is concerned, I certainly agree that 
we must describe what we actually observe. But to divorce this from the 
theory would be counterproductive. After all, the theory is meant to account 
for what we observe. Imagine a theoretical physics that is divorced from 
observational physics - that would be absurd! There may be - indeed there are 
- discrepancies between theory and observation, both in physics and in game 
theory; but fundamentally, the aim must be to bring theory and observation 
together, not to distinguish between them. And as we have already discussed, 
in game theory we are doing quite well in this regard. 

By the way, the evolutionary interpretation does not support only sub- 
game perfect equilibrium, or, for that matter, trembling hand perfection. It 
also supports non-perfect equilibria. One beautiful example of that is in the 
ultimatum game. One can understand what is observed in the ultimatum 
game^^ on the basis of non-perfect equilibria. This is not an empty predic- 
tion. As you know, of course, every possible offer is a part of an equilibrium 
in the ultimatum game, so one could say that this does not mean very much; 
but that is not correct, because when you say that you have an equilibrium, 
then you say that this is in a sense a norm of behavior. That can be checked, 
and it was checked in the work of Roth, Prasnikar, Okuno-Pujiwara and 
Zamir^^, and they found regularities of behavior in various different places 
in the world, which were different in the different places. That is a beauti- 
ful verification of the evolutionary interpretation of Nash equilibrium. Far 
from the ultimatum game being some kind of rejection of game theory, it is 
a beautiful verification of it, a corroboration! 

Q: You said yesterday that you thought that this was the direction to go in 
strategic game theory, that is, to further develop the evolutionary approach. 
At present, the models that we have are based on agents who do not have any 
cognition at all. Do you view this as a first step towards other models in which 
there is limited cognition, towards richer models of bounded rationality? 



Giith, W-, R. Schmittberger, and B. Schwarze (1982), “An Experimental Analy- 
sis of Ultimatum Bargaining,” Journal of Economic Behavior and Organization 
3, 367-388. In this experiment, two players were asked to divide a considerable 
sum (varying as high as DM 100). The procedure was that PI made an offer, 
which could be either accepted or rejected by P2; if it was rejected, nobody 
got anything. The players did not know each other and never saw each other; 
communication was a one-time affair via computer. The only subgame perfect 
equilibria of this game lead to a split of 99-1 or 100-0. But there are many Nash 
equilibria, of the form “PI offers x, P2 rejects anything below x.” 

“Bargaining and Market Behavior in Jerusalem, Ljubljana, Pittsburgh, and 
Tokyo: An Experimental Study,” American Economic Review 81 (1991), 1068- 
1095. Roughly speaking, it was found in this experiment that in each of the four 
different venues, the accepted offers clustered around some specific x, which 
differed in the different venues. 




20 



A: No, actually not. That is muddying the waters. The evolutionary interpre- 
tation is best understood as standing on its own legs, not as a stepping stone 
leading to rational or cognitive interpretations. That would not be a good 
way to go. We have here two separate - but completely parallel, completely 
isomorphic - interpretations of the same mathematical system. A popula- 
tion equilibrium in the evolutionary interpretation corresponds precisely to a 
strategic equilibrium in the cognitive interpretation. In a population equilib- 
rium, each genotype that constitutes a positive proportion of the population 
is best possible given the environment. Similarly, in a strategic equilibrium, 
each pure strategy with positive probability is best possible given the mixed 
strategies of the others. In both cases we have full maxima; there is no way 
that the equilibria in the evolutionary interpretation could somehow be im- 
proved if we made them “more rational.” 

In brief, one should not try to transform an evolutionary model first into 
a boundedly rational model and then into something like a full rationality 
model; that’s not the way to go. One should have both models in mind. Both 
are important. 

Actually, the formal similarity between the two models probably reflects 
an important conceptual relationship. Rational thinking may have evolved be- 
cause it is evolutionarily efficient. Let me clarify that. Evolutionary biologists 
keep asking about everything: What is the evolutionary function of this, what 
is the evolutionary function of that? One question, raised by Dawkins in his 
book^^ is: What is the evolutionary function of consciousness? What I’d like 
to suggest is that consciousness evolved because it serves a very important 
evolutionary function, namely that it enables people to think rationally in 
the sense of utility maximization. You are not only a computing machine; you 
have goals to achieve, utilities to maximize. For that you must be conscious, 
you must be able to experience, you must be able to say this is pleasurable, 
this is painful. 

Just an example of that: What is the function of taste in food? Why must 
it have a taste? Well, it must have a taste because if it did not taste good we 
would not be motivated to eat it and then we would starve to death. We have 
to be conscious for that, we can’t taste something without being conscious. 
The consciousness enables us to experience pleasure and pain. It has long 
been recognized that pain has a very important evolutionary function. I’m 
suggesting that pleasure also does. 

Another example is, of course, sex. Why is sex pleasurable? What is the 
evolutionary function of that? Not of sex itself, that’s obvious, but of the 
pleasure in sex? It is to induce the organism to have sex. On a deeper level, 
then, it is quite possible that the combination of the logical thinking in the 
brain and the consciousness which enables us to have utility, to have desires, is 
an outcome of an evolutionary process. So rational thinking, purpose oriented, 
utility maximizing behavior is an outcome of the evolutionary process. Not 

The Selfish Gene, Oxford: Oxford University Press, 1976. 




21 



the other way around - like we usually say that evolution is “as if” organisms 
were thinking! No, the thinking comes because of the evolution. What Fm 
trying to say is that in the end the two interpretations of Nash equilibrium 
really could be two sides of the same coin. 

In his book, The Growth of Biological Thoughi?'^, Ernst Mayr distinguishes 
between two kinds of explanation in biology, corresponding to the questions 
“how” and “why”. Sometimes people confuse these terms. For example, the 
question “why do we see” might be answered in two ways. One is, “we see 
because we have eyes with lenses and retinas and neural connections to the 
brain.” The other is, “we see because it helps us enormously in getting along 
in the world.” Mayr says that the first answer really responds to the question 
^^how do we see;” the term “why” is more appropriate to the second answer. 
“How” refers to the mechanism, “why” to the function. Needless to say, both 
questions are legitimate, and both answers are correct. 

What I’m suggesting is that to the extent that it really occurs, conscious, 
rational utility maximization is a mechanism, an answer to the “how” ques- 
tion. Consciousness, the desire to maximize, the ability to calculate - these 
are traits that have evolved, like eyes. Their function is to enable us hu- 
man beings to adapt, in a flexible way, to the very complex environment in 
which we operate. Because of its complexity and variability, it would be much 
more difficult to operate in this environment with genetically predetermined, 
“hard- wired” modes of behavior. 

Q: Let us now turn to the questions that Werner Giith prepared in writing. 
The first one is this: One can distinguish two major tasks of game theory, 
namely 

— formally to represent situations of strategic interactions, i.e., to define ad- 
equate game forms, and 

— to develop solution concepts specifying the individually rational behavior 
in such games. 

A: Well, let me take issue with that right away. It is not only strategic in- 
teraction. Yesterday we were talking about cooperative and noncooperative 
game theory and I said that perhaps a better name for cooperative would be 
“outcome oriented” or “coalitional,” and for non-cooperative “strategically 
oriented” . The way Giith’s question is set up already points to the strategic 
direction. That, of course, is a very important part of game theory, but it 
is just one side of it. Game Theory develops not only solution concepts that 
specify rational behavior of individuals, but also solution concepts that spec- 
ify outcomes. It is not just a question of behavior, but also of outcomes. And 
this is very very important, because in order to do game theory you would 
be very restrictive if you confined yourself to situations that are strategically 
well defined. In applications, when you want to do something on the strategic 

Cambridge, Mass.: Belknap Press, 1982. 




22 



level, you must have very precise rules; for example, like in an auction. An 
auction is a beautiful example of this, but it is very special. It rarely happens 
that you have rules like that. What other situations do you have that are 
strategically well-structured? OK, sometimes you have a parliament where 
you have rules of order and rules of introducing bills; Ferejohn^^ has actually 
done some work on applications of noncooperative theory to political science 
in political situations that are strategically well-defined. Another example is 
strategic voting^"^. Those situations are basically strategic and there really 
the non-cooperative side is very important. But such situations, with very 
precisely defined rules, are relatively rare. 

Cooperative theory allows the rules of the game to be much more amor- 
phous. If you wanted to do the Roth labour market, you could not do that 
non-cooperatively, the rules are not sufficiently well specified. If you wanted 
to do a non-cooperative analysis of coalition formation, you can’t do it, it is 
not sufficiently well-specified. Who talks first, who talks second? It matters 
enormously in noncooperative game theory. 

To sum it up, it is not just behavior that matters; it is also outcomes, 
and at least as much outcomes as behavior. So I disagree with the phrasing 
of the question. 

Let’s go on and read the second sentence of Werner’s question. 

Q: “In the social sciences one presently relies nearly exclusively on non- 
cooperative game models.” 

A: But that’s absolutely incorrect. I can’t disagree more. There have been 
important analyses of noncooperative models, but as the discussion up to 
now has shown, many of the models that we have been talking about are 
cooperative and many of them are successful ones. Roth’s work on labour 
markets^^ is mindblowing, this is game theory at its best. If anything, the 
cooperative theory has played a more important role in applications than the 
non-cooperative theory. 

Q: Maybe social sciences should be narrowed down to economics and the 
question has been motivated by the fact that there are books about game 
theory that don’t refer to the cooperative branch at all. 

A: That is true, but it is unfortunate. The fact that there are books published 
doesn’t mean that that is what the world is about. The people who write 
these books are missing some very important sides of game theory; they are 
representing only their own knowledge and their own interests. These books 
do not give a balanced summary of what game theory has accomplished. 



See, e.g., Ferejohn, J. and J. Kuklinski (eds.) (1990): Information and Demo- 
cratic Processes^ Urbana: University of Illinois Press. 

Of course, as we mentioned, voting also has a cooperative side, 
op. cit. (Footnotes 8 and 9). 




23 



By the way, there is an excellent book, recently published, by Osborne 
and Rubinstein^®. These authors made most of their theoretical contributions 
on the strategic side, and yet they devote a nice portion of their book to 
cooperative game theory. I recommend this book highly, it is beautifully 
done, and it recognizes the importance of the cooperative theory. 

Incidentally, the Rubinstein alternating offers modeP^ is an example of a 
“bridge” : a noncooperative, behavioral model leading to the Nash bargaining 
solution, which, as we know, is a cooperative, outcome-oriented concept. Not 
only in TU-situations; it also leads to the Nash solution in an NTU-setup, as 
Binmore hcts shown^®. 

Summing up, one should look at what is significant, not at what is fash- 
ionable. 

Q: Perhaps we can now move to the third and last part of this question. 
It reads: Although the noncooperative focus brought about a very thorough 
analysis of many institutional aspects (e.g. of economic markets), it seems 
that nearly every phenomenon can be explained by designing an adequate 
model (e.g. by relying on an infinite time horizon as in the case of the so- 
called Folk Theorems). Do you see this more as a problem (we need more 
selective predictions!) or as an advantage (we have to find out the relevant 
institutional details!)? Is there some hope for a revival of cooperative game 
theory, which avoids the specification of many facets? 

A: There are several things to say to that. First, a word about institutions. 
Institutions can be treated as exogenous or endogenous. Giith’s question (“we 
have to find out the relevant institutional details”) treats them as exogenous: 
Given this or that institution, what are the equilibria? Though this has some 
interest, it is more interesting to treat institutions endogenously. Why did a 
given institution come about? What are the underlying forces that led to its 
formation? 

This dichotomy is discussed, for example, in the work by Jacques Dreze 
and me about rationing^^. Before this work, rationing was treated exoge- 
nously: Given the institution of rationing, what parameter values would bring 
the economy into equilibrium? Dreze and I asked a different question: What 
led to the institution of rationing? If it is a matter of excess demand or 
supply, why specifically rationing, and what form of rationing? Could other 
institutions handle this? 



A Course in Game Theory^ Cambridge, Mass.: MIT Press, 1995. 

Rubinstein, A. (1982): “Perfect Equilibrium in a Bargaining Model,” Econo- 
metrica 50, 97-109. 

“Nash Bargaining Theory II,” in The Economics of Bargaining^ K.G. Binmore 
and P. Dasgupta, eds., Oxford: Basil Blackwell 1987, 61-76. 

“Values of Markets with Satiation or Fixed Prices,” Econometrica 54 (1986), 
1271-1318. 




24 



An even more basic example is the equivalence theorem^® itself. Rather 
than taking the institutions of money and prices as given, the equivalence 
theorem predicts the emergence of these institutions. 

Note that both these examples come from cooperative game theory, so 
we are led naturally to your question about a “revival” of the cooperative 
theory. Cooperative theory is actually doing quite well. I’ve already said in 
this interview that many of the most interesting applications of game theory 
come from the cooperative side. In his invited talk at this conference^^, Mike 
Maschler discussed over thirty significant contributions to the cooperative 
theory that have been produced over the last few years. Yes, I suppose one 
could call that a “revival.” And I agree with you that an important advantage 
of the cooperative theory is that it takes a broader, more fundamental view, 
that it is not so obsessed with procedural details; its fundamental parameters 
are the capabilities of players and coalitions. 

Adam Brandenburger, who teaches at the Harvard Business School, has 
told me that the students there consider the cooperative theory a lot more 
relevant to business than the non-cooperative theory. In my opinion, both 
are important, perhaps for different kinds of applications. 

Let’s come now to your question about “nearly every phenomenon can be 
explained by designing an appropriate model.” First of all, as you say, that 
happens less with cooperative models. But beyond that, the importance of 
a model depends on the sum total of its applications, on how the specific 
kind of model of this kind has been applied in other places. One has to get a 
feel for the applications that are covered by a certain kind of model. Perhaps 
you could design a model for anything, but that does not mean that it is an 
interesting model. It is interesting if you design it, and then it has applications 
A, B, C, D; it applies to labour, it applies to auctions, it applies to search, 
it applies to signalling, it applies to discrimination. When you have a lot 
of applications, then you can start saying, well, this is an important model. 
This is what I tried to point out yesterday in my lecture on relationships, 
that when you get something which gives you important things in many 
different applications, like the Shapley value, or the nucleolus, then it gets 
some credibility. 

Q: Your answer brings up a related issue. In the survey that you wrote for The 
New Palgrave^^ you describe the game theoretic approach as being very dif- 
ferent from the classical approach in economics. The game theoretic approach 
is unifying: the same concept is applied in various contexts. In contrast, in the 
economic approach, a model and solution concept is tailored to the situation 
at hand. What we currently see in game theoretic applications to economics 
is something like a half-way house: There is a unifying solution concept - 

That is, the equivalence between competitive equilibrium in markets and game 
theoretic concepts like core and value. 

“Games ’95,” Jerusalem, 
op. cit. (Footnote 18). 




25 



Nash equilibrium - but there is a great deal of freedom in constructing the 
extensive or strategic form model. There is a lot of flexibility. 

A: Well, that applies to the non-cooperative theory. On the cooperative side, 
there are three or four central solution concepts - value, core, nucleolus, stable 
sets - but much less flexibility in constructing the model. The model is much 
better defined. 

Even on the non-cooperative side, it’s not clear that Nash equilibrium can 
be called a “unifying” solution concept. What about all the refinements and 
other variations? We still have a way to go, the smoke still has not cleared on 
the refinement battle field, or maybe it is not a battle field, but a refinement 
factory. I don’t know what the product there is. We have a lot of refinements; 
which is the right one? Eric, actually what do you think? Now it is 1995, we 
are more than a decade into the refinement business, almost two decades. Do 
you see one or two or three refinements emerging as the accepted ones? What 
do you think? 

Q: I think that some, like backwards induction, subgame perfect equilibrium, 
are there to stay. Others, like proper equilibria probably not, since they don’t 
satisfy properties like invariance. As far as the variety of stability-like con- 
cepts, that were introduced by Kohlberg and Mertens^^, is concerned, it is 
still unclear, the dust hasn’t settled yet. For example, there is still discussion 
about the admissibility requirement, is it necessary for strategic stability 
or not? Should one really look at strategy perturbations as, for example, 
Nash did in his original work on bargainings"^? The discussion is still open, 
I think. And then there is the concept of persistent equilibria, due to Kalai 
and SametS^, which I think belongs to a different category; it might show 
up in a learning context, in what Nash calls the “mass-action” interpreta- 
tion of equilibria. That is, a learning process might converge to a persistent 
equilibrium. 

A: Your answer is very useful; it underscores the difference between your 
viewpoint and mine, as representatives of different schools of thought. 

For example, in discussing refinements just now, you were concerned about 
assumptions: admissibility is too strong, too weak, and so on. That is not my 
concern. I have never been so interested in assumptions. I am interested in 
conclusions. Assumptions don’t have to be correct; conclusions have to be cor- 
rect. That is put very strongly, maybe more than I really feel, but I want to be 
provocative. When Newton introduced the idea of gravity, he was laughed at, 
because there was no rope with which the sun was pulling the earth: Gravity 
is a laughable idea, a crazy assumption, it still sounds crazy today. When I 
was a child and was told about it, I could not believe it. It does not make 

“On the Strategic Stability of Equilibrium,” Econometrica 54 (1986), 1003-1034. 

“The Bargaining Problem,” Econometrica 18 (1950), 155-162. 

“Persistent Equilibria in Strategic Games,” International Journal of Game The- 
ory 13 (1984), 129-144. 




26 



any sense; but it does happen to yield the right answer. In science one never 
looks at assumptions; one looks at conclusions. Where do these refinements 
Zead? Take an application, what do Kohlberg/Mertens^® have to say about 
this application? What do Kreps/Wilson^^ have to say about this applica- 
tion? What do Kalai/Samet^® have to say about this application? What do 
Cho and Kreps^^ have to say about it? I don’t care what the assumptions 
are. So, I get a feel for what the concept says not by its definition but by 
where it leads. This is a fundamental difference between what I am trying to 
promote and the school that you are representing (which, of course, also has 
its validity). 

Beyond that I agree with you that what’s here to stay is subgame perfect 
equilibrium. That says a lot and we know what that says, but we don’t know 
it from the definition. We know it from the applications. What else is here 
to stay, I don’t know. I agree entirely with what Giith says that, given the 
situation, there is a lot of room for constructing a game tree, and it matters 
very much how you construct the game tree and I think that that is one 
of the problems. And that limits the applicability of the strategic approach 
altogether. It does not eliminate it, it has very important applications, but 
it limits it. 

Q: As far as applications are concerned, one can point to signalling games. 
For example, take Spence’s original model of education as a signal^®. The 
game model of that situation has many equilibria, but the stable equilibrium 
outcome corresponds with the unique one that was originally singled out by 
Spence. I believe we can empirically verify the predictions of this equilib- 
rium, for example, the overinvestment in the signal. I think we could also 
verify these predictions in other signalling contexts, for example, in financial 
markets. 

A: That’s the kind of result I am looking for. When you ask what are the 
right refinements, you have to say: What are the refinements that give us the 
Spence signalling equilibrium? That’s what we have to ask. 

Q: Well, in order to get that answer you have to apply strong solution con- 
cepts, you need basically the full strength of the Kohlberg/Mertens^^ 1986 
Econometrica stability concept. 

A: Fine, we need more applications like that. We need to have applications 
out there, not just Beer-Quiche; that’s very important, but it is an example. 

op. cit. (Footnote 33). 

Kreps, D. and R. Wilson (1982): “Sequential Equilibrium,” Econometrica^ 50, 

863-894. 

op. cit. (Footnote 35). 

“Signaling Games and Stable Equilibria,” Quarterly Journal of Economics 102 

(1987), 179-221. 

Spence, A. (1974): Market Signaling. Cambridge, Mass: Harvard University 

Press. 

op. cit. (Footnote 33). 




27 



We need a class, a kind of model to which to apply these things. By the 
way, there is an excellent article of Kreps and Sobel^^ on signalling in the 
Handbook. There are a lot of important applications out there, there is no 
question about it. But, as I said, the smoke has not cleared yet. It hasn’t 
cleared on the assumptional side and it hasn’t cleared on the conclusional 
side. 

Q: What is definitely a drawback of some of these solution concepts is that 
they are very difficult to work with. Just as it is difficult to work with Von 
Neumann-Morgenstern stable sets in the cooperative theory, it is also difficult 
to see what the implications of the Mertens^^ stability concept are. 

A: That is a very important point and it is one that I endorse entirely. It is 
something that I also stressed in the article “What is Game Theory Trying to 
Accomplish?” : It is important to have an applicable model. It sounds a little 
like the man who had lost his wallet and was looking under the lamppost for 
it. His friend asked him: Why do you look only under the lamppost? And he 
answered: That’s because there is light there, otherwise I wouldn’t be able 
to see anything. It sounds crazy, but when you look at it more closely it is 
really important. If you have a theory that somehow makes a lot of sense, 
but is not calculable, not possible to work with, then what’s the good of it? 
As we were saying, there is no “truth” out there; you have to have a theory 
that you can work with in applications, be they theoretical or empirical. 

Incidentally, there is a man whom I love and respect very much and with 
whom I have worked closely and extensively, but who disagrees with me on 
this issue, and that is Mike Maschler. He has the opposite opinion. He wants 
to know what the “truth” is, and if it is difficult to calculate, he doesn’t care, 
he has to find the truth. I don’t have this notion of truth. It is really odd 
to see him wanting to find the “right” bargaining set. He has this idea that 
some of the bargaining sets are wrong and some of them are right. I don’t 
have this idea. I have the idea of usefulness, not right or wrong. So, he would 
disagree with me. He says, if it is hard to calculate, it is just too bad, but 
we must find out what it is. My own viewpoint is that inter alia, a solution 
concept must be calculable, otherwise you are not going to use it. 

Q: Whereas Reinhard Selten has tried to refine the equilibrium concept, as 
originally proposed by Cournot and Nash, you have developed your well- 
known coarsening idea of correlated equilibria^"^ . Do you think that solution 
concepts should allow for more behavioral possibilities rather than select more 
and more specifically? 

A: Well, my response to that is that the word “should” is out of place. I can’t 
answer the question because I disagree with its formulation. This is again a 

op. cit. (Footnote 14). 

op. cit. (Footnote 16). 

“Subjectivity and Correlation in Randomized Strategies,” Journal of Mathe- 
matical Economics 1 (1974), 67-96. 




28 



somewhat different viewpoint from that of many people. Game theory is not 
a religion. In religion one says: One should observe the Sabbath, one should 
give charity. But game theory is not a religion, so the word “should” is out of 
place. I love correlated equilibria and I also love subgame perfect equilibria; 
I have written papers, which I hope the world will enjoy, on both subjects. 
A few years ago I wrote about correlated equilibria, and recently I published 
a paper^^ called “Backward Induction and Common Knowledge of Rational- 
ity” . Backward induction, of course, is a subgame perfect equilibrium. So I 
do both, and I think that one “should” not do this or that exclusively, but 
one should develop both concepts and see where they lead. There are impor- 
tant things to say about correlated equilibria and there are important things 
to say about subgame perfect equilibria. This idea that correlated equilibria 
represent the truth, or that subgame perfect equilibria represent the truth, 
is an idea that I reject. 

Q: If game theory is not religion, is it a tool? 

A: Yes, absolutely, it is science. It is a tool for us to understand our world. 

Q: Let us now move on to the notion of consistency, which played an im- 
portant role in your talk yesterday. It is a property that is satisfied by many 
solution concepts, among them Nash equilibrium: If we fix some of the players 
in the game at their equilibrium strategies, then the combination of equilib- 
rium strategies for the remaining players constitutes an equilibrium of the 
reduced game. When writing your paper for The New Palgrave^® you were 
apparently already aware that this property could be used to axiomatize the 
concept. I would like to discuss this property in connection with equilibrium 
selection. There is a paper^^ that shows that no solution concept that assigns 
a unique equilibrium to each finite strategic game can be consistent. I don’t 
know how to deal with this. 

A: Well, that is certainly a very interesting result, and it is also important. 
But I can’t say that it is distressing. If there is one thing that we have 
learned from axiomatics, it is that if you write down all things that are 
obviously something that you would want, then you almost always find a 
contradiction. So let me start by saying that I won’t lose sleep over that. 

Having said that, let me respond substantively. Equilibrium selection in 
the style of Harsanyi and Selten^® is a beautiful venture, it is very important. 
It is important for two reasons. One is, that from one point of view, equilib- 
rium selection is an important facet of the background of equilibrium. Game 
theory is not a religion, and, equilibrium does not have to be “justified” ; but 

Games and Economic Behavior 7 (1995), 6-19. 
op. cit. (Footnote 18). 

Norde, H., J. Potters, H. Reynierse and D. Vermeulen (1996), “Equilibrium 
Selection and Consistency,” Games and Economic Behavior 12, 219-225. 
Harsanyi, J. and R. Selten (1988): A General Theory of Equilibrium Selection 
in Games, Cambridge, Mass.: MIT Press. 




29 



for those people who do want to justify it, including Harsanyi and Selten, 
selection plays a fundamental role. (Let me say that though I hold certain 
methodological viewpoints, that does not mean that I reject the importance 
of other people’s methodological viewpoints. I have said that assumptions are 
not important, but I can understand also people who argue that assumptions 
are important, and who want to justify equilibria, and who do think that 
game theory is a way of life, that’s OK also.) So in that respect - to “justify” 
equilibria - selection theory is important. And it has other important sides 
to it, byproducts: risk dominance is a very important by-product of selection 
theory, and so is the tracing procedure. 

On the other hand, selection theory has a sort of fantastic aura to it. 
Hundreds of pages, filled with extremely complex instructions as to how one 
picks one equilibrium in each game. And, you know, for 20 years before they 
published the book, they had a different version every year or two. And now 
that they have published the book, they still come out with new versions. It 
is a beautiful abstract structure, and it has important concrete implications, 
but it has also this fantastic Rube Goldberg aura to it. 

In brief, that one cannot find a consistent selection is not terribly dis- 
tressing or surprising. If anything, it is an additional criticism of selection 
theory, not of consistency. Though it is not devastating, it is something that 
you have to chalk up against selection theory. 

Q: In your foreword to the book of Harsanyi and Selten, you write that one 
rationale of Nash equilibrium, that a normative theory that advises people 
how to play has to prescribe an equilibrium since otherwise it is self-defeating, 
essentially relies on the theory making a unique prediction, hence, one needs 
a selection to justify the equilibrium concept in this way. 

A: Yes, that is precisely what I meant when I just said that selection plays 
a fundamental role in justifying equilibrium. Thank you for clarifying that. 
What was your question? 

Q: It was a remark, that, if you need selection for this “justification” of the 
equilibrium concept and if there is no consistent selection, then this “justifi- 
cation” might be problematic. 

A: Yes, it is indeed a little problematic. As I just said, the whole selection 
project is a little problematic, and this is a mark against it. But if you want 
to “justify” equilibrium from a cognitive point of view, then equilibrium 
selection is not the only way of doing it; you can do it in other ways also. 
Have you seen the recent paper^^ by Brandenburger and myself? It is a 
different cognitive approach to Nash equilibrium, not using selection. You 
get equilibrium when certain informational assumptions are satisfied; the 
equilibrium does not have to be unique. 

“Epistemic Conditions for Nash Equilibrium,” Econometrica 63 (1995), 1161- 

1180. 




30 



So, selection is just one way of “justifying” equilibrium. That it is incon- 
sistent with consistency is unfortunate, or regrettable, but it is not the end 
of the world, certainly not the end of the world for equilibrium, not even the 
end of the world for selection, we often have inconsistencies, it is something 
one learns to live with, both as a physicist and as a game theorist. 

There is one more point to be made in this connection. Harsanyi and 
Selten base their selections on the mathematical form (strategic or perhaps 
extensive) of the game. Thus in two games with the same mathematical 
form, the same equilibrium must be selected. Now that \s surely needed to 
“justify” the idea of equilibrium from the point of view of advising the player. 
When you are giving advice, you do know about the real-life context of the 
game, and you can base your advice on that. You could easily select different 
equilibria in different contexts. For example, in my controversy with Roth and 
Shafer about the NTU Shapley value, I discussed two games with precisely the 
same mathematical form - one in a trading context, the other in a political 
context - whose “natural” or “expected” equilibria are quite different^^. For 
a simpler example, think of a two-person game where each player must choose 
“L” or “R” ; each gets 1 if they choose the same, 0 otherwise. Suppose they 
are driving toward each other, and the “L” or “R” represents the side of the 
road on which they drive. If it’s in England, I would select (L,L); if in the 
Netherlands, (R,R). Harsanyi and Selten suggest randomizing, which seems 
a little crazy^^. 

In brief, there’s no good reason to base the selection on the mathematical 
form only; the context, the history also matter. 

Q: Werner also had a question about consistency. Could we turn to the last 
part of that question, which reads: Could you comment on the extremely 
opposite justifications of Nash equilibrium which partly require perfect ratio- 
nality and partly deny any cognition? 

A: I have already responded to that at some length in this interview. They 
are just different approaches. There is no problem with that; on the contrary, 
it sheds light on them. It helps to fortify, to corroborate the concept, to 
validate it. Giith thinks of them as one “versus” the other^^; he thinks if 
one is “right,” the other must be “wrong.” I think they are both right, they 
both illuminate the concept, from different angles. And as I have said, I don’t 



“On the Non- Transferable Value: A Comment on the Roth-Shafer Examples,” 
Econometrica 53 (1985), 667-677; see specifically Section 8, 674-675. 

They might answer that I’m making too much of a degenerate, non-generic 
example. But that misses the point. Suppose we perturb the payoffs very slightly, 
so that the Harsanyi-Selten selection is pure. I would still select (L,L) in England 
and (R,R) in the Netherlands. They, of course, would select the same in both 
countries. That’s almost as crazy. 

Giith, W., and H. Kliemt “On the Justification of Strategic Equilibrium — Ra- 
tionality Requirements Versus Conceivable Adaptive Processes,” DP 46, Eco- 
nomics Series, Humboldt University, Berlin, 1995. 




31 



believe in justifications; it’s not religion and it’s not law; we are not being 
accused of anything, so we don’t have to justify ourselves. 

Q: Let’s move on to the next question on Giith’s list: “In my view, the notion 
of a strategy with all its partly counterfactual considerations seems already 
a too demanding concept for a descriptive theory. Correlated equilibria are 
of an even more complex nature.” 

A: Well, both assertions sound strange. It’s true that the notion of a strat- 
egy is an abstraction, but it’s conceptually a simple object. Anyway, if you’re 
going to challenge that, then you’re challenging the whole conceptual basis 
of the noncooperative theory, including, of course, the notion of Nash equi- 
librium. 

As for correlated equilibria, conceptually they are fairly simple objects. 
The set of correlated equilibria is always a convex, compact polyhedron with 
finitely many extreme points, it is a very simple object, very easy to work 
with, much easier than the set of Nash equilibria. Nash equilibria are alge- 
braically very complex objects. Harsanyi once wrote an article describing the 
deep algebraic geometry of Nash equilibria^^. By contrast, a schoolchild can 
work out the correlated equilibria. They are much simpler objects. 

Q: By now I have understood that assumptions do not count. 

A: It’s not that assumptions don’t count, but that they come after the con- 
clusions; they axe justified by the conclusions. The process goes this way: Sup- 
pose you have a set of assumptions, which logically imply certain conclusions. 
One way to go is to argue about the innate plausibility of the assumptions; 
then if you decide that the assumptions sound right, then logically you must 
conclude that the conclusions are right. 

That’s the way that I reject, that’s bad science. 

The other way is not to argue about the assumptions at all, but to look 
at the conclusions only. Do our observations jibe with the conclusions^ do the 
conclusions sound right? If yes, then that’s a good mark for the assumptions. 
And then we can go and derive other conclusions from the assumptions, and 
see whether they Ye right. And so on. The more conclusions we have that jibe 
with our observations, the more faith we can put in the assumptions. 

That’s the way that I embrace, that’s good science. Logically, the con- 
clusions follow from the assumptions. But empirically, scientifically, the as- 
sumptions follow from the conclusions! 

Q: Let’s move to the next question on Giith’s list: Like in traditional evo- 
lutionary biology, in evolutionary game theory one often assumes a “geneti- 
cally" determined behavior. In modern ethology this is rather debated (apes, 
for instance, are known to have well developed cognitive systems). Should 

Unpublished. See also Schanuel, S.H., L.K. Simon, and W.R. Zame (1991), “The 
Algebraic Geometry of Games and the Tracing Procedure,” in Reinhaxd Selten 
(ed.), Game Equilibrium Models //, Berlin: Springer. 




32 



one not allow for a more continuous transition from no cognition at all (e.g. 
when studying primitive organs like plants) to more or less rational behavior 
(when studying the behavior of apes and humans) to resemble the rather 
gradual evolutionary process as, for instance, measured by DNA-differences? 

A: We discussed that above. No, one should not allow for that. We don’t have 
to talk about apes, we can talk about humans. Humans undoubtedly have 
well developed cognitive systems and, in spite of that, much human behavior 
fits into the evolutionary paradigm. Most human actions are not calculated. 
I know that I don’t calculate. Maybe you think that I don’t have a well 
developed cognitive system: be that as it may, I hardly ever calculate when 
making decisions. Much of behavior is not calculated; it is either genetically 
determined or it is determined by experience, by learning - which is very 
similar to genetic determination, in the sense of Dawkins’s memes^"^. A meme 
and a gene behave mathematically almost the same. Moreover, as I said 
above, to the extent that we do calculate, it may be a result of evolution. 

Q: You have been very influential in developing a rigorous definition of com- 
mon knowledge of rationality (CKR) and elaborating its implications. Re- 
cently it has been argued that CKR is self contradictory where one, of course, 
relies on more centralised players like in the normal form and denies “trem- 
bles” in the sense of Selten’s perfectness idea of equilibria. When do you think 
is the assumption of CKR justified? 

A: Well, to answer this last question, it is almost never justified. It is a far 
reaching assumption. Please refer to my paper in the Hahn Festschrift^^, “Ir- 
rationality in Game Theory” , which indicates how very very small departures 
from CKR can lead to behavior that is very different from that of CKR, for 
example in the centipede game. The departures from CKR are small both in 
the sense of being tiny probabilities and also in that the failure is at a high 
level of CKR. In other words, you get mutual knowledge of rationality to a 
high level, and after that level CKR fails only by a very small probability; 
and nevertheless, the results are very different from those under CKR. So, 
CKR is not “justified”; it does not happen. But that does not mean that 
CKR is unimportant. It is still very important to know what CKR says and 
what it implies, and to understand the connection between it and backward 
induction. Just like it is important to know how a perfect gas behaves, though 
there are no perfect gases; and it is important to know what perfect compe- 
tition does, although perfect competition does not exist. It is important to 
know what happens in the ideal state, although ideal states don’t exist. 

No, CKR never happens. 

Q: Is it self-contradictory? 

The Selfish Gene^ op. cit. (Footnote 21). 

In Economic Analysis of Markets and Games, Essays in Honor of Frank Hahn, 

P. Dasgupta, D. Gale, O. Hart and E. Maskin, eds., Cambridge, Mass.: MIT 

Press, 1992, 214-227. 




33 



A: No, that is simply a mistake. The idea that CKR is self-contradictory was 
due to an inadequate model. If you build your model carefully and correctly, 
then CKR is not self-contradictory. Admittedly, I had to think for three years 
before coming up with the right model. It is a very very confusing business, 
it is very subtle, it takes a lot of thought. The reader is referred to my article 
on “Backward Induction and Common Knowledge of Rationality”^®. That 
article should settle this issue, or go a long way to settle it. 

No, CKR is not self-contradictory in games of perfect information. 

Q: Well, I heard you lecture about this paper and I must admit I didn’t 
understand the argument fully. I haven’t read the paper yet, but I am going 
to do so. 

A: Good. Read the paper; it is not difficult and there is a careful conceptual 
discussion at the end of the paper. The structure is not very elaborate. It has 
three building blocks: rationality, knowledge, and commonality of knowledge. 
You first examine each one separately, define it carefully; then you put the 
three together, and you get backward induction. So first you have to ask 
yourself: What is rationality? What does it mean to be rational? And that 
has a simple definition. Not: What does it mean to have common knowledge 
of rationality, just what does rationality mean. And then, what does it mean 
to have knowledge? What is the exact model of knowledge? And you ask that 
independently of what rationality is, and what common knowledge is. Just, 
what does knowledge mean? And you find a satisfactory answer to that. And 
then you have to ask yourself: What is common knowledge? And there you 
use the accepted definition, you just iterate the knowledge operator. The key 
is to think about knowledge separately, and about rationality separately, and 
only then, in the end, to go to common knowledge of rationality; and then 
you get the result. It is to keep the ideas separate, not to confuse them. That 
is what mathematics is about, that’s where mathematics helps. 

Q: Shall we conclude the interview here or is there something that you would 
like to add? 

A: I would like to express my tremendous admiration for Selten, as a human 
being, and as a pioneer in a number of fields. First of all, in the equilibrium 
refinement business. The first refinements, and among those that are sure to 
survive, are the subgame perfect equilibrium and the ordinary perfect equi- 
librium, the trembling hand perfect. This will survive, this is a monumental 
idea. Another very important contribution is the idea of doing experiments. 
I don’t put quite the faith in experiments that Selten does, but one very 
important function of experiments, which he realizes himself, is that to run 
an experiment, you really have to see what the rules of the game are. You 
really have to think about the game very very carefully. Just like when you 
write a computer program for anything, you have to ask yourself: what am I 



op. cit. (Footnote 45). 




34 



doing here? Now, when you run an experiment it forces you to say: What is 
this game? What the subjects actually do is perhaps not all that significant 
(although it can be suggestive), but when you are designing an experiment it 
is very important to design it right. So, that is an input that Selten recognizes 
and emphasizes, and it bears more emphasis. 

And, as I said last Sunday evening when we were talking about the No- 
bel Prize winners, a very important contribution of Selten is his work on 
the evolutionary approach. He has made some important scientific^^ contri- 
butions on this, but the main importance is to bring this into the world of 
game theory. To bring in the work of Maynard Smith^® and his associates. 
Another thing to be mentioned is that he had an important influence also on 
John Harsanyi in value theory. His thesis^^ was basically about the Harsanyi 
value®®, and that is a very important concept. 

On a different level, another important idea is Selten ’s umbrella. Fm not 
sure that all the people at the dinner on Sunday understood what this is 
about. Selten lives in a world where rain is unpredictable; it can rain any 
day. Rather than continually investing mental energy in the question “should 
I take an umbrella or shouldn’t I?” , he has made it a rule of life always to 
take an umbrella. Now, if he starts thinking that when he comes to Israel he 
really does not need an umbrella because it never rains in the summer, then 
he is contravening this rule of life which is not to put thought into it. So he 
has this meme for always carrying an umbrella. He himself is an embodiment 
of the evolutionary approach to game theory. 

So, I have tremendous admiration for Reinhard and I am very happy that 
he got the Nobel Prize and I wish him continued success in the future and 
he seems to be strong and going well. 

Q: Thank you very much for this interview! 



“A Note on Evolutionarily Stable Strategies in Asymmetric Animal Conflicts,” 
Journal of Theoretical Biology (1980), 93-101; and “Evolutionary Stability in 
Extensive Two-Person Games,” Mathematical Social Sciences {MSS) 5 (1983), 
269-363 (see also “Correction and Further Development,” MSS 16 (1988), 223- 
266). 

Maynard Smith, J. (1982): Evolution and the Theory of Games. Cambridge: 
Cambridge University Press. 

Selten, R. (1964): “Valuation of n-Person Games,” in Advances in Game Theory^ 
Annals of Mathematics Study 52, R.D. Luce, L.S. Shapley and A.W. Tucker, 
eds., Princeton: Princeton University Press, 577-627. 

Harsanyi, J. (1963): “A Simplified Bargaining Model for the n-Person Cooper- 
ative Game,” International Economic Review^ 4, 194-220. 




Working with Reinhard Selten 

Some Recollections on Our Joint Work 1965-88 

John C. Harsanyi 

University of California at Berkeley 

Reinhard Selten and I first met in Princeton in 1961 at a game-theory confer- 
ence sponsored by Oskar Morgenstern. The earliest thing I remember about 
him is that he presented an excellent paper at the conference about the dif- 
ficulty of defining a value concept for games in extensive form. 

In 1965 both of us joined a small group of young game theorists study- 
ing the game-theoretic aspects of arms control as part-time consultants for 
the U.S. Arms Control and Disarmament Agency (ACDA) but formally em- 
ployed by the Princeton research organization Mathematica founded by Mor- 
genstern. Other members of our group were Bob Aumann, Harold Kuhn, 
Mike Maschler, Jim Mayberry, Herb Scarf, Martin Shubik, and Dick Stearns. 
(Sometimes also some other people joined our group for short periods.) Be- 
tween 1965 and 1969 the members of our group met fairly regularly about 
three times a year in Washington, D.C., or nearby to discuss the work each 
of us did for the ACDA project since our last meeting. 

As far as I remember, close cooperation by the two of us really started 
at the First International Workshop in Game Theory organized by Aumann 
and Maschler, and held in Jerusalem in 1965. It lasted only for about three 
weeks and had only 16 or 17 participants (Aumann, 1987, p. 476). But its 
small size facilitated a very lively and fruitful exchange of ideais among the 
participants. 

It was at this Workshop that I first outlined my probabilistic model for 
games with incomplete information (see Harsanyi, 1967-68). As a result, I 
received many helpful comments and suggestions from the other participants, 
including a suggestion by Reinhard on how to extend my model to cases where 
the different subjective probability distributions about alternative states of 
the world entertained by different player types axe inconsistent (cf. Harsanyi, 
1967-68, section 15). 

Both of us were great admirers of Nash’s (1950 and 1953) theory of two- 
person bargaining games. Thus, we decided already in Jerusalem to work to- 
gether on the problem of extending Nash’s theory to two-person bargaining 
games with incomplete information. In undertaking this work, one of our ob- 
jectives was to gain better game-theoretic understanding of the arms-control 
negotiations between the United States and the Soviet Union conducted at 
the time, and studied by our ACDA group. 

Our first paper on this work was our ”A Generalized Nash Solution for 
Two-Person Bargaining Games with Incomplete Information” , published in 
1968 as Chapter V of the volume The Indirect Measurement of Utility, Final 




36 



Report of ACD A/ST-143. ^ It has been reprinted as Chapter 3 of Mayberry 
et al. (1992). 

After writing this paper, we continued our work on this problem. We made 
particularly good progress during the academic year 1967-68, when Reinhard 
was a Visiting Professor in our Business School in Berkeley. We reported our 
results in a second paper^ with the same title as that of our first paper of 
1968. It was published in Management Science, vol. 18 (January 1972, Part 
II), pp. 80-106 (see Harsanyi and Selten, 1972). 

Soon after writing this paper, we came to the conclusion that, instead of 
defining specific solution concepts for particular classes of games, we should 
try to define a general solution concept for all games, both those with com- 
plete and those with incomplete information. As we agreed with Nash’s (1951, 
p. 295) suggestion that cooperative games should be modeled as noncooper- 
ative games of special kinds, our formal objective was to define a solution 
concept applying to all noncooperative games. 

This in turn meant finding rational criteria for selecting one specific Nash 
equilibrium as the solution of any given noncooperative game. In other words, 
it meant to suggest a credible general theory of equilibrium selection for all 
noncooperative games, including cooperative games modeled as noncoopera- 
tive games. 

In the end it took us 16 years of hard work to come up with a theory we 
felt to be worth presenting to the game-theorist community (see Harsanyi and 
Selten, 1988). Now, seven years after the publication of our book reporting 
our conclusions, I for one certainly do not want to claim that we have said 
the last word about the problem of equilibrium selection. Yet, I do think 
that we have made some real contributions to this problem - as well as some 
worth-while incidental contributions to the resolution of some other game - 
theoretic problems. 

Most of our work on the equilibrium selection problem was done in the 
1970s. Reinhard visited Berkeley for two or three periods of several weeks 
each during the decade. I visited him in Bielefeld for two full academic years, 
in 1973-74 and in 1978-79. We did a small amount of work also by correspon- 
dence and by telephone calls. 

Over time we first accepted and then rejected at least three different 
theories of equilibrium selection, before we adopted our final theory described 
in our 1988 book. 

Our first theory was based on the concept of diagonal probabilities. Let 
s = (si,S 2 , ...,5n) be a Nash equilibrium of an n-person game G. Suppose 
that at first all n players use their s-strategies. Then, all (n — 1) players 

^ Chapter VI of the same ACDA volume is another paper co-authored by Rein- 
hard on a closely related topic. It is: Reinhard Selten and John P. Mayberry, 
’’Application of Bargaining I-Games to Cold- War Competition” . Reprinted as 
Chapter 4 of Mayberry et al., op. cit. 

^ Our second paper deals only with incomplete-information bargaining games 
with fixed threats (Nash 1950). 




37 



j other than i deviate from their 5-strategies with the same probability p. 
The smallest deviation probability p that will destabilize player i from his 
5 -strategy Si will be called his diagonal probability di. Then, we can measure 
the stability of s by the product tt = Ilidi of these diagonal probabilities. 
This stability measure can be regarded as an analog of the Nash product 
(Nash, 1950). 

Yet, we rejected this theory based on these probability products when we 
found that it was not invariant to splitting a given player into two identical 
players. 

Another theory of equilibrium selection we considered was based on trying 
to measure the influence that any given player had on the strategies used by 
the other players. (This ’’influence” concept was meant to be a generalization 
of the probability that a given type ol player i will be present in a game 
with incomplete information.) 

A third theory we considered was based on the concept of formations (see 
Harsanyi and Selten, 1988, p. 195). This theory represented an important 
step toward the theory described in our 1988 book, in which the concept of 
formations, and particularly that of primitive formations, plays an important 
role. 

I am very sorry that I no longer remember many of the details of the 
theories we later discarded. I did not make any notes of our discussions. 
Wulf Albers was kind enough to send me all Working Papers of the Institute 
of Mathematical Economics at the University of Bielefeld published during 
the relevant period. But unfortunately they did not contain any information 
about our pre-1988 theories. Neither did our Working Papers published in 
Berkeley. 

I am reasonably sure that Reinhard kept much better records than I did 
about different stages of our work, and will eventually write them up for 
publication or will make them available to interested scholars. 

Let me finish this short paper with the following remarks. I feel very lucky 
indeed that I have had Reinhard as a close personal friend and as a close fel- 
low worker on many scientific problems of great importance over so many 
years. There are many scientific problems we managed to solve together, 
which I could not have done alone without his help. We often disagreed, as 
is inevitable in any serious scientific work, but our disagreements have never 
weakened our friendship. I enjoyed discussing many intellectual and social 
problems with him, even problems quite unrelated to our work. I likewise en- 
joyed his willingness, which I fully shared, to take views contrary to majority 
opinion when he felt to have good reasons to do so. 

I feel particular gratitude to him for the special interest he showed in the 
intellectual and emotional development of my son, Tom, and all the three 
of us are very grateful for the friendship that both Reinhard and his wife 
Elisabeth have shown for us over so many years. 




38 



References 



Aumann, R.J. (1987), ’’Game Theory”. In J. Eatwell et al. (eds.). The New Palgrave: 
A Dictionary of Economics. Macmillan Press, London; pp. 460-482. 

Harsanyi, J.C. (1967-68), ’’Games with Incomplete Information Played by Bayesian 
Players”, Parts I, II, and III, Management Science^ 14, 159-182, 320-334 and 
486-502. 

Harsanyi, J.C., and Selten, R. (1972), ”A Generalized Nash Solution for Two- 
Person Bargaining Games with Incomplete Information” , Management Science^ 
18 (January 1972, Part II), 80-106. 

Harsanyi, J.C., and Selten, R. (1988), A General Theory of Equilibrium Selection 
in Games. MIT Press: Cambridge, MA. 

Mayberry, J.P., et al. (1992), Game- Theoretic Models of Cooperation and Conflict. 
West view Press: Boulder, CO. 

Nash, J.F. (1950), ’’The Bargaining Problem”, Econometrica^ 18, 155-162. 

Nash, J.F. (1951), ’’Non-cooperative Games”, Annals of Mathematics^ 54, 286-295. 

Nash, J.F. (1953), ”Two-Person Cooperative Games”, Econometrica, 21, 128-140. 




Introduction and Survey 

Werner Giith 



Understanding strategic interaction can be interpreted in more than one way: 
One is to explain interaction phenomena by modelling them as games and 
derive their solutions by relying on perfectly rational players. Let us refer to 
this as the rationalistic approach to understanding strategic interaction. 

Another possibility is the behavioristic approach which requires only 
bounded rationality and which usually relies on empirical observations. Par- 
alleling this methodological dualism, which is so characteristic of Reinhard 
Selten’s work, the contributions in this volume can be partitioned accord- 
ingly, e.g. by distinguishing the applications of game theory from experimen- 
tal studies. 

A third approach is that of evolutionary stability, i.e. one views the ways 
in which the individual members of some population interact as the stable 
constellation for some evolutionary dynamics. Again Reinhard Selten was 
among the pioneers who clearly understood how the concept of evolutionary 
stability is related to the equilibrium concept and how it can be generalized, 
e.g. to games in extensive form. 

The contributions in this volume can also be distinguished into conceptual 
discussions and applications of established concepts where one may, further- 
more, subdivide each group into smaller categories. 

Although the volume is not partitioned into sections of related contribu- 
tions, this introduction offers a way how the various contributions could be 
arranged. Actually the papers were ordered accordingly. We admit, however, 
that other ways of subgrouping and ordering the papers are possible. Our 
attempt is simply to supply a more structured survey than just describing 
the contents of the various contributions. 

Conceptual Aspects of Game Theory 

The major task of game theory is first to formally describe situations of 
strategic interaction - one usually speaks of game forms - and second to 
develop solution concepts describing more or less specifically how rational 
players decide in such a formally described situation. The contributions in 
section I are partly addressing restrictions which one should impose on the 
game form (the papers by Binmore and by Harstad and Phlips) and partly 
discussing solution concepts and their properties. 

One standard assumption of most game theoretic models is that of perfect 
recall, i.e. every player remembers everything he knew before, especially his 
own previous moves. Clearly, the assumption is unrealistic. It can be defended 
normatively by arguing that perfectly rational players - unlike human beings 
— possess this ability. Inspired by the “absent-minded driver’s problem” Ken 
Binmore discusses various ideas how to incorporate imperfect recall in game 




40 



theoretic models. The essential idea of the absent-minded driver is that of an 
information set like: If player i goes R(ight) at the left node, he immediately 
forgets this and therefore does not know after this choice whether he is in the 
right or left decision node. Binmore discusses how the difficulties to define 
rational behavior for such games can be overcome by relying on repeated 
decision making or on a team approach. A more recent study (Aumann, 
Hart, and Perry, 1996) claims, however, that this may not be needed. 

According to the so-called Harsanyi-doctrine incomplete information 
should be consistent - individual beliefs about others’ types are the marginal 
distributions of the type vector generating distribution - if all players are 
perfectly rational. Prom the very beginning Reinhard Selten has suggested 
that one might want to allow for inconsistent incomplete information when 
studying actual situations where beliefs will not be revised until they are con- 
sistent. The idea is to introduce fictitious chance moves with vector- valued 
instead of single-valued probability assignments where, as usual, the proba- 
bility assignments are commonly known. Ronald M. Harstad and Louis 
Phlips go one step further by allowing for doubly inconsistent priors on a 
futures market: agents entertain inconsistent beliefs about the inconsistent 
beliefs of their trading partners who all can either be optimists believing in 
a high futures price or pessimists who believe that this price will be low. 
Although ex-post-inefficiences occur, they show that an ex-ante-efficient out- 
come can be reached by subgame perfect equilibria. 

Inconsistent incomplete information is also considered by Michael 
Maschler who suggests representing inconsistent incomplete information in 
extensive form by one game tree for each player (instead of the vector valued 
type generating distribution suggested by Reinhard Selten, 1982). Of course, 
to compute an equilibrium such that no type of no player can profitably 
deviate, a player has to know also the trees of the other players. 

Strict equilibria - every player loses by deviating unilaterally - satisfy the 
most crucial stability requirements like being immune against small trem- 
bles of strategies and/or payoffs. Unfortunately, they do not always exist. 
The refinement approach, inspired by Selten ’s concept of perfect equilib- 
ria, essentially tries to find out which crucial properties of strict equilibria 
a refinement, whose existence is guaranteed, should preserve and on which 
properties one should not insist. Robert Wilson focusses on admissibility 
- an equilibrium should not rely on dominated pure strategies - and demon- 
strates that requiring admissibility may be harmful when selecting among 
equilibrium components although it may be useful when selecting within a 
component. In spite of the deep substructures of games explored the argu- 
ments of Wilson’s thorough and - in our view - informative discussion can 
be easily understood since he illustrates them by special games. 

Unlike in their earlier versions of general theories of equilibrium selection 
Harsanyi and Selten (1988) have used two very different ideas to discrimi- 
nate among equilibria, namely risk dominance and payoff dominance. Their 




41 



mutual inconsistency is, furthermore, resolved by granting priority to payoff 
dominance which they did not require at all in their earlier versions. The 
intuition for this is that players will naturally focus attention on equilibria 
which everybody prefers. Eric van Damme considers team games in which 
all players always receive the same payoff since, in his view, the unique pay- 
off undominated equilibrium in such a game is clearly focal. What he shows 
is that even in this special class of games risk dominance and payoff domi- 
nance are inconsistent and that the payoff dominant equilibrium may be risk 
dominated. 

Roger B. Myerson is interested in “sustainable” equilibria which, if 
once culturally established, cannot be easily destabilised as, for instance, 
certain mixed strategy equilibria. After demonstrating his intentions with 
the help of some basic games like the battle of sexes - or stag hunt games 
he argues that persistent equilibria might be the best suited refinement for 
selecting “sustainable” equilibria. The attempt is rather to describe an open 
problem than to offer yet a rigorous definition. 

Evolutionary Game Theory 

It has been a great inspiration both for evolutionary biology as well as for 
game theory that under certain conditions evolutionary stability can be de- 
scribed by evolutionarily stable strategies. Peter Hammerstein tries to 
demonstrate in a non-technical way the conflict aspect of some important 
stages in the evolution of life, e.g. in the evolution of cells, why chromosomes 
have evolved, or in the origin of sexes. Together these examples illustrate why 
conflict analysis, i.e. evolutionary game theory, may be helpful in revealing 
the evolutionary logic in the design of life. 

One way to justify the assumption of perfectly rational players is to ab- 
stract from cognition at all, e.g. by relying on genetically predetermined be- 
havior as it is - until now - mostly assumed in evolutionary biology, and to 
assume that the frequency of the possible ways of behavior is governed by 
evolutionary dynamics, i.e. a more successful type of behavior becomes more 
frequent. What has to be discussed then is, of course, what kind of “evo- 
lutionary” dynamics are needed to justify certain ideas of rational behavior. 
Jonas Bjornerstedt, Martin Dufwenberg, Peter Norman, and Jorgen 
W. Weibull explore whether strictly dominated strategies can survive in the 
long run when certain assumptions for the evolutionary dynamics in discrete 
and continuous time are met. For an overlapping-generations version of the 
well-known replicator dynamics it can be shown that the survival of strictly 
dominated strategies depends on the “generational overlap”. Furthermore, 
it is shown that certain evolutionary dynamics in continuous time do not 
guarantee elimination of strictly dominated strategies. 

Based on Reinhard Selten’s extension of evolutionarily stable strategies 
to asymmetric animal conflicts Avi Shmida and Bezalel Peleg analyse an- 
imal conflicts with payoff irrelevant asymmetries. Reinterpreting Aumann’s 




42 



lottery device in the definition of correlated equilibria as nature’s choice al- 
lowing phenotypes to condition their behavior the authors show that strict 
and symmetric correlated equilibria can be viewed as distributions of evolu- 
tionarily stable strategies, i.e. a chance device in nature allows phenotypes 
to behave diflFerently in different states in spite of the same genotype. 

Applications of Non-Cooperative Game Theory 

The recent success of game theory, which is by now the dominating method- 
ology in economic theory, and perhaps also in the social sciences, is due to the 
insights of the rationalistic approach. One now understands more clearly the 
importance of certain institutional aspects like the time order of decisions, 
the exact informational conditions etc. The predominant tradition of the ra- 
tionalistic approach has, furthermore, relied on the so-called noncooperative 
approach of game theory as suggested by Nash (1951) and more thoroughly 
discussed later on by Harsanyi and Selten (1988). It essentially means to 
model a social conflict situation as a strategic game and to solve it then by 
determining its equilibria. 

Matthew Jackson and Ehud Kalai define a recurring bully game which 
can be informally described as a recurring chain store paradox with incom- 
plete information where all previous decisions can be observed. The bully 
— the analogue of the chain store - can be of two types, namely a ratio- 
nal one who would yield when being challenged and an irrational one who 
would fight in such an event. The bully’s type is chosen randomly accord- 
ing to one of several type generating distributions where challengers only 
know the probabilities of the various distributions, but not the one which 
has been actually selected. They, however, may draw inferences about this 
distribution by updating their beliefs in view of earlier decisions. Jackson and 
Kalai are especially interested in exploring whether the crazy perturbation 
or reputation- “solution” of the chain store pardox can be maintained. They 
therefore assumed that the selected distribution guarantees rational bullies 
with probability 1. With trembles their clearcut result is that in the long 
run rational bullies never fight and therefore will always be challenged re- 
gardless of the initial beliefs in the sense of the probabilities for the various 
type generating distributions. Thus if trembles ensure learning, building up 
a reputation of craziness requires craziness, i.e. irrational bullies must result 
with positive probability. 

In repeated games many types of behavior are possible, purely adaptive 
forms of behavior and more rational kinds of strategy selection. Thomas 
Quint, Martin Shubik, and Dicky Yan compare so-called dumb bugs - 
who stubbornly believe that others repeat their previous choices regardless of 
any contradictory evidence - with various sorts of more or less rational types 
of behavior (equilibrium and maxmin-players) as well as with malicious anti- 
players. The basic idea is to compare how monomorphic - all players rely 
on the same type of behavior - populations fare when they do not play 




43 



just one game but a whole universe of games. Since a “universe” is defined 
by a class of games and a probability distribution over this class, it could 
be interpreted as a complex stochastic game where players are allowed to 
react to the characteristics of the selected game. For a specific example it 
is demonstrated that a society of bugs may not fare worse than a society of 
equilibrium players. 

Eric J. Friedman and Thomas Marschak discuss the problem whether 
an organization with autonomous members will communicate less than the 
organization as a team which maximizes the sum of individual profits. Based 
on a class of special models, justified by “classic information theory”, the 
authors show that team members may communicate too little as well as too 
much when they become independent choice makers. An interesting further 
conclusion is that the customary role of mixed behavior in assuring existence 
of equilibria may be lost when communication effort is measured in a way 
that depends on probabilities. 

As in some of his earlier studies Eyal Winter analyses how the agenda 
determination is related to the results of bargaining. One model he explores 
assumes that the agenda is negotiated by the committee members before 
dealing with the actual issues; according to the other model each proposer 
can propose an alternative on any unsettled issue at any time. Whereas for 
the first model efficiency of the bargaining outcome cannot be guaranteed, 
this is possible for the second model if no more than two issues have to be 
decided and if members can be restricted to vote for or against an issue 
(yes-no voting). 

A central theme of human cooperation is how to prevent the inefficient 
results which would prevail if everybody behaves opportunistically. The ap- 
proach of Akira Okada is to allow players to change the “rules of the game” , 
e.g. by establishing a costly enforcement agency in order to limit free-riding. 
Based on an n-person prisoners’ dilemma he presents a game model whose 
rules describe how players by their free will can found an organization - a 
coalition of players - whose enforcement agent tries to discourage free-riding 
by the members of the organization. The multistage game is closely related 
to Reinhard Selten’s distinction between small (not more than 4 sellers) and 
large (not less than 6 sellers) markets (Selten, 1973). It is shown that an or- 
ganization with a prohibitive punishment policy has to be large enough, but 
- due to free-riding in forming the organization - usually no general cooper- 
ation will be achieved. In the case of symmetric players the only symmetry 
invariant equilibrium implying cooperative results is therefore one in mixed 
strategies. Okada also analyses a game with heterogeneous players and dis- 
cusses two applications, namely environmental pollution and the emergence 
of the state. 

Classic philosophers have studied the difficulties and possibilities to sub- 
stitute anarchy by social order when all individuals are rational. Werner 
Giith and Hartmut Kliemt try to relate some of these philosophical ideas 




44 



to the conceptual innovations by Reinhard Selten. The problems discussed 
and illustrated by simple games are Bodin’s attempts to understand what 
the lack of commitment power of the sovereign implies, de Spinoza’s analysis 
of opportunism in trust relations, Locke’s ideas to justify legal institutions by 
consensus and Hobbes’ related efforts. It is shown that some of the concep- 
tual and philosophical insights by Reinhard Selten were already anticipated 
although the general principles were not available what, in our view, illus- 
trates their intuitive appeal. 

Both, Elisabeth Selten and Reinhard Selten, are engaged in the Esperanto- 
movement and they both like to rely on this non-ethnic language whenever 
others speak Esperanto too. Together with Jonathan Pool Reinhard Selten 
has, furthermore, analysed classes of linguistic games (Selten and Pool, 1995) 
where communication can also be ensured when interaction partners learn 
some planned language which nobody has learnt so far and whose learning 
costs are much lower than those of natural languages. Werner Giith, Mar- 
tin Strobel, and Bengt-Arne Wickstrom try to resolve a typical problem 
in such linguistic games, namely the multiplicity of their strict equilibria, by 
applying the theory of equilibrium selection (Harsanyi and Selten, 1988). The 
major intention is, of course, to find out under which conditions international 
communication should rely on Esperanto rather than on learning complicated 
ethnic languages like English. 

Relating Cooperative and Non-Cooperative 
Game Theory 

Although the rationalistic approach has predominantly used the noncooper- 
ative way of modelling and solving strategic interaction, one certainly has 
experienced and will experience some frustrations when relying on the non- 
cooperative approach because of its enormous flexibility. By now it seems 
commonly accepted that nearly any type of interaction phenomena can be 
justified by carefully designing an appropriate noncooperative game model 
and focussing on one of its equilibria. This is most dramatically illustrated 
by the so-called Folk Theorems which, under appropriate conditions, can jus- 
tify any individually rational behavior or by the so-called crazy perturbation 
approach which introduces special forms of incomplete information. 

In view of these likely frustrations it seems important to keep in mind what 
cooperative game theory has to say about the results of strategic interaction. 
The cooperative approach does not specify the strategic possibilities of the 
interacting parties and may therefore imply a more robust type of results, 
provided that they are specific enough to offer some guidance. 

Benny Moldovanu illustrates how graph-theoretical methods can be 
used to study the relations of stable demand vectors and core allocations.The 
basic idea of stable demands is that every player forms an aspiration of what 
he requires when joining a coalition. Such demands are stable if no player 
can raise his demand without excluding himself from all coalitions that meet 




45 



the demands by all its members and if nobody depends on somebody else 
(an agent depends on another agent if any coalition, which can guarantee 
his demand, includes the other agent, but not vice versa). The author con- 
centrates on NTU (non transferable utility)-games whose players are buyers 
and sellers. Here each stable demand vector induces a graph whose vertices 
are the agents on the market and where a buyer and a seller are connected if 
their demands are compatible, i.e. if they agree on the sales price. 

Using non-atomic exchange economies Guillermo Owen discusses three 
properties of efficient allocations: the well-known competitive allocations, 
consistent allocations based on equal weighted contributions as well as ho- 
mogeneous allocations (coalition members receive the same vectors of net 
trade if their coalition becomes larger or smaller without changing its com- 
position) . It is shown that homogenity and consistency of efficient allocations 
correspond to competitiveness. 

Starting with the early and truly impressive approach of Edgeworth an 
important and influential justiflcation of competitive allocations is that they 
are the only core-elements when the number of agents becomes very large. 
Joachim Rosenmiiller establishes the equivalence of the core and the set 
of competitive allocations by an approach, based on “non-degenerate” char- 
acteristic functions. Unlike previous positive results relying on transferable 
utility his aim is to tackle also non-transferable utility economies. The study, 
which is seen as a first step in this direction, provides a partial answer: for 
certain classes of exchange economies with piecewise linear utilities a finite 
convergence-theorem can be proved. 

Philip J. Reny and Myrna Holtz Wooders refer to units of an orga- 
nization as states and to the organization itself as a commonwealth if certain 
stability requirements are met. They study the stability of the organization if 
its units can threaten to leave it. The model employed is a cooperative game 
with side payments. A threat (against a payoff vector) of a unit to secede is 
called credible if no group of players can achieve their payoffs by leaving and 
if this would harm at least one other player. A payoff vector and a partition 
of the organization into states is a commonwealth if the payoff vector is in 
the core of the game, if in each state all members need each other and if no 
states can credibly threaten to secede. 

Principles of Behavioral Economics 

Unlike the rationalistic research tradition the behavioristic approach is usu- 
ally based on empirical observations rather than on philosophical ideas of 
what rationality requires. By now there is a lot of evidence that people do 
not always choose the alternative which serves their own interests best. Very 
influential concepts trying to account for non-optimizing behavior are the 
norm of reciprocity and the theory of satisficing (instead of optimizing) and 
of aspiration adjustment. 




46 



Colin Camerer tries to discuss the major differences between main- 
stream experimental economics and the considerably longer experimental 
tradition in (social) psychology. Although he concedes that this distinction 
is not sharp, he states some crucial differences like a stronger orientation 
toward testing formal models or theories in experimental economics and the 
more thorough ways of experimental psychologists in analysing not only how 
people decide, but also why they do so. In our view, Camerer rightly concludes 
that these differences would have been sharper without the experimental work 
of Reinhard Selten who, since the beginning of his experimental work, has 
always tried to combine the two approaches. Both research traditions are 
needed to promote the behavioristic alternative of the still dominating homo 
oeconomicus. 

The behavioral norm of reciprocity, whose importance has been demon- 
strated by many experimental studies, is discussed by Elisabeth Hoffmann, 
Kevin McCabe, and Vernon L. Smith. Here reciprocity means to react in 
kind, i.e. one rewards kind and punishes mean partners by actions which do 
not serve one’s own interests in the best way. The authors stress the crucial 
role of reciprocity in organizing human cooperation and division of labor and 
outline several approaches to explain the evolution of reciprocity. Instead of 
speculating wildly they develop their arguments by relying on previous exper- 
imental results. Their article is therefore both a selective survey of influential 
experimental studies pointing out how relevant reciprocity is and an outline 
of how one might try to explain this norm. 

One of the most influential ideas of the theory of bounded rationality is 
the theory of aspiration adaptation (Sauermann and Selten, 1962). Unlike in 
normative theory decision makers are not assumed to search for the best, but 
to satisfy certain aspirations levels where these aspiration have to be updated 
in the light of own and others’ experiences. Reinhard Tietz has observed 
aspiration adaptation experimentally by asking the participants to engage 
in decision preparation by stating certain critical points, e.g. first demand, 
planned bargaining goal, attainable agreement, planned threat to break off, 
planned conflict limit in bilateral bargaining. He mainly relies on the results 
of his well-known Kresko-experiments where collective wage bargaining is 
based on a complex macroeconomic model. His major result seems to be the 
principle of aspiration balancing stating that both bargaining parties reach 
equally high aspirations. 

Most theories of characteristic function bargaining - a very prominent 
topic in cooperative and experimental game theory with pioneering contribu- 
tions by Reinhard Selten - simply state conditions for proposals (a coalition 
structure together with a proposed allocation of payoffs) to qualify as stable. 
Sometimes - like in the sequence “proposal, objection, count erobject ion” on 
which the bargaining set is based - the dynamics of bargaining enter the pic- 
ture, but only in a rudimentary way and often without affirmative empirical 
evidence. Based on own and other experimental studies Wulf Albers tries 




47 



to outline a more comprehensive model of characteristic function bargain- 
ing by experienced players who, based on previous experiences, do not try 
out anymore deviations with worse terminal results, but can correctly antici- 
pate future consequences. An attempt is made to incorporate the prominence 
structure of the number system, the stronger loyalty of players with similar 
roles as well as the attribution of aspiration levels (attributed demands) from 
previously rejected proposals. 

Common-pool resources (CPRs) are defined by Elinor Ostrom, Roy 
J. Gardner, and James Walker as natural, e.g. fisheries, or man-made, 
e.g. irrigation systems, resources in which exclusion is difficult and yield is 
subtract able. Because of the latter property common-pool resources are en- 
dangered by congestion and even destruction. The continuing and impressive 
research program is based on formal models of various institutional arrange- 
ments including rules how to assign users’ rights and monitor users as well as 
on field and experimental observations of how these arrangements affect the 
sustainability of the common-pool resources. Field studies have to be supple- 
mented by laboratory experiments since the latter allow for a better control 
of the relevant variables. The ideal combination of game theoretic modeling 
and related field and laboratory studies sets a high standard for model-based 
empirical research. 

Experimental Studies 

The behavioristic approach to understand strategic interaction should be 
based on principles which boundedly rational decision makers want to use 
and can apply. It is therefore important to provide empirical evidence guiding 
our attempts to develop behavioral theories of strategic interaction. Since 
quite often empirical field studies cannot offer the data needed for specifying 
behavioral theories, one has to rely on experimental studies to test hypotheses 
or — in case of more explorative experimental studies - to suggest hypotheses 
for boundedly rational decision making. 

Although game theory started out by solving two person zero sum games, 
the question if and in which ways boundedly rational players deviate from the 
obvious maxmin-strategies in such games is still open. The inspiring exper- 
iment by Ariel Rubinstein, Amos Tversky, and Dana Heller explores 
two person zero sum games as well as two person coordination and discoor- 
dination games. The experimental setup is always a hider and seeker-design 
employing pictorial and verbal items of which one is always distinctively fo- 
cal. Whereas maxmin-behavior requires that all items will be selected with 
equal probability in strictly competitive games, the observed behavior is sig- 
nificantly biased, e.g. by avoiding the endpoints or by relying on the focal 
item. The results further suggest that seekers tend to avoid endpoints more 
than hiders who, in turn, avoid focal items more often. Concerning the non- 
zero sum games players relied on focal items to coordinate whereas focal 




48 



items were nearly as likely as other items in discoordination games where 
both players are interested in choosing different items. 

Two standard paradigms, demonstrating doubts about the predictive 
power of game theoretic solution concepts like (subgame) perfect equilibria, 
are the finitely repeated prisoners’ dilemma game and the chain store para- 
dox as originally introduced by Reinhard Selten (1978). Whereas the role 
of players is more or less the same in repeated prisoners’ dilemma games, 
there is a crucial asymmetry in the chain store paradox since the chain store 
wants to deter entry of its many potential competitors. James A. Sundali 
and Amnon Rap op or t report on experiments testing whether the intuitive 
deterrence idea (the chain store fights in case of early entries to discourage 
further entry) or the game theoretic backward induction solution is more in 
line with the experimental observations. Although an increase in the num- 
ber of entrants from 10 to 15 increases deterrence, even this higher level of 
deterrence could not prevent most entrants from entering. It seems that the 
intuition underlying deterrence requires a more substantial number of poten- 
tial entrants. The authors conclude “... Selten’s intuition in choosing m = 20 
(as the number of potential entrants) was right ...” 

Intergroup conflicts are special team games where at least two groups 
compete and where the group outcomes are public goods which are non- 
excludable concerning the group members. Gary Bornstein, Eyal Win- 
ter, and Harel Goren consider a situation where each player can either 
contribute or not and where the game within a team and the game between 
teams are prisoners’ dilemma games. In their experiment two groups of three 
players either played an intergroup prisoners’ dilemma or each of the two 
groups played an independent prisoners’ dilemma game resembling the pos- 
sible intergroup games of the intergroup conflict. As would have been pre- 
dicted by psychological concepts like group or corporate identy intergroup 
competition increases cooperation. This effect, however, becomes weaker in 
later rounds when the games are played repeatedly. 

Consider an experimental situation with the player roles described by the 
rules of the game. If two different individuals have to decide in the same role 
of such a game, their behavior will usually not only depend on the rules, 
but also on their individual characteristics. Axel Ostmann and Ulrike 
Leopold-Wildburger focus attention on bargaining styles which experi- 
mental participants employ in face to face-bargaining and which can be de- 
tected by recordable data. The basic idea is that bargainers quickly develop 
partner images to which - of course, also to the rules of the game - they 
adopt their communicative style. Relying on the 3-dimensional SYMLOG- 
space (dominance, friendliness, task orientation) the authors try to assess 
bargaining types (insistent, leadership, cooperation, low profile) based on 
videotapes and protocols. It can then be assessed which bargaining styles are 
more successful. 




49 



Based on the empirical evidence from the “Iowa Electronic Markets” 
Joyce Berg, Robert Forsythe, and Thomas Rietz discuss what makes 
markets predict well. In addition to field data the observations of the experi- 
mental markets offer another and for theoretical purposes more adequate way 
to analyse how markets reveal information. More specifically, the large-scale 
experimental markets provide data about traders’ individual characteristics 
which is not available in field studies. The authors rely on the data from 
the US political markets to identify factors determining when these markets 
accurately predict election results. While the experimental markets predict 
election results quite well, the prediction error depends on certain factors like 
the number of major candidates and the pre-election market volumes. 

Roy J. Gardner and Jurgen von Hagen define the budget process, 
e.g. of national governments, as the system of rules which govern the process 
like, for instance, the sequence of budget decisions for which two extreme pos- 
sibilities exist: Bottom up-budgeting starts by collecting requests which are 
individually decided by voting, top down-budgeting starts with voting on the 
total budget which then is split up. The hypothesis the authors want to test 
experimentally claims that top down-budgeting leads to greater fiscal disci- 
pline. The interesting experimental design distinguishes four treatments, i.e. 
four distributions of ideal points of the five voters, and compares the predic- 
tive success of more or less normative theories (subgame perfect equilibrium 
as a point and as an area prediction and efficiency) . 

Charles R. Plott and Theodore L. Turocy speak of intertemporal 
speculation if one buys commodities now to resell them later. In their ex- 
periment future demand was unknown to non-speculating agents and spec- 
ulators who could place unlimited orders without knowing the number of 
competitors nor their orders. Participants played repeatedly a two-period- 
game which allowed speculators to carry over inventory from the first to the 
second period. Markets were organized as computerized multiple unit double 
auctions and contained always 4 potential buyers, 4 sellers and 4 speculators 
where the latter could only gain by speculation in one treatment. In the other 
treatment speculators could act also as normal agents, i.e. buyers or sellers. 
Although there were only four experimental sessions, the rational expecta- 
tions hypothesis seems to be supported what, once again, demonstrates the 
surprising information efficiency of the double auction procedure. We agree 
with the authors that one needs to explore more thoroughly the mechanism 
of information transfer in double auction procedures. 

One of the so-called anomalies is the endowment effect according to which 
the same commodities are higher evaluated if they are part of one’s endow- 
ment as compared to the situation where one still wants to acquire them. 
The typical empirical evidence is that one would sell one’s own commodity 
only at a higher price than the price one is willing to accept when acquir- 
ing it. Whereas most of the previous experimental studies have used riskless 
commodities Graham Loomes and Martin Weber rely on risky assets. 




50 



They observed the buying prices for state contingent claims by auctioning 
such claims by a value-revealing auction mechanism. The way to detect an 
endowment effect in such a setup is to auction off various claims and to 
analyse the vector of bids. If, for instance, two state contingent claims taken 
together constitute a sure payoff or dividend, an endowment effect could be 
detected when the two bids for the two claims add up to less than that 
amount. In addition to endowment effects Loomes and Weber also test for 
framing effects (the same endowment is once framed as a small amount of 
money plus a state dependent claim and once as a larger amount of money 
minus a complimentary state dependent claim). 

Altogether these contributions provide a broad picture of how one can model 
and analyse strategic interaction. It hopefully provides guidance and inspira- 
tion for scholars who want to follow the traces of Reinhard Selten. Further- 
more, it should be clear by now that such scholars could originate from many 
scientific disciplines. 

Of course, it seems difficult to equal Reinhard Selten’s capabilities: his 
philosophical insights, mathematical abilities, profound knowledge of the lit- 
erature and last, but not least his enormous creativity in modelling and 
analysing strategic interaction. But he also offers us many ways to join him 
on his way of improving our understanding of strategic interaction: It can be 
done by developing new concepts, by applying such concepts rigorously, by 
empirical research, especially experimentation, and by formulating behavioral 
theories based on empirical observations. 

Join the crew if you have not yet done so! 



References 

Robert J. Aumann, Sergin Hart, and Motty Perry (1996): The Absent-Minded 
Driver, Discussion Paper No. 94, Center for Rationality and Interactive Deci- 
sion Making, The Hebrew University of Jerusalem 
John C. Harsanyi and Reinhard Selten (1988): A General Theory of Equilibrium 
Selection in Games^ MIT Press, Cambridge, Massachusetts 
John F. Nash (1951): Non-Cooperative Games, Annals of Mathematics^ Vol. 54, 
286-295 

Heinz Sauermann and Reinhard Selten (1962): Anspruchsanpassungstheorie der 
Unternehmung, Zeitschrift fur die gesamte Staatswissenschaft, 118, 577-597 
Reinhard Selten (1973): A Simple Model of Imperfect Competition where 4 are Few 
and 6 are Many, International Journal of Game Theory, Vol. 2, Issue 3, 141-201 
Reinhard Selten (1978): The Chain Store Paradox, Theory and Decision, 9, 127-159 
Reinhard Selten (1982): Einfiihrung in die Theorie der Spiele mit unvollstandiger 
Information, Schriften des Vereins fur Socialpolitik, Vol. 126, Information in der 
Wirtschaft, 81-147 

Reinhard Selten and Jonathan Pool (1995): Enkonduko en la Teorion de Lingvaj 
Ludoj - Cu mi lernu Esperanton (in Esperanto and German with extended 
abstracts in English, French, Italian, and Polish) Akademia Libroservo, Berlin 
& Paderborn 




A Note On Imperfect Recall* 



Ken Binmore 

Economics Department, University College London Gower Street, London WCIE 
6BT, England 



Abstract. The Paradox of the Absent-Minded Driver is used in the liter- 
ature to draw attention to the inadequacy of Savage’s theory of subjective 
probability when its underlying epistemological assumptions fail to be satis- 
fied. This note suggests that the paradox is less telling when the uncertainties 
involved admit an objective interpretation as frequencies. 



Ce que j’ai appris, je ne le sais plus. Le 
peu que je sais encore, je I’ai devine. 

Chamfort, Maximes et Pensees, 1795 



1. Introduction 

Von Neumann and Morgenstern [17] proposed modeling bridge as a two- 
person, zero-sum game in which each partnership is one of the two players. 
Modeled in this way, bridge becomes a game of imperfect recall, because 
the players forget things they knew in the past. For example, when bidding 
as North, the player representing the North-South partnership must forget 
the hand he held when bidding as South. A bank choosing a decentralized 
lending policy for its hundred branches confronts a similar problem. The boss 
can imagine himself behind each manager’s desk as he interviews clients, 
but then he must forget the earlier lending decisions he made while sitting 
behind other managers’ desks. The decision problem he faces is therefore one 
of imperfect recall. 

Since Kuhn [9] pointed out that mixed and behavioral strategies are in- 
terchangeable only in games of perfect recall, little attention has been paid 
to the difficulties that can arise when recall is imperfect.^ The orthodox ap- 
proach has been to regard a person with imperfect recall as a team of agents 
who have identical preferences but different information. In the style of Selten 
[15], each agent is then treated as a distinct player in the game used to model 
the problem. For example, the orthodox approach models bridge as a four- 
player game with two teams and the banking problem as a hundred-player 
game with one team. 

Support from the Economic and Social Research Council under their “Beliefs 
and Behaviour” Programme L 122 251 024 is gratefully acknowledged. 

^ Notable exceptions are Isbell [7] and Alpern [1]. 




52 



As Gilboa [6] confirms, the consensus in favor of the team approach is 
very strong. Nevertheless, a recent paper of Piccione and Rubinstein [12] has 
revived interest in the problem of decision-making with imperfect recall. Their 
emphasis in this paper is on the straightforward psychological fact that most 
people know that they will forget things like telephone numbers from time 
to time. The orthodox approach dismisses such folk as irrational and thereby 
escapes the need to offer them advice on how to cope with their predicament. 
But I agree with Piccione and Rubinstein that people who know themselves 
to be absent-minded can still aspire to behave rationally in spite of their 
affliction. However, my own interest in the imperfect recall problem is mostly 
fuelled by the difficulties with the team approach outlined in Binmore [2]. 

Players stay on the equilibrium path of a game because of their beliefs 
about what would happen if they were to deviate. But how do players know 
what would happen after a deviation? The orthodox approach treats a player 
as though he were a team of agents, one for each information set at which he 
might have to decide what action to take. Such a viewpoint de-emphasizes 
the inferences that a player’s opponents are likely to make about his thinking 
processes after he deviates. One agent in a team may have made a mistake, 
but why should that lead us to think that other agents in the same team 
are liable to make similar mistakes? Traditionalists see no reason at all, and 
hence their allegience to backward induction and similar solution concepts. 
However, a theory that treats a player as a team of independently acting 
agents is unlikely to have any realistic application, because we all know that 
real people are liable to repeat their mistakes. If I deviate from the equilibrium 
path, it would therefore be stupid for my opponents not to make proper 
allowance for the possibility that I might deviate similarly in the future.^ 
The team approach is therefore not without its difficulties even in games 
of perfect recall. It therefore seems an unlikely panacea for imperfect recall 
problems. 

Piccione and Rubinstein [12] do not claim to provide a theory of rational 
decision-making under imperfect recall. They seek only to comment on some 
of the issues that would need to be addressed in formulating such a theory. 
I am even less ambitious in that I shall simply be commenting on some of 
their comments. The problem of time-inconsistency raised by the one-player 
game that they aptly describe as the Paradox of the Absent-Minded Driver 
is particularly interesting. 



^ Binmore [2] argues that one needs an explicit algorithmic model of the reasoning 
processes of a player in order to take account of such considerations. Finite 
automata have been used for this purpose in a number of papers. However, 
as Rubinstein [13] notes, one cannot model players as finite automata without 
introducing imperfect recall problems. 




53 



2. Paradox of the Absent-Minded Driver 

The general issue of imperfect recall in games is discussed in my Fun and 
Games (Binmore [3]). Such textbooks explain how to interpret representa- 
tions of imperfect recall problems like that shown in Figure 2. 1(a). ^ One may 
imagine an absent-minded driver who must take the second turning on the 
right if he is to get home safely for a payoff of 1. If he misses his way, he will 
find himself in an unsafe neighborhood and receive a payoff of 0. 

The driver’s difficulty lies in the fact that he is absent-minded. At each 
exit, he forgets altogether what has happened previously in his journey. Since 
both exits look entirely the same, he is therefore unable to distinguish between 
them."^ 



0 

k 



d,/l 



-R- 



; R 






(a) 




Fig. 2.1. The absent-minded driver’s problem 



^ Von Neumann and Morgenstern [17] excluded cases like this by requiring that 
no play of a game should pass through an information set more than once. 
However, it has now become customary to accept such cases as particularly 
challenging examples of games of imperfect recall. 

One can, of course, invent ways in which he could supplement his memory. 
For example, he might turn on the radio on reaching an exit. He could then 
distinguish between the exits by noting whether his radio is on or off. But the 
introduction of such expedients is not allowed. 




54 



The driver has two pure strategies for this one-player game of imperfect 
recall, R and S. The use of either results in a payoff of 0. It follows that the 
same is true of any mixed strategy. However, in such games of imperfect recall, 
one can achieve more by using a behavioral strategy. A behavioral strategy 
requires the driver to mix between R and S each time he finds himself called 
upon to make a decision.^ Let b(p) be the behavioral strategy in which R is 
chosen with probability 1 — p and S is chosen with probability p. A driver 
who uses 6(p) obtains an expected payoff of p(l — p), which is maximized 
when p = i. According to this analysis, his optimal behavioral strategy is 
therefore 6(|), which results in his receiving an expected payoff of 

The paradox proposed by Piccione and Rubinstein [12] hinges on the time 
at which the driver chooses his strategy. The argument given above takes for 
granted that the driver chooses h{\) before reaching node d \ , and that he can 
commit himself not to revise this strategy at a later date. But such an atti- 
tude to commitment is not consistent with contemporary thinking in game 
theory. In particular, Selten’s [15] notion of a perfect equilibrium, together 
with all its successors in the literature on refinements of Nash equilibrium, 
assumes that players will always be re-assessing their strategy throughout the 
game.® More precisely, the orthodox view is not that players cannot make 
commitments, but, if they can, their commitment opportunities should be 
modeled as formal moves in the game they are playing. However, once their 
commitment opportunities have been incorporated into the rules of the game, 
then the resulting game should be analyzed without attributing further com- 
mitment powers to the players. 

So what happens in the absent-minded driver’s paradox if the driver is 
not cissumed to be committed to 5(|)? Following Piccione and Rubinstein 
[12], let us assume that he reaches the information set I and remembers that 
he previously made a plan to choose 6(|) on reaching I. He then asks himself 
whether he wants to endorse this strategy now that he knows he has reached 
the information set I and hence may either be at d\ or d^. If he attaches 
probability 1 — ^ to the event of being at d\ and probability q to the event 
of being at ^ 2 , then choosing b{p) at I results in a payoff 

7T = (l-9)p(l-p)+9(l-p)- (2-1) 

This payoff is maximized when p = (1 — 2q) /2(1 — g). The driver will therefore 
only choose p = | at / if he believes that g = 0. That is to say, in order that 
a time-inconsistency problem not arise, it is necessary that the driver deny 
the possibility of ever reaching the second exit. But to deny the possibility of 
reaching the second exit is to deny the possibility that he can ever get home! 

® By contrast, a mixed strategy requires a player to randomize over his pure 
strategies once and for all before the game is played. 

® McClennen [10] is one of a number of philosophers who insist that rationality 
includes the facility to commit oneself to perform actions in the future under 
certain contingencies that one’s future self would regard as suboptimal. 




55 



3. Whence g? 

To make progress with the Paradox of the Absent-Minded Driver, it is nec- 
essary to atsk how the driver came to believe that the probability of being at 
the second exit (I 2 is q. This question forces us in turn to face a philosophical 
question about the nature of probability. In the terminology of Binmore [5, 
p.265], is the driver’s probability theory logistic, subjective or objective? A 
logistic theory treats a probability as the rational degree of belief in an event 
justified by the evidence. The subjective theory of Savage [14] is the basis for 
the familiar Bayesian orthodoxy of economics.^ An objective theory regards 
the probability of an event as its long-run frequency. 

The most satisfactory interpretation for q in the Paradox of the Absent- 
Minded Driver would be logistic, in the style attempted by Keynes [8]. How- 
ever, I think it uncontroversial that no theory of this type has yet come near 
being adequate. My guess is that most economists would take for granted that 
q is to be interpreted as a subjective probability a la Savage [14]. But there 
are major difficulties in such an interpretation. In the first place, Bayesian 
epistomology — as described in Chapter 10 of Binmore [3] — fails in the absent- 
minded driver’s problem.® Secondly, if the postulates of Savage’s theory are 
to make sense, it is vital that the action space A, the space B of states of 
the world, and the space C of consequences have no relevant linkages other 
than those incorporated explicity in the function / : AxB C that deter- 
mines how actions and states together determine consequences (Binmore [5, 
p.310]). But it is of the essence in the Paradox of the Absent-Minded Driver 
that states are not determined independently of actions. A rational driver’s 
beliefs about whether he is at the first exit or the second must surely take 
account of his current thinking about the probability at which he would turn 
right if he were to reach an exit. 

Personally, I think that the most important role for paradoxes like that 
of the absent-minded driver is to focus attention on the inadequacies of our 
current logistic and subjective theories of probability. However, I have nothing 
particularly original to propose on either front, and so follow Piccione and 



^ Notice that I do not identify Bayesianism with Savage’s theory. I distinguish 
those who subscribe to Savage’s view from followers of Bayesianism by calling 
the former Bayesians and the latter Bayesianismists (Binmore [4, 5]). Bayesian- 
ismists argue that rationality somehow endows individuals with a prior prob- 
ability distribution, to which new evidence is assimilated simply by updating 
the prior according to Bayes’ Rule. Such an attitude reinterprets the subjective 
probabilities of Savage’s theory as logistic. This may sometimes be reasonable 
in a small- world context, but Savage [14] condemns such a procedure as “ridicu- 
lous” or “preposterous” in a large- world context. 

® The epistomology taken for granted by Bayesian decision theory is simply that 
a person’s knowledge can be specified by an information partition that becomes 
more refined as new data becomes available. Binmore [3, p.457] discusses the 
absent-minded driver’s problem explicitly in this connexion. 




56 



Rubinstein [12] in this paper by turning to the interpretation of gr as an 
objective probability. 

If q is to be interpreted objectively, one must imagine that the driver 
faces the same problem every night on his way home from work. After long 
enough, it is then reasonable to regard the ratio of the number of times he 
arrives at the second exit to the total number of times he arrives at either 
exit as a good approximation to the probability q. Of course, this frequency 
will be determined by how the driver behaves when he reaches an exit. If the 
driver always continues straight on with probability P at an exit, then the 
number of times he reaches d 2 will be a fraction P of the number of times he 
reaches d \ . It follows that 

q = P/{l-\-P) and 1 - g = 1/(1 + P) . (3.1) 

One school of thought advocates writing the values for q and 1 — g from 
(3.1) into (2.1) and then setting p — P to obtain tt = 2p(l — p)/(l +p). The 
result is then maximized to yield the optimal value p = \[2— 1. However, such 
a derivation neglects the requirement that a decision-maker should maximize 
expected utility given his beliefs. In what follows, I therefore always treat 
a player’s beliefs as fixed when optimizing, leaving only his actions to be 
determined. However, if (2.1) is maximized with q — P/(l-fP) held constant, 
then the maximizing p satisfies p = (1 — P)/(l-hP). A time-inconsistency 
problem then arises unless p = P. Imposing this requirement leads to the 
equation + 2p — 1 = 0, whose positive solution is p = V2 — 1 as before. In 
the next section, I plan to defend this result as the resolution of the Paradox 
of the Absent-Minded Driver in the case when it is possible to interpret q as 
a frequency. 



4. Repeated Absent-Mindedness 

One of the things that game theory has to teach is that difficulties in an- 
alyzing a problem can sometimes be overcome by incorporating all of the 
opportunities available to the decision-maker into the formal structure of his 
decision problem. The need to proceed in this manner has been recognized for 
a long time in the case of precommitment, and I think it uncontroversial to 
assert that the orthodox view among game theorists is now that each oppor- 
tunity a player may have to make a precommitment should be built into the 
moves of a larger game, which is then analyzed without further commitment 
powers being attributed to the players. 

If the absent-minded driver’s decision problem is prefixed with such a 
move — at which the driver commits himself to choosing a probability p with 
which to continue straight ahead whenever he reaches an exit— then we have 
seen that the problem reduces to choosing the largest value of p(l — p). 
However, if the problem is presented without a formal commitment move, as 




57 



in Figure 2.1(a), then the convention in game theory is to seek an analysis 
that does not attribute commitment powers to the driver. 

It seems to me that the same should go for Piccione and Rubinstein’s [12] 
assumption that the driver is able to remember a decision made in the past 
about which action he planned to take on encountering an exit. If there are 
pieces of information that are relevant to the decisions that might be made 
during the play of a game, then these should be formally modeled as part 
of the rules of the game. Otherwise, my understanding of the conventions of 
game theory is that an analysis of the game should proceed on the assumption 
that the unmodeled information is not available to the players. In particular, 
we should analyze the absent-minded driver’s decision problem as formulated 
in Figure 2.1(a) without assuming that the driver remembers anything at 
all that is relevant to his decision on arriving at /. One could, of course, 
introduce a new opening move at which the driver makes a provisional choice 
of behavioral strategy for the problem that follows. However, personally I 
think that this issue is something of a red herring. As we all know when we 
dip our toes in the sea on a cold morning, the plans we made earlier when 
getting changed are not the plans that determine what we actually do. What 
we do now is determined by the decision we make now. 

If I understand correctly, Piccione and Rubinstein [12] want the driver to 
remember his original provisional plan so that they have grounds for attribut- 
ing beliefs to him about whether he is at di or But why should his original 
plan determine his beliefs if he has no reason to believe that he will carry 
out his original plan once he reaches /? In my view, this and other issues 
in the case when q is to be interpreted as a long-run frequency are clarified 
by explicitly modeling the situation as a repeated decision problem, rather 
than leaving the repetitions to be implicitly understood. One is then led to 
present the problem as shown in Figure 2.1(b), where it is to be understood 
that time recedes into the infinite past as well as into the infinite future. In 
order to avoid the type of time-inconsistency problems pointed out by Strotz 
[16], it will be assumed that the driver discounts time according to a fixed 
discount factor S (0 < 5 < 1). 

The label attached to an edge representing an action in Figure 2.1(b) will 
also denote the time the driver spends between the nodes joined by the edge. 
Nothing very much hinges on this point, but I think it helpful to imagine that 
no time at all is spent at a node, but that the driver does all his thinking while 
driving between nodes. The information set I in Figure 2.1(b) is therefore 
not quite the same as its cousin in Figure 2.1(a), since it includes not only 
the nodes d\ and ^ 2 ? but also the open edge that joins do and di and the 
open edge that joins d\ and ^ 2 - 

I envisage the driver moving through the tree reviewing his plan of cam- 
paign as he goes. As noted above, whether he actually remembers his previous 
plan or not seems irrelevant to the question since he will be reiterating all 
of the considerations each time he reviews his situation. When he reaches an 




58 



exit, he implements whatever plan he currently has in mind. To keep things 
simple, I assume that there is a probability of 1 that he will review his plan 
at least once on each edge of the tree. Notice that, as the problem is set up 
in Figure 2.1(b), the driver always remembers what happened when he faced 
the same problem in the past. Only within I does he forget something, and 
then the only relevant matter of which he is unaware is his location in I. (He 
is, of course, not allowed to carry a clock or to employ any other device that 
would help him to reduce his uncertainty while in I.) 

Within this formulation, two simple points seem apparent to me. While 
suffering dreadfully with a hangover, I can recall swearing never to drink 
unwisely again. I can also recall making similar resolutions repeatedly in the 
past. But plans for the future made under such circumstances remain relevant 
only while the memory of the hangover is sharp. Once the memory has faded, 
the joys of convivial company again outweigh the anticipated suffering and 
one drinks too much again.® In brief, if we want to know what someone who 
is not a regular tippler will do when tempted to overindulge, we need to know 
how he will view the matter when he is tempted — not immediately after he 
has overindulged in the past. Similarly, in the problem of the absent-minded 
driver as modeled in Figure 2.1(b), it seems to me that only the plan the 
driver makes while in the set I is relevant. 

The second point concerns his beliefs while in the set I. The driver always 
remembers all his past history up to his most recent entry into the set I. He 
can therefore compute the frequency P with which he continued straight 
ahead at exits in the past. This information does not tell him for certain 
what probabilities he should be using to estimate his location in / now, but 
it is the strongest possible evidence he could possibly have on this subject.^® 
Let V be the expected payoff to the driver, given that he is at node di 
and will always play h{p) at an exit now and in the future. Let w be the the 
expected payoff when he is to use h{p) but is now at node c ?2 • Then 

u = (1 — -^pwS^^ , 

w = (1 — p)d^ + (1 “ -h pvS ^^ , 

® Alcoholics Anonymous recommend permanent attendance at group sessions. Is 
this to keep memories sharp by renewing them vicariously through the experi- 
ence of others? 

One might argue that we should model the driver as a computing program. 
Nothing would then prevent this program from using itself as a subprogram 
and hence simulating its own thinking processes. Could it not therefore dispense 
with external information and simply use an introspective analysis to determine 
what it must have done in the past? Binmore [2] points out that difficulties 
arise in such cases because of the Halting Problem for Turing Machines. In 
simple terms, a program that decides what to do on the basis of a prediction of 
what it is about to do will get into an infinite loop. However, the driver in the 
formulation given in the text has no need to face the problems that modeling 
such introspection creates, because he is provided painlessly with the data that 
the attempt at introspection is intended to generate. 




59 



from which it is easy to calculate v and w. One can experiment with various 
relative values of the time intervals of the model. In some fairly natural cases, 
it turns out that the driver’s informational state is irrelevant, since he would 
make the same decision at d\ and d 2 even if he knew his location. The issue 
is also trivialized by considering the case when the time intervals are fixed 
and S 1. However, I plan to consider the limiting case when 5 is fixed 
but n = r 2 = S 2 = T and T oo. This removes the infiuence of the 
future repetitions of the problem on the driver’s current behavior while still 
allowing him access to his decisions in the same situations in the past. Under 
this simplifying hypothesis, 

V = p(l — , 

w = (1 — p)S^ . 

If the driver knew he were between do and di , he would want to maximize 
V and so would choose p = If he knew he were between di and d 2 , he would 
want to maximize w and so would choose p = 0. But once he is within the set 
I he does not know his location. But he can easily compute the conditional 
probability q that the latter case applies to be = Ps\I{sq + Psi). I take 
5o = si = 5, so that this formula reduces to q = P/{P 1) as in the 
analysis of Piccione and Rubinstein [12] discussed in the preceding section. 
The optimizing calculation for p then proceeds precisely as in their analysis, 
so that the maximizing p is (1 — P)/(l + P). 

I see no more paradox in the fact that this value of p differs from that 
the driver would choose outside I than the fact that it differs from the p 
he would choose if he knew he were approaching d\ or d 2 . In both cases, he 
knows less at the time of decision than in the circumstances with which his 
actual decision is to be compared. In particular, within the set /, he does not 
know whether he is involved in a decision problem Di that starts at di or 
a decision problem D 2 that starts at ^ 2 . On the other hand, if any plan he 
made outside I stood a chance of still being in place when di is reached, then 
he would be choosing in the knowledge that the problem to be solved is Di . 

It remains to argue that p = P. In accordance with my doubts about back- 
ward induction (Binmore [2]), I do not believe that it is possible to tackle this 
question adequately without considering out-of-equilibrium behavior. How- 
ever, in the absence of an algorithmic model of the driver’s thinking processes, 
it is only possible to offer a sketch of how a full argument would go. 

Begin by considering the possibility that p ^ P. The driver now has a 
problem because the behavior that his calculation recommends for the infinite 
future is not consistent with his summary of his behavior in the infinite past. 
He therefore needs some theory of “mistakes” to explain why he did the wrong 
thing in the past or why he may not actually do what he has calculated to be 
optimal in the future.^^ When p ^ P, the driver will presumably accept that 

I hope that it is uncontroversial to suggest that Sel ten’s theory of the “trembling 

hand” is too simple a story to be appropriate in the current context. 




60 



either P is flawed as a prediction of p or P/(l H- P) is flawed as a prediction 
of q. He will then need to employ more complicated functions of his history 
to generate the estimates he needs to make a sensible decision. If these 
functions move the driver’s estimates towards consistent values over time, 
then we have an equilibrium story to tell. In particular, on the equilibrium 
path we will And that p = P, so that p = \/2 — 1. 



5. A Team Analysis 

My doubts about modeling players as teams of agents with identical prefer- 
ences have already been mentioned. But it cannot be denied that the method- 
ology has its advantages in games of perfect recall in which no player ever 
moves more than once. In the presence of the latter proviso, it is irrelevant 
what would be inferred about a player’s future play from a counterfactual 
deviation on his part from the equilibrium path. At the same time, the “single 
improvement property” for games of perfect recall ensures that there is no 
loss of efficiency in allowing agents to make decisions one-by-one (Piccione 
and Rubinstein [12]). 

Gilboa [6] offers the orthodox case for modeling the the absent-minded 
driver as a team problem. The driver is treated as though he had a multiple 
personality, with two personalities or agents to be called Alice and Bob. They 
have the same preferences but different information. When one of the two 
agents is called upon to make a decision, the agent does not know whether 
he is at the first exit or the second, but he does know that the other agent will 
be making an independent decision at the other exit (should it be reached). 
Figure 5.1(a) shows a representation of the game of perfect recall that must 
then be played between Alice and Bob. Its opening move is a chance move 
that assigns control of the driver at the first exit to Alice or Bob with equal 
probability.^^ 



For example, on his nth entry into the information set I he might perhaps 
arbitrarily estimate the probability that he has yet to reach node d\ as the 
discounted sum 

Qn = (1 ~ ^) ^{Qti-1 + ^qn-2 + Qn-Z + ' ' ') j 



where, for each k < n, qk = Pife/(1 +Pfc) and pk is the probability with which 
he actually went straight ahead at intersections during his A;th entry into I. 
On the assumption that Qn is correct, he can then choose pn optimally to 
be (1 — 2Qn)/(l + 2Qn)‘ Then qn is defined in terms of its predecessors by 
= Pn/(1 4- Pn)- The question is then whether the sequence (qn) converges. 
Although nothing in the specification of the Paradox of the Absent-Minded 
Driver would seem to provide a strong justification for setting the probability r 
with which the opening chance move assigns control to Alice equal to 1/2. But 
if r ^ 1/2, we shall not be led to the satisfying conclusion that Alice and Bob 
should each choose R oi S with probability 1/2. 




61 



Chance 

move 





{b} 



Fig. 5.1. Splitting the driver’s personality 



Figure 5.1(b) shows the strategic form of the game. Its unique equilibrium 
is easily calculated. Alice and Bob should each independently choose R or 
S with probability 1. The outcome is then the same as when the driver 
commited himself to the behavioral strategy 6(1) before reaching I in the 
analysis of Piccione and Rubinstein [12]. In particular, both Alice and Bob 
receive an expected payoff of 1. 

In spite of the welcome conclusion to which this analysis leads, I believe it 
to be wrong, because it neglects the fact that what a person does in certain 
circumstances must surely be evidence about what he will do if placed in 
exactly same circumstances later on. It is certainly true that arguments that 
take account of such reasoning can easily go astray. Somehow a line must be 
drawn between valid uses of the principle and invalid uses that lead authors 
like Nozick [11] to twinning arguments which supposedly demonstrate that 
cooperation is rational in the Prisoners’ Dilemma (Binmore [5, p.205]). As in 
much else, my own view is that the right way or ways to proceed will remain 
mysterious until we have satisfactory algorithmic models of the players we 
study (Binmore [2]). 



6. Conclusion 

This paper has done no more than comment on the comments made by 
Piccione and Rubinstein [12] on their Paradox of the Absent-Minded Driver. 
It claims that the paradox is illusory when the driver has frequency data 



14 



This is not an accident. Whatever payoffs are assigned to the possible outcomes 
in the game, the two analysis yield the same result. 





62 



to support his beliefs. But, as Gilboa [6] aptly observes, what counts as 
a paradox depends on the viewpoint of the observer. As for the paradox 
in the general case, I have nothing useful to say beyond the observation 
that it highlights the need for an adequate theory of decision-making under 
uncertainty to supplement the current Bayesian orthodoxy. 



References 



1. S. Alpern. Games with repeated decisions. SIAM Journal of Control and 
Optimization^ pages 468-477, 1988. 

2. K. Binmore. Modeling rational players, I and II. Economics and Philosophy^ 3 
and 4:179-214 and 9-55, 1987. 

3. K. Binmore. Fun and Games. D. C. Heath, Lexington, Mass., 1991. 

4. K. Binmore. Debayesing game theory. In B. Skyrms, editor. Studies in Logic 
and the Foundations of Game Theory: Proceedings of the Ninth International 
Congress of Logic, Methodology and the Philosophy of Science. Kluwer, Dor- 
drecht, 1992. 

5. K. Binmore. Playing Fair: Game Theory and the Social Contract I. MIT Press, 
Cambridge, Mass., 1994. 

6. I. Gilboa. Technical report. 

7. J. Isbell. Finitaxy games. In Contributions to the Theory of Games III. Prince- 
ton University Press, Princeton, 1957. 

8. J. M. Keynes. A Treatise on Probability. Macmillan, London, 1921. 

9. H. Kuhn. Extensive games and the problem of information. In H. Kuhn and 
A. Tucker, editors. Contributions to the Theory of Games II. Princeton Univer- 
sity Press, Princeton, 1953. 

10. E. McClennen. Rationality and Dynamic Choice. Cambridge University Press, 
Cambridge, 1990. 

11. R. Nozick. Newcomb’s problem and two principles of choice. In N. Rescher, 
editor. Essays in Honor of Carl G. Hempel. Reidel, Dordrecht, Netherlands, 
1969. 

12. M. Piccione and A. Rubinstein. On the interpretation of decision problems with 
imperfect recall. Working Paper 9-94, Sackler Institute, Tel Aviv University, 
1994. 

13. A. Rubinstein. Finite automata play the repeated prisoners’ dilemma. Journal 
of Economic Theory, 39:83-96, 1986. 

14. L. Savage. The Foundations of Statistics. Wiley, New York, 1951. 

15. R. Selten. Reexamination of the perfectness concept for equilibrium points in 
extensive-games. International Journal of Game Theory, 4:25-55, 1975. 

16. R. Strotz. Myopia and inconsistency in dynamic utility maximization. Review 
of Economic Studies, 23:166-180, 1973. 

17. J. Von Neumann and O. Morgenstern. The Theory of Games and Economic 
Behavior. Princeton University Press, Princeton, 1944. 




Futures Market Contracting 

When You Don’t Know Who the Optimists Are 



Ronald M. Harstad^ and Louis Phlips^ 

* Rutcor Eind FOM, Rutgers University, New Brunswick, NJ 08903-5062, USA 
^ Department of Economics, European University Institute, 1-50016 San Domenico di 
Fiesole, Italy 

Abstract. We have built models of speculative futures trading based upon 
inconsistent priors, analyzing games of inconsistent incomplete information. 
These models have assumed that the inconsistent priors are themselves common 
knowledge. In this paper, we explore the game-theoretic implications of treating 
doubly inconsistent incomplete information, in that inconsistent priors are private 
information, and traders attach inconsistent assessments to the probability that a 
trader will be an optimist. 

The result is not arbitrary: the logic of a separating equilibrium can be specified 
via backwards induction. It is unlikely that subgame-perfect equilibria will 
exhibit pooling. The volume of speculative trading is reduced by informational 
constraints, but a sense is specified in which no ex ante agreed-upon Pareto 
improvements over separating equilibrium behavior can satisfy the information 
constraints. 

1. Introduction 

Reinhard Selten's most widely known publications ([1965], [1975], [1978], and 
the [1988] book with Harsanyi) naturally did not give a complete picture of his 
research interests. Like nearly all game-theoretic models with some dynamic 
structure, our analysis uses the concept of subgame-perfect equilibrium ([1965]). 
However, we rely more heavily on a less standard concept, games of inconsistent 
incomplete information, first formalized in Selten [1982]. Also, our paper points 
to a less-widely known feature of Selten's scholarly career: his continuing 

emphasis on the relevance of game-theoretic techniques to the modeling of 
applied problems (indeed, the original 1965 paper focuses on oligopoly models 
with repeat purchase loyalty). Our application concerns futures markets for basic 
commodities. 

The sort of futures markets we seek to model exist for many commodities, 
including metals and crude oil. The market in mind when selecting model 



* Reinhard Selten first suggested the topic of this paper, while the authors were visiting the 
Center for Interdisciplinaiy Research, Bielefeld, Germany. We are grateful for the Center's 
support, and for stimulating suggestions from Joel Sobel and Eric van Damme. 




64 



simplifications is the London International Petroleum Exchange for Brent Blend 
(a standardized mixture of North Sea crude oil, cf. Phlips [1992] for a 
description). It bears emphasis that this market (like many futures markets) does 
not offer opportunities to purchase today a quantity for delivery at some future 
date. It is a purely financial or “paper” market: a buyer of a futures contract pays 
a seller today the agreed-upon price of the contract; in return, the seller makes a 
purely financial payment to the buyer at the specified maturity date, with the 
amount of that payment determined by the spot market price prevailing on the 
maturity date. (There is no option to demand delivery of the commodity at 
maturity, nor any constraint that the total outstanding contracts must be less than 
the deliverable amount of the commodity.) A trader’s “position” or, more 
precisely, net short position, is his total sales less his total purchases of futures 
contracts. In most trades, the buyer is simply gambling that the spot price will 
constitute a good return on the money invested in buying the futures contract, 
while the seller is gambling that it will not. 

The standard literature on futures markets assumes all agents active in these 
markets have common priors. This assumption is needed to construct rational 
expectations models in which the futures price is an unbiased predictor of the spot 
price at maturity. Instead, we use the more realistic assumption that agents have 
different priors: their beliefs are inconsistent, reflecting differences in opinion, 
not in information. For concreteness, each agent is one of two types: either an 
“optimist” or a “pessimist.” The optimist expects the spot price at maturity to be 
high; the pessimist expects it to be low. As these expectations are due to 
differences in opinion, in our model, an optimist would not adjust his estimate of 
the spot price upon learning that another agent is a pessimist. His estimate has 
already taken into account the possibility that other agents might have 
expectations he views as incorrect. 

In addition, we suppose that agents know only their own type and “don't know 
who the optimists are.” They assign subjective probabilities to other agents' types, 
and these probabilities are also inconsistent. There is thus a double inconsistency, 
in that agents have inconsistent beliefs about inconsistent beliefs of their trading 
partners. Our purpose is to show that a game-theoretic analysis of such a 
situation, which is described by Selten [1982] as one of inconsistent incomplete 
information, is possible and provides interesting insights into the working of 
futures markets. 

We have studied futures markets for natural resources in circumstances 
featuring extractors with market power in the cash market (Phlips and Harstad 
[1991], Harstad and Phlips [1990], Phlips and Harstad [1992], listed in the order 
they were written). Those papers assumed inconsistent priors, which enabled 
explanations of some stylized facts about futures markets that are inconsistent 
with mainstream futures models, principally a nonzero volume of rational 
speculative trade. Here we address a key limitation of those papers, exploring the 
considerable complications that arise under the more reasonable assumption that 
the fact of inconsistent priors is common knowledge, but the priors themselves are 




65 



private information. When simplifying assumptions sidestep other issues, we will 
refer to one of the other papers addressing them. 

We basically use the three-player, two-stage noncooperative game studied in 
Phlips and Harstad [1991]. Two risk-averse producers of an exhaustible resource 
simultaneously set market supplies in each of two successive spot markets. Prior 
to the first spot market, they and a risk-neutral speculator can trade futures 
contracts for the resource, contracts which mature as of the second spot market. 

We show that the double inconsistency of beliefs has no impact on the 
producers' spot market activity, except through the futures contracts signed. 
Nonetheless, the speculator can (and must) evaluate the impact of the producers' 
spot market behavior on the expected profitability of his own futures position. 

Subgame-perfect equilibria were found to reach Pareto-efficient outcomes when 
the players' types were common knowledge. Under the assumptions made here, 
the desired equilibrium outcomes are also located on contract curves. However, 
these curves are based on subjective probabilities. Consequently, the contracted 
futures positions may be larger or smaller than the “objectively” efficient ones. In 
a world characterized by uncertainty, however, an allocation mechanism is 
appropriately judged by its efficiency relative to the information available at the 
time the mechanism is put in place. In that sense, the game analyzed here reaches 
an ex ante efficient outcome in subgame-perfect equilibrium. Nonetheless, ex 
post, objective inefficiencies occur. 

We emphasize that the behavior we analyze below is fully rational, and indeed, 
assumes players correctly handle extremely complex calculations. It is simply 
behavior that is not conditioned on adjusting one's own beliefs for the beliefs of 
other traders. 

2. The Order of Moves 

The game is complex despite our efforts to avoid tangential complications. To 
sort out detail, the order of moves is presented here, with the nature of incomplete 
information specified in the next section, and spot market cost and demand 
conditions following in section 4. Hence, specification of the producers' 
payoffs — certainty equivalent levels of sums of profits from futures market and 
spot market transactions — and the speculator's payoff — expected profit from his 
futures dealings — are delayed until equations (3) and (5) in section 4. 

The set of players is {A,B,S}. To begin, a futures market for the natural 
resource is opened at the beginning of period 1. The game unfolds as follows: 

Step 1: Nature determines whether a player is optimistic or pessimistic, and 
privately informs each player of his type. Nature also determines a “commitment 
order,” that is, a permutation of the order {A, B, 5), which is common 
knowledge. 

Step 2: Each player simultaneously announces one futures contract offer, either 
a sale or a purchase offer. Each offer consists of a {price, quantity) pair, and is 
an all-or-nothing offer (neither price or quantity is negotiable). Once announced, 
the offers are irrevocable. 




66 



Step 3.- The player designated first in the commitment order irrevocably accepts 
or rejects each contract offered by another player. ^ 

Step 4: The player designated second in the commitment order irrevocably 

accepts or rejects each contract offered by another player and not yet accepted. 
(A contract accepted by one player is not available for later acceptance by another 
player.) 

Step 5- The player designated third in the commitment order accepts or rejects 
each contract offered by another player and not yet accepted. The futures market 
then closes. 

Step fi: The extraction subgame occurs, that is, producers A and B 

simultaneously determine what fraction of their stock of the resource to supply 
before the maturity date of the futures contracts. Technically, this ends the game. 

Step 7: The market demand level is revealed and determines the spot price at 
maturity, given the extraction decisions in step 6. 

Our analysis throughout will be limited to consideration of pure strategies, 
presumably not a serious restriction when a continuum of actions is available at 
step 2. Several simplifications follow from a traditional assumption: that a player 
indifferent between accepting and rejecting a contract in steps 3-5 accepts. 

3. Inconsistent Incomplete Information 

There are two sources of uncertainty. The first is about the level of aggregate 
demand: each player has an expectation % {i = A, B, S). These expectations are 
inconsistent, since drawn from different distributions. For simplicity, these 
expectations are of two types: either = p^ or p-=p^< p ^ — a player is either 
an optimist or a pessimist. All optimists have the same mean expectation p^ . All 
pessimists have the same mean expectation p^. The values p^ and p^ are 
common knowledge. An economic interpretation of p is delayed until analysis of 
the extraction subgame in Section 4. 

Second, each player knows his own type but does not know the types of the 
other players. They know that Nature determines whether a player is an optimist 
or a pessimist with probabilities A and (1 — A). However, each player has his 
own personal evaluation of A, Aj, i = A,B,S. 

If a “true” A were commonly known, all conditional subjective probabilities that 
any other player is an optimist or a pessimist would be equal to A or 1 — A, 
respectively. To illustrate, consider the probability matrix of type combinations 
given in Table 1. An optimistic S is represented as S^, and a pessimistic S as 
with types of A and B analogously represented. The sum of its 8 elements is 1 . 
This matrix is uncorrelated only if A = 0.5. If correlated, however, any given 
player, say S, with belief A5, still attaches a conditional probability A^ that any 
other given player, say A, is an optimist. 



^ That is, he chooses to accept both contracts, accept either contract, or reject both 
contracts offered. 




67 



Table 1. Probability Matrix of Type Combinations 









A^ 


A^ 








B^ 


B^ 


<S" 




A^(l-A) 


A^(l-A) 


A(l-A)^ 


5^ 


A^(l-A) 


A(l-A)^ 


A(l-A)^ 


(1-A)’ 



Consistent incomplete information is defined by the condition that all players 
know the true A and therefore have the same subjective probabilities. Inconsistent 
incomplete information allows each player to have a different theory about how 
Nature selects types and, therefore, a different A. Then we may have 
A 5 ^ Aa ^ Afc. Notice, however, that the values A 5 , Aa, Xh and the corresponding 
probability matrices are common knowledge in the inconsistent as well as in the 
consistent case. 

4. The Extraction Subgame 

This section concludes the game’s description by specifying conditions of the 
extraction subgame, step 6 , and then begins analysis with that step. The two 
producers have exogenous initial stocks Sao and Sbo, which are common 
knowledge. In period 1, each extracts and sells qn units (i = A, B). In period 2, 
that is, at maturity, they extract and sell the remainder 



Qi2 — ^tO Qil- 


( 1 ) 


In both periods, the instantaneous inverse demand function is 




Pt(9at + 96t) (9ot + 96t), t=l,2. 


( 2 ) 


The intercept a > 0 is unknown. The price pt is a present price, net of extraction 
cost. Risk aversion implies the certainty equivalent payoffs 


Wa = Ea[7Ta] - ^ {SaO ~ fab ~ fas )^ 


(3.1) 


Wb = EtlTTi] - f ( S60 + fab - fbs )^ 


(3.2) 



where 

TTa = P2( qa2 + 962 ) ( 9a2 ~ fab ~ fas) + Pl( 9al + 961 ) 9ol 
“t” Pabfab Pasfasy 



(4.1) 




68 



7T6 = P2{qa2 + ^62 ) ( qb2 + fab ~ fbs) + Pl( ^al + qbl ) qbl 

Pah fab H“ Pbsfbsf (^*^) 

fab > 0 is a futures position such that A sells to B (A “goes short” and B “goes 
long”) at the futures price pab agreed upon, and similarly for fas and fbs- Of 
course, fba = — fab, so the futures market revenue terms in (4.1) and (4.2) reverse 
the sign on pab- All futures positions are closed out at price p2 in period 2. Thus, 
equations (4.1) and (4.2) simply sum spot-market revenues and futures-market 
net revenues. Equations (0.1) and (0.2) use the mean-variance model (with 
constant absolute risk aversion parameters Ma,Mt) and the fact that 
P 2 {qa 2 qb 2 ) and pi{qai-^qbi) have a common uncertain parameter a. The 
variance of each producer's belief about a is normalized to 1, to simplify. There 
is no covariance term. Notice that a producer's degree of risk aversion affects his 
payoff so long as his net futures position leaves some of his initial stock 
unhedged. 

To complete the specification of the game, the risk-neutral speculator's payoff is 
simply his net revenue on the futures market: 

0s ~ ^s[(,P2 Pas ) fas “1“ ( P2 Pbs ) fbs]- (5) 

We now analyze behavior in the extraction subgame. With given futures 
positions, simultaneous maximization of Wa in qai and Wb in qbi (given (1)) gives 
the unique Cournot equilibrium strategies 

^ ^qO /a6~i~ fas i fbs ~ fab . ^ ^aO I /ab~f~ fas fbs fab . 

qal — 2 3 ^ 6 ’ — 2 ^ 3 6 ’ 

/ 7 , , fbs— fa b I fab-^fas . ^ I fbs — fab fah + fas 

qbl — 2 3 ^ 6 ’ ^^2 — 2 ^ 3 6 * 

Producer ^'s net short futures position is fab -f- fas- Producer B's net short 
position is fbs — fab = fbs + fba- Each producer is seen to adjust the amount 
extracted prior to maturity (qa) for the net futures position he has taken and to 
partially counteract the adjustment his rival makes. If fab + fas > 0 and 
fbs — fab <2{ fab + /as ), then A makes his net short position more profitable by 
shifting extraction to at-maturity supply, driving down the spot price at maturity, 
and with it the value of the futures contracts he has (net) sold. 

The equilibrium spot prices are 

Pi = P + I (fas + fbs); P2 = P-I(fas + fbs ), (7) 

where ( fas + fbs ) is the producers' combined net short position (or the 
speculator's net long position) and 



( 6 . 1 ) 

( 6 . 2 ) 




69 



p = 

This parameter can now be given a straightforward economic interpretation: p 
is the spot price that would prevail in both periods in Cournot equilibrium, if the 
producers were both inactive in the futures market. It is a natural benchmark. 
Since it is also a linear function of a, it is convenient to express beliefs in terms of 
p rather than a. The players' beliefs are thus expressed as the mean expectations 
Pa J Pb ^^d Pg . 

It should be noted that, given futures market positions, the equilibrium 
extraction rates do not depend upon the producers' beliefs about a. A trading 
partner of a producer on the futures market can therefore predict how the producer 
will shift extraction in reaction to contracts accepted without knowing the 
producer's expectations. It remains to be seen whether he can predict if the 
producer will accept the contract. 

5. Subgame-Perfect Acceptable Contracts 

In determine subgame-perfect actions in steps 5 to 2, we first incorporate the 
extraction subgame's equilibrium strategies (6.1)-(6.2) and the equilibrium prices 
(7) in the payoff functions Wa and Wb of the producers and (ps of the risk-neutral 
speculator. The result is 



— ^aO Pa ^5 (^*1) 

Wb = SboPb + ( 8 . 2 ) 

— {Ps Pas ) fas + {Ps Pbs ) fbs 5 ( fas H~ fbs {^-^) 



where 



— {Pab Pa) fab {Pas Pa) fas f^^) 

18 



Ma 



(^aO fab fas) ? 



(9.1) 

Vb — {Pa Pab)fab {Pbs ~~ Pbjfhs 77 H” fbs) f^b fbs) • 

18 2 

(9.2) 

We focus on K? and because the expected value of the stock, S{q%, is 
unaffected by futures trading or by extraction rates. 

5.1. Calculation of Barely Acceptable Contract Terms 

When announcing a particular offer (a price and a quantity), a player aims at 
obtaining contract terms that are most favorable to himself and yet acceptable to 
the player for whom the announced contract is intended. Each player therefore 




70 



has to determine which terms are just acceptable to the other side of the market, 
namely, the highest price acceptable to the other party if an offer to sell is being 
announced, and the lowest price acceptable to the other party if an offer to 
purchase is being announced. Player B, say, can do this by solving 

Va-Va\ ^.2 

If B wants to purchase from A (fab > 0), any price equal to or higher than the 
price that satisfies the equality is acceptable to A, since 

is the contribution of the futures position fab to ^'s expected profit and any 
nonnegative contribution will be preferable to not trading with B. The solution is 

Pab I [Pa - Ma{SaO ~ fas)] + ^ fab 3S fab^ 0, (10.1) 

The boldfaced italic subscript indicates the player for whom the price is 
acceptable. By the same reasoning, 

Pab^lPb - Mb(SbO - fbs)]- ^ fab as fab^O, (10.2) 

Pas ^ [Pa - Ma{SaO ~ fab) ~ J fbs] + {^ ~ ^)fas 3S fas^O, (10.3) 

Pas I [Ps - 5 /6s] - g fas aS fas^O, (10.4) 

Pbs ^ [Pb - MbiSbO + fab)- \ fas] + (f‘ ~ ^)fbs aS fbs ^ 0, (10.5) 

Pbs I [p., - 5 fas] - I fbs as fbs^O, (10.6) 

The difficulty for B, in evaluating pab, for example, is that B does not know 
whether A is an optimist (p^ = or A^ for short) or a pessimist 
The two possibilities define two boundaries for the price-quantity combinations 
just acceptable to A. The same is true when A evaluates the terms acceptable to 



2 This equation assumes that B has not accepted and will not accept any contract offered by 
A. Violations are pathological, and analyzed similarly; the only simplification lost is the 
linearity of what would otherwise be parallel quadratic curves in Figures 1 and 2. 




71 



B according to (10.2). For each of the possible contracts, a figure similar to 
Figure 1 can be drawn. 



Figure 1 




5.2. The Basic Geometry of Acceptable Contracts 

Figure 1 illustrates the case of a sale by .4 to 5 {fas > 0). The two upward- 
sloping lines represent the lowest selling prices type can accept (i.e., for 
Pa = p^) and (lower) can accept. The two downward-sloping lines represent 
the highest prices at which types and are each willing to buy. Slopes of the 
lines simply reflect risk aversion. The vertical distance between pairs of parallel 
lines is p^ — p^ (for both pairs). The relative positioning of the pairs of lines 
reflects sizes of stocks, extent of risk aversion, and, most importantly, positions on 
contracts with the third player (S, in the case shown where a contract between A 
and S is being analyzed), as indicated in equations (10). When a sufficiently 
large share of the potential gains to trading futures between A and S have been 
usurped by contracts one or the other player reaches with B, this positioning may 
become tight enough to switch from Figure 1 to Figure 2. If information 
limitations can be surmounted, trading is possible whenever an upward-sloping 
and a downward-sloping line cross, and for price-quantity combinations within 
the triangle formed by these lines and the vertical axis. 





72 



Figure 2 




6. Contract Acceptances in the JiBS Order 

Description of subgame-perfect equilibrium, and the associated inferences about 
types, quickly gets complex. Nonetheless, analytic steps have geometric 
interpretations, given below. As the qualitative properties do not depend upon the 
commitment order, we will present the analysis only for the order ABS. This 
substantially reduces repetition and adds some useful concreteness; the cost is a 
suggestion of higher payoff for A and lower payoff for S, in general, than occurs 
across commitment orders. Where it aids concreteness, we will assume parameter 
configurations with the property that, were it common knowledge that all three 
players were of the same type (all optimists or all pessimists), then A would take 
a net short position, S a net long position, and B an intermediate position. This 
would be accomplished by giving A a larger stock than B, and setting their risk 
postures close approximations to each other, relative to stocks. While these 
parameter configurations are not essential to the analysis, they simplify 
presentation, and fit with the intuitive value of imagining fab,fas,fbs all being 
positive. (In particular, they underlie presentation of figures with a positive 
horizontal axis.) 

6.1. Notation for Pooling or Separating Announcements 
The analysis of subgame-perfect equilibrium proceeds, as is usual, backwards 
from Step 5. Before beginning, however, it is essential to note forward inferences 
resulting from Step 2 behavior. It is common in games of incomplete information 
to distinguish between “pooling” and “separating” equilibria: in a pooling 

equilibrium, a player’s type is not revealed by his behavior, as both types choose 





73 



the same strategy in equilibrium. In a separating equilibrium, each type selects a 
different strategy, and no uncertainty about a player's type remains after his action 
is observed. 

Intermediate “semi-pooling” cases can arise here. Specifically, one or two 
players could choose announcements which reveal their type, with the rest of the 
players choosing pooling announcements in Step 2. During Steps 3, 4 and 5, all 
players will face no uncertainty in predicting the behavior of a player who made a 
separating announcement, but may be unsure of which contracts, if any, will be 
acceptable to a player who chose a pooling announcement. To handle all 
possibilities, the following notation will be employed. Player i's announcement 
made in step 2 will be denoted Ci = (pi,/t), i = where /* is a short 

position: fi>0 implies i is offering to sell a quantity fi of futures contracts at 
price Pi, while i is offering to buy a quantity — fi of futures contracts (go long) at 
price Pi if fi < 0. The information available in step 3 about players' types as 
revealed in equilibrium announcements c=(ca,Cfc,C 3 ) is summarized by 
/ = where takes the value if i made a separating 

announcement revealing type (optimist), P^ if i made a separating 
announcement revealing type and P^ if both types of i made the same 
announcement. 

6.2. Summary of Futures Contract Acceptance Behavior 

This subsection summarizes subgame-perfect behavior in steps 5, then 4, then 3.^ 
Whether A in step 3 or B in step 4 gains by accepting an available contract 
depends on the anticipated contract acceptance behavior to follow. Only if a 
player has identified his type via a separating announcement can his contract 
acceptance behavior be predicted with certainty. 

PROPOSITION 1 : Under doubly inconsistent incomplete information, any set 
of contract announcements yields a unique subgame-perfect equilibrium 
continuation. 

Uniqueness depends upon the convention that a player indifferent between 
accepting and rejecting a contract accepts. The point, though, is that doubly 
inconsistent incomplete information does not create a game in which any behavior 
whatsoever can be rationalized. Contract acceptance behavior is as uniquely 
determined by the rationality postulates supporting subgame-perfect equilibrium 
as in a game of consistent incomplete information. Probabilistically, it is also Just 
as determinable as if information were consistent. 

PROPOSITION 2: In subgame-perfect equilibrium continuation following any 
set of contract announcements, no player alters his contract acceptance behavior 
to avoid revealing his type. 

Separating equilibrium behavior in steps 3-5 stems from two sources: 
independence of extraction behavior from beliefs, and independence of payoffs 
from rivals' types, given the contracts that are accepted by rivals. 



^ Propositions stated here are available to the reader in Harstad and Phlips [1993], the 
fuller version of this paper. 




74 



PROPOSITION 3: Contract acceptance behavior of any player i when any 

rivals deciding after i have revealed their types is essentially a matter of 
anticipating which contracts will be accepted following any behavior by i, and 
then simply deciding whether i would prefer the pattern of acceptances that 
includes his acceptance of an available contract to the pattern that includes his 
rejection of that contract. 

PROPOSITION 4: Contract acceptance behavior of any player i when some 
rival deciding after i has not revealed his type involves anticipating the 
subjectively expected pattern of acceptances following any behavior by i, and 
then simply deciding whether i would prefer the stochastic pattern of acceptances 
that includes his acceptance of an open contract to the stochastic pattern that 
includes his rejection of that contract. When the player committing first makes 
this determination, he anticipates the behavior of the player committing second by 
anticipating the behavior of the player committing third using the second player's 
prior beliefs (with which he disagrees), and then using his own prior beliefs to 
predict both rivals' behavior. 

7. Step 2 Announcements in the ABS Order 

It is almost a tautology that the details of subgame-perfect decisions about which 
contracts to offer cannot be less complex than the details of acceptance behavior. 
We again offer a summary: 

PROPOSITION 5: In any subgame-perfect equilibrium, the player committing 
first makes a separating announcement in step 2, revealing his type. 

Since the announcements are simultaneous, A's rivals in the ABS order cannot 
gain in step 2 from learning As type. Proposition 0 has already shown that A 
will choose to separate, revealing his type, in step 3 before his rivals take their 
next actions. So in a pooling announcement, at least one type of player A would 
be sacrificing payoff. 

Proposition 5 prevents the existence of a “completely pooling” equilibrium in 
which all three players make pooling announcements. The principal results of this 
section characterize “completely separating” equilibria in which all three players 
announce separating contract offers: 

PROPOSITION 6: If the second (J) and third (k) players in the commitment 
order make separating announcements (as does the first player i), then i: 

[a] examines each pair of rivals' types for which an offer is accepted by k with 
some given probability and determines the conditionally efficient contract for 
each pair; 

[b] determines, among these efficient contracts with k, the one that has the highest 
expected profit; 

[c] repeats steps [a] and [b] to determine an efficient offer to j; 

[d] finally selects the offer (directed to j or k) that provides the highest expected 
profit. 

Players j and k follow essentially the same logic. 




75 



PROPOSITION 7: Proposition 6 implies that targeting any announcement to the 
player with whom the greatest gains from trade are at stake is typically best 
response behavior. In general, the resulting equilibrium is not unique.^ 

7.1. Remarks on Existence of a Separating Equilibrium 

To our knowledge, the literature on existence of a separating equilibrium has not 
offered a proof covering the case of an extensive form with three players 
simultaneously choosing from an unordered continuum of signals prior to a 
sequential-move subgame; this paper will not change that situation. By and large, 
adding realism to the deliberately demonstrative model considered here would 
surely not ease these complications. 

It is also possible that the nonconvexities constitute the sort of “controlled 
discontinuities” for which Dasgupta and Maskin [1986] offer existence theorems. 
At least generic use of such an approach may be possible via the following sort of 
truncation argument. For arbitrary behavior in the futures market, there exists a 
unique subgame-perfect continuation in the extraction subgame, so the game can 
be truncated by replacing this subgame with its payoffs. For arbitrary contract 
offers in step 2, the convention that offers are accepted when the player is 
indifferent makes the subgame-perfect continuation uniquely determined. Thus, 
payoffs can in principle be specified so that the game can be truncated after step 
2. The resulting game is a simultaneous-move game in which each player's 
strategies form a compact subset of R 2 ,^ and payoffs are quadratic functions of 
strategies except on sets of measure zero. 

We found a one-parameter family of subgame-perfect equilibria in our similar 
model where inconsistent beliefs were common knowledge (Phlips and Harstad 
[1991]); this is also encouraging for the existence of a separating equilibrium 
here. 

7.2. Remarks on Existence, Nature and Plausibility of a Semi-Pooling 
Equilibrium 

Proposition 5 has shown the nonexistence of completely pooling equilibrium, as 
the player who commits first does not pool. Thus, the issue discussed here is the 
possibility of a semi-pooling equilibrium, in which both types would make the 
same contract offer announcement, and then separate in their contract acceptance 
behavior, for one or both of the players committing last. In this section, for this 
model, that is what the term “pooling equilibrium” will mean. 



^ The reader interested in the detailed logic of equilibrium announcements is referred to 
Harstad and Phlips [1993]. 

^ Compactness is no problem. Since the Vi functions all involve quadratic terms in futures 
positions with negative coefficients, there must exist a finite number which is a nonbinding 
absolute bound on the /» coefficient of any best response. With the fi's artificially but 
harmlessly bounded, it is then similarly possible to obtain a nonbinding, finite upper bound 
on the Pi component of best responses. Without loss of generality, the strategy sets can be 
altered to incorporate these bounds. 




76 



The canonical signaling games (see van Damme [1987], ch. 10, for an 
introduction) achieve pooling equilibria by having the receiving player's payoff 
depend directly on the signaling player's type, and by having the receiver respond 
to an out-of-equilibrium message based upon an assumption about sender's type 
that is unfortunate for one type of sender. As a result, that type of sender's 
predominant interest is in imitating the equilibrium message of the other type of 
sender. 

It does not require the sort of subtle thinking debated in the refinements 
literature (van Damme [1987] and references cited) to restrict acceptance behavior 
following out-of-equilibrium contract offer announcements. A player's payoff 
depends upon a rivals' type only through the contracts that are accepted; no player 
changes his own contract acceptance behavior directly as a result of his belief 
about a rival's type. The only possible impact is that during step 3 or 4, a player 
may alter his contract acceptance behavior in response to a belief that a rival's 
type will lead to particular acceptance behavior in following steps; in what 
follows, we call this the “pooling impact.” Acceptance behavior that serves 
directly to reduce the payoff of a rival whose type is uncertain, or is believed to be 
revealed by an out-of-equilibrium move is clearly disequilibrium behavior in this 
model. 

Thus, there is no particular incentive for types of a player to pool in order to 
avoid revealing their types, other than the pooling impact. Its possible benefit is 
limited by the binary nature of the sequential decisions. It seems most likely, in a 
pooling equilibrium scenario, that a pooling impact beneficial to one type of a 
player is harmful to that player's other type. It follows that type will not benefit 
from an effort of type t to imitate type so t' will only alter his contract 
announcement from a separating equilibrium choice if and only if doing so 
separates. Hence, fs benefit from the pooling impact would have to increase his 
equilibrium payoff more than the decrease that resulted from announcing the 
contract that was a best response for t' rather than for t. We have no proof that 
such pooling equilibria exist only for unlikely parameter constellations, but this 
seems the most natural conclusion to conjecture. 

Should both a separating and a pooling equilibrium exist, refinements that are 
only well-defined for finite games or single-signaler, single-receiver games will 
not directly offer useful distinctions. However, we can envision an argument of 
the following sort. Any pooling equilibrium supported by beliefs that an out-of- 
equilibrium message was sent by the type benefiting from the pooling impact, 
when that message could be a sensible announcement for the type who does not 
benefit from the pooling arrangement is likely to seem less plausible than the 
separating equilibrium, under the same sort of arguments raised in refining 
pooling equilibria of finite signaling games. 

8. Market Efficiency 

We briefly discuss the efficiency of trading in this model, assuming that behavior 
is characterized by a subgame-perfect separating equilibrium. Initially consider 




77 



the singly-inconsistent case: whether each player is optimistic is his own private 
information, but the subjective prior odds are consistent: Aa = At = A5. Each 
player is offering a take-it-or-leave-it contract which maximizes his share of gains 
from exchange, given the behavior of the others and equilibrium continuation. 
Thus, each has incentives to take efficiency into account. In particular, a social 
planner faced with the same informational constraints as the players could not 
suggest ex ante an alternative pattern of behavior that would yield a higher 
expected payoff for one type of one player without yielding a lower expected 
payoff for some type of some player (possibly the other type of the same player). 
In this sense, the singly-inconsistent case reaches an outcome that is 
informationally constrained, ex ante efficient. 

It is difficult to arrive at a satisfactory definition of an efficient outcome for the 
doubly inconsistent case, the general case analyzed above. Players disagree about 
the strength of demand for an exhaustible resource at the maturity date of a 
futures market; neither optimists or pessimists revise their beliefs even if certain 
that others hold different beliefs. This is essential for rational players to strictly 
prefer some speculative trades over foregoing any speculative trade which is 
acceptable to the trading partner. In addition, here, we have allowed players to 
have inconsistent prior beliefs about the probability that another player is an 
optimist; A does not revise his estimate of the odds that S is an optimist upon 
learning that B attaches higher odds. 

Let the rationality of each player i evaluating uncertainty over types via A^ be 
accepted for the purposes of a welfare analysis that is at least a distant analogue to 
efficiency. Then the following sort of characterization holds for the separating 
equilibrium. There is no alternative pattern of offer and acceptance behavior 
which a social planner, subject to the same informational constraints and to 
subgame-perfect equilibrium actions guided by the individual Aj's, could suggest 
that would yield an outcome in which some type of some player viewed himself 
as attaining a higher expected payoff, without the other type of the same player or 
some type of another player viewing himself as attaining a lower expected payoff. 

In sum, despite doubly inconsistent incomplete information, a logic of specific 
rational behavior that was straightforward, step-by-step, emerges. It predicts a 
nonzero volume of purely speculative activity, roughly akin to ex ante efficiency. 
Inconsistent priors increase the difficulties of calculating equilibrium strategies by 
at least an order of magnitude; doubly inconsistent priors add little further 
difficulty. No “anything can happen” sort of Pandora's box is opened. 




78 



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Duncker and Humblot, Berlin. 




Games of Incomplete Information: 

The Inconsistent Case* 

Michael Maschler 

The Hebrew University of Jerusalem, e-mail: maschler@kineret.huji.ac.il 



1. Introduction 

In his well known series of papers Harsanyi [1967-68] develops the theory of 
noncooperative games of incomplete information, in which such games are 
replaced by games of complete information involving chance. He emphasizes 
the consistent case, in which the various types of players in a game of incom- 
plete information jT, as generated from the various beliefs, can be thought as 
being derived from a joint probability matrix. In this case Harsanyi constructs 
a game G of perfect recall and in extensive form, whose players represent the 
various types of the players in F. They are called agents and G itself is called 
the agent- form game representing F. 

The game G has the following properties: 

(i) Each agent, when he comes to play in G, has the same beliefs as the 
type in F that he represents. 

(ii) Each agent, when he comes to play in G, has the same choices as the 
type he represents has in F. 

(iii) If the players agree in F to play a Nash equilibrium of G, no type will 
find it beneficial to deviate, because deviating alone will not increase his 
payoff. 

It is sometimes erroneously believed that Harsanyi ’s theory will not work 
if the beliefs are inconsistent. The purpose of this paper is to dispel this 
belief. One example which indicates how this can be done already appeared 
in Aumann and Maschler [1995]. The present paper generalizes that example 
and supplies a formal proof. 



2. The Data 

To keep the presentation simple, we restrict ourselves to a game F with 
incomplete information, involving two players I and IF The generalization 

* I express my gratitude to the Spanish Ministry of Education for a financial 
support via grant SAB95-0050 DGICYT for the research done while this paper 
was written. I also want to thank Eric Van Damme whose comment lead to the 
inclusion of Theorem 4.2. 




80 



to more players is straightforward. Each player in F can be one of several 
types, denoted and //i, 7 / 2 , . . . , For each combination of 

types {Ip, I la) there corresponds an m x n bimatrix game (afj, 
which is the payoff bimatrix of the players, if the real types that play are Ip 
and I la . The beliefs of the various types of player 7 on the probabilities that 
he is facing one of the types of player 77 are given by a matrix (p^)rxs 5 P = 
1, 2, . . . , r, cr = 1, 2, . . . , s. Thus, player 7 of type p believes that he is playing 
against player 77 of type a with probability p^. Naturally, Pa — 

all p, p = 1, 2, . . . ,r. Similarly, the beliefs of each type of player 77 on the 
types of player 7 are summarized by the matrix )rxs? where Y^^p=i — 
for all cr, cr = 1, 2 , . . . , s. 

It is assumed that the matrices (dij), (Pa) nnd (q^) are common knowl- 
edge. Harsanyi shows that many classes of games with incomplete information 
can be cast into the above framework. 



3. An ^^Extensive Form” Representation of the Data 

We now construct two game trees — one for each player. The trees will be 
identical, except for the numbers representing the probabilities assigned to 
the choices of chance and the payoffs at the endpoints. Briefly, both trees start 
with chance choices for all possible matching of a type of player 7 with a type 
of player 77. The continuation yields m moves for player 7, and information 
sets are placed to show that this player knows his type but does not know 
the type of his rival. The tree continues with n moves for player 77 and 
information sets placed to indicate that this player knows his own type, but 
does not know neither the type of player 7 nor the move he took. 

In more details, each tree will start with r x s arcs emanating from the 
root, which is a vertex of chance. These correspond to the various possibilities 
that chance can match an agent for player 7 and an agent for player 77. To 
be specific, we assume that the s leftmost arcs correspond to a matching of 
agent I\ with the agents 77i, 772, • • - , 77^, in this order. The next s arcs will 
correspond to a matching of I 2 with the s types of player 77, again in the 
order 77i , II 2 , • • • , Hs^ etc. 

We now partition the endpoints of these arcs into r consecutive informa- 
tion sets, each containing s vertices, representing the fact that Player 7 in 7" 
knows his own type, but does not know the type of player 77 against whom 
he is playing. 

We now partition the m x s x r new endpoints into s information sets 
each containing r x m vertices. This should be done in a way that matches 
the fact that player 77 does not know neither the actions of player 7, nor 
his type in F. Thus the m leftmost arcs emanating from each leftmost vertex 
in an information set of player 7 should be grouped together to make an 
information set for 77i, etc. 




81 



To complete the trees we now issue n arcs from every new endpoint. 
These correspond to the n choices that player II has. We end up with r x 
s X m X n endpoints, each corresponding to an outcome in which a certain 
type of player I and a certain type of player II have each made some move. 
Looking at the corresponding payoff matrix, we can now assign payoffs to the 
endpoints. In the tree for player I we assign the payoffs of player I and in 
the other tree we assign the payoffs of player II. 

It remains to assign probabilities to the choices of chance at the roots, 
this is done as follows: 

We choose arbitrary positive numbers ai, 0 : 2 ? • • • , adding up to 1, and 
arbitrary positive numbers /3i, • • • , also adding up to 1. For the tree 
for player /, we assign the probabilities aip\,aip\ . . . ,aip\ in this order, 
to the s leftmost choices of chance (which end up in the information set of 
agent /i). We assign the probabilities 02 ^ 1 , 02 ^ 2 ? • • • in this order, to 

the next s choices of chance (which end up in 72 ’s information set), etc. Note 
that these probabilities add up to one. For the tree for player II we assign 
the probabilities Piq\,(3iq2^ . . . in this order to the choices of chance 

whose continuations intersects the information set of agent 77i. Similarly, 
we assign the probabilities p2qh ^ I^^Qr order to the choices of 

chance whose continuations intersect the information set of agent II 2 , etc. 
These probabilities too add up to 1. 

Example 3.1. Player 7 can be one of two types and so can player 77 be. The 
bimatrix games that correspond to the various matching of the types are 
given below: 



Player 77i 




Player II 2 




The beliefs of the types of players 7 and 77 are given by the following 
tables:^ 



^ Tables open to the right represent beliefs of player 7. Tables open to the bot- 
tom represent the beliefs of player 77. I owe this convention to Harsanyi (oral 
communication) . 




82 





Ih Ih 


ffi 


Ih 


h 


2 3 T 

5 5 


3 

7 


1 

2 


h 


— — r 

4 4 ^2 


4 

7 


1 

2 



The corresponding trees are drawn in Figures 1, and 2. 



Chance 




Figure 1: The Tree for Player I. 



4. Recommendation 

The trees described in the preceding section possess the same pure strategies; 
therefore, they can be converted into a single bimatrix game — call it G. 
Note that in general it will be a nonconstant-sum game even if all the games 
associated with the types happen to be constant sum. The strategies in G 
are exactly those available to the players in F. We say that the pair of trees, 
as well as the game G represent the game with incomplete information F. 

The trees have another important property: Each agent, when he comes 
to play in G, he is actually located in one of the information sets of the tree 
of the player to whom the agent belong — the one that corresponds to the 



83 



Chance 




Figure 2: The Tree for Player II. 



type in F that he represents. At that information set he has exactly the same 
beliefs about the types of his opponent as the corresponding type has in the 
game F. These facts imply the following theorem: 

Theorem 4.1. If the players agree in F to play a Nash equilibrium point of 
the game G, then none of them, regardless of his type, will find it beneficial 
to deviate alone in F. 

Proof. Let {u, v) be an equilibrium pair of strategies for the game G. Suppose 
that a certain type t of player /, playing in F, finds it profitable to deviate 
from his part ut in u, to a strategy ut, assuming that his opponent, whatever 
his type may be, is playing as his part in v. Consider the pair of strategies 
{u-t,ut^v) played in G. The payoffs to all matchings other than those in 
which agent agent It plays will be the same and the payoffs to the remaining 
matching will yield a higher payoff to It. Since a* > 0, this implies that 
(u_t, iit , u) yields in G a higher expected payoff to player I. This contradicts 
the fact that (u, v) was an equilibrium point. A similar contradiction obtains 
if we assume that a certain type of player II finds it profitable to deviate. ■ 

The converse theorem is also true, and it shows that the choice of the a^’s 
and is not important, as long as they satisfy the requirements specified 
in Section 3. (However, some choices may lead to a simpler game G, as can 
be seen in the next section.) 



84 



Theorem 4.2. If a set of strategies is specified for each type, and together 
they have the property that no player, regardless of his type, can benefit by 
deviating, given that all the other types play as specified, then these strategies, 
when placed at the corresponding information sets in game trees, combine to 
form behavior strategies which constitute an equilibrium point in the game G. 

Proof. If a player in G can benefit by deviation alone, than this deviation 
induces deviations in some of his information sets in the tree games. In at 
least one of them he will benefit, since his expected payoff is a positive lin- 
ear combination of the expected payoffs of his types (given the strategy of 
the other player). This means that at least one of his types will benefit by 
deviations, which is contrary to the data, because the expected payoff of a 
type does not depend on what other types of the same player do (they do 
not exist, anyhow). ■ 



5. The Consistent Case 



Suppose now that the types were derived from a joint probability matrix 
imj)rxs^ Then 



P) = 



w 






Wii -f- Wi2 + . . . H- 



Wi 



4 = 



'^ij 

+ . . . + tt; 



sj 



and therefore, if we choose = wn + W {2 + . . . -h Wis,i = 1, 2, . . . ,r and 
Pj — '^ij + W 2 j 4- . . . + Wrj,j = 1, 2, . . . , s, we find that aip'j = Wij = Pjq{. 
Thus, in this case, both trees will have the same probabilities assigned to the 
choices of chance. These trees can therefore be combined into a single tree 
(with vector payoffs), which is, in fact, Harsanyi’s tree. 



References 



1. Aumann, R.J., Maschler M. (1995): Repeated games with incomplete informa- 
tion. The MIT Press, Cambridge, MA. 

2. Haxsanyi, J.C. (1967-68): Games with incomplete information played by 
bayesian players, parts I-III. Management Sci 14 159-182; 320-334; 486-502 




Admissibility and Stability 

Robert Wilson 

Stciiiford Business School, Stcinford CA 94305-5015 USA 



Abstract. Admissibility is a useful criterion for selecting among equilibria, 
but I argue that enforcing admissibility dilutes the power of stability crite- 
ria to select among equilibrium components, and this accounts for anomalous 
examples of stable sets. Therefore, admissibility should be invoked only when 
selecting equilibria within a component that is immune to payoff perturba- 
tions. 



The purpose of this essay is to examine the role of admissibility in for- 
mulating criteria for stability of equilibria. Recall that admissibility excludes 
an equilibrium that uses a dominated pure strategy. In particular, each pure 
strategy used (i.e., with positive probability) must be a best response to some 
completely mixed strategy. Reinhard Selten’s (1975) criterion of perfection 
ensures admissibility by requiring that a pure strategy is used only if it is a 
best response to a sequence of completely mixed strategies. Admissibility wcis 
adopted subsequently by Kohlberg and Mertens (1986) and Mertens (1989) 
as a primary desideratum in their identification of stable sets of equilibria. 

My thesis here is that admissibility plays two roles, only the first of which 
advances the program of defining stability for games in extensive form. The 
first role is that in selecting among equilibria within a component, admis- 
sibility strengthens the criterion of stability. However, in its second role in 
selecting among equilibrium components, admissibility weakens the criterion 
of stability — as examples will illustrate. I argue, therefore, that admissibil- 
ity criteria should be deleted when selecting among equilibrium components; 
that is, the selected components should be essential^ in the sense that they 
are stable with respect to all perturbations of normal- form payoffs. This lends 
credence to Kohlberg and Mertens’ original definitions of ‘hyperstable’ and 
‘fully stable’ sets of equilibria, provided they are restricted to lie in essential 
components. It suggests modifying Mertens’ (1989) homological definition of 
stability to invoke payoff perturbations rather than strategy perturbations. 

Section 1 summarizes technical aspects that are the basis in Section 2 for 
a comparison of stability criteria derived from perturbations of strategies and 
payoffs. All examples are collected in the Appendix. For a general survey of 
this topic see van Damme (1991). 



1. Background 

To simplify, we consider only games in extensive form with two players, called 
I and II. Each player has a finite number of pure strategies; each has perfect 
recall; and all the data of the game are common knowledge. Thus the normal 
form of a game has two matrices with rows labeled by I’s pure strategies and 
columns labeled by IPs pure strategies. For each pair of their pure strategies, 
the corresponding entry in one matrix describes I’s payoff, and in the other 




86 



matrix IFs payoff, each expressed in terms of that player’s von Neumann- 
Morgenstern utility. We use to denote player k’s payoff from Fs pure 
strategy i and IFs pure strategy j . Recall that each pure strategy assigns 
a feasible action to each information set of that player; that is, to each event 
in which that player takes an action. Each player’s feasible strategies are his 
mixed strategies, i.e., the probability distributions over his pure strategies. 
If I uses the strategy x = (x,) and II uses the strategy y = (yj) , where 
a; > 0 and ^ X{ = 1 and similarly for y , then k’s expected payoff is 
XiU^jVj . A Nash equilibrium is a pair (ar,y) of strategies for the 
two players such that each player uses only pure strategies that are optimal 
responses to the other player’s strategy; that is, 

Xi > 0 only if ^ u{^yj = max^ , 



and similarly for player II. 

Selten (1965, 1975) argued convincingly that only a subset of the Nash 
equilibria predicts behaviors by rational players. He proposed restricting equi- 
libria to those that are perfect. One version of his 1975 (normal-form) def- 
inition says that an equilibrium (x^y) is perfect if it is the limit of a se- 
quence of completely mixed strategies (i.e., every pure strategy is used), say 
(^/i y/i) ^ Q converging to (x,y) , such that 

Xi > 0 only if ^ ujjy^ = max^ , 

3 3 

and similarly for player II. Because each used pure strategy is an optimal 
response to a completely mixed strategy, it must be undominated. A perfect 
equilibrium therefore satisfies admissibility. 

One interpretation of Selten ’s construction is that each strategy used in a 
perfect equilibrium must remain optimal against some ^tremble’ by the other 
player; that is, small probabilities of using strategies that the equilibrium 
predicts will otherwise be unused. His other, equivalent definition empha- 
sizes this aspect: an equilibrium (x,y) of a game G is perfect if there is 
a sequence (x^,y^;G^) converging to (a:, y; G) , for which each (x^,y^) is 
an equilibrium of the perturbed game . In this version, G^ is the strat- 
egy perturbation of G obtained by adjoining the constraints x^ > and 
y^ > j where (e^, J^) ^ 0 and lim/i^oo(^^, <^^) = 0 . This interpretation 
in terms of perturbations of the game had a profound effect on the subsequent 
development of theories of equilibrium refinements. 

The prominent feature of Selten ’s formulation is that it considers per- 
turbations of mixed strategies. This approach was well- adapted to the spec- 
ification of Kreps and Wilson’s (1982) definition of sequential equilibrium, 
adapted from Selten ’s 1965 definition of subgame-perfect equilibrium, to ad- 
dress the games without proper subgames that motivate Selten ’s 1975 article. 
This criterion imposes the requirement of sequential rationality; that is, at 
each information set a player’s continuation strategy is optimal with respect 
to a probability distribution (over possible histories) consistent with the game 
and Bayes’ Rule. A perfect equilibrium meets this requirement, using the 
limit of the conditional probabilities induced by the sequence of perturbed 
games. The perfect equilibria are therefore the sequential equilibria satisfying 




87 



a strong form of admissibility.^ Indeed, in generic extensive games the sets 
of perfect and sequential equilibria coincide (Blume and Zame, 1994). One 
version of this fact is that allowing all perturbations of payoffs (rather than 
strategies) produces exact coincidence between the resulting ‘weakly perfect’ 
equilibria and the sequential equilibria. The different style of reasoning in- 
volved in using payoff perturbations to select equilibria is illustrated in the 
Appendix via the well-known Beer-Quiche game. 

In their study of hyperstability, Kohlberg and Mertens allowed all pertur- 
bations of normal-form payoffs. As they recognized, this enlarges the set of 
allowed perturbations. Each perturbation of strategies has the same net effect 
as a corresponding perturbation of payoffs. This can be seen by interpreting 
the trembles as chance moves whose small probabilities perturb the expected 
payoffs from each pure strategy specifying intended actions at information 
sets. Kohlberg and Mertens advanced Selten’s agenda towards identifying 
‘essential’ or ‘strictly perfect’ equilibria, those immune to all perturbations 
in a neighborhood of the game, not just a single sequence of strategy per- 
turbations. Because such ideal equilibria need not exist, this step required 
reconsideration of the unit of analysis.^ They demonstrated that each game’s 
equilibria are partitioned into a finite number of closed connected compo- 
nents; and for generic extensive-form games, equilibria in the same compo- 
nent have the same outcomes, namely, they differ only off the equilibrium 
path and therefore their probability distributions over histories are the same. 
Combining these two results indicates that an equilibrium outcome immune 
to payoff perturbations is identified generically by a component that is es- 
sential; that is, every nearby game has an equilibrium near the component.^ 
They also proved that every game has essential components.^ Moreover, some 
are invariant in the sense that they are essential in every game with the same 
reduced normal form obtained by deleting redundant strategies (those strate- 
gies whose payoffs for all players are convex combinations of other strategies’ 
payoffs) . 



^ Admissibility cind normal-form perfection of equilibria are equivalent in two- 
person games. The analogous extensive-form definition of perfection allows use 
of dominated strategies. In Mertens’ (1995) example, the dominant-strategy 
equilibrium is not extensive-form perfect, and remarkably, every extensive-form 
perfect equilibrium uses a dominated strategy. Similar examples led van Damme 
(1984) to propose cm ciltemative definition of quasi-perfection. 

^ Kohlberg and Mertens (1986, Appendix D) show that an equilibrium that is 
extensive-form perfect in every extensive game with the same reduced normal 
form is strictly perfect, provided all best responses cire used. For many interest- 
ing games, however, this proviso is not satisfied. 

^ Genericity is mostly immaterial here because a nongeneric case has a component 
that is the union of components induced by necirby games; a case where none 
of these subcomponents is essential would be doubly rare. 

^ That such a component exists follows from their demonstration that the pro- 
jection of the graph of the Nash equilibrium correspondence is homotopic to a 
homeomorphism. This is called the ‘rubber sphere’ theorem, because it shows 
that when the space of games is mapped onto the surface of a sphere via a 
one-point compatification, the equilibrium graph is mapped to a deformation 
of a sphere of larger radius. Consequently, above every game lies a component 
for which every necirby game has an equilibrium ne 2 irby. 




88 



Kohlberg and Mertens showed that an invariant essential component has 
several desirable properties. 

Invariance: Because an invariant component depends only on the re- 
duced normal form, it does not depend on which among the many strate- 
gically equivalent extensive forms is used. 

Backward Induction: It contains perfect and even proper equilibria.^ 
Such equilibria satisfy Selten’s 1965 criterion (each strategy remains op- 
timal in every subgame) and Kreps and Wilson’s generalization to ‘ex- 
tended subgames’ in games with imperfect information. Proper equilib- 
ria are perfect, and therefore use only admissible strategies. Moreover, 
a proper equilibrium induces a sequential equilibrium in every extensive 
form with the same reduced normal form. 

Iterated Dominance: Its projection onto the equilibrium graph of a 
smaller game, obtained by deleting dominated strategies, contains an 
invariant essential component. 

Forward Induction: Its projection onto the equilibrium graph of a 
smaller game, obtained by deleting a strategy that is suboptimal at all 
its equilibria, contains an invariant essential component.^ 

Further, within an invariant essential component is a minimal closed set 
(called hyperstable) with analogous properties; inside that is a minimal closed 
set (called fully stable) immune to perturbations of any finite set of pure or 
mixed strategies; and inside that is a minimal closed set (called stable) im- 
mune to perturbations of pure strategies. The latter (though omitting min- 
imality) allows a homological definition proposed by Mertens (1989). In the 
next section we re-examine Kohlberg and Mertens’ and Mertens’ rejection 
of all these except stable sets, on the grounds that only stable sets’ equilib- 
ria exclude inadmissible strategies. We shall see that this has unfortunate 
consequences, because some stable sets reside in inessential components. 

This brief review omits a vast literature emanating from Selten’s insights, 
but it includes some main themes. Extensive games require equilibrium refine- 
ments. Intuitive criteria such as admissibility, backward induction, and con- 
sistent conditional probabilities can be founded on consideration of strategy 
perturbations. Combining these with iterative elimination of dominated or 
suboptimal strategies meets further criteria, such as rationalizability (Pearce, 
1984) and forward induction. 

To convey the scope of extensions not described above, we mention only 
the models of reputation effects in finitely-repeated and centipede games. 
These models are derived from Selten’s (1978) study of the chain-store game. 
In repeated games (without or with stopping options), a perturbation of a 



^ To be proper, the sequence justifying a perfect equihbrium must assign a prob- 
ability of lower order to one pure strategy that is inferior to another (Myerson, 
1978). 

® This property is motivated by games involving signcding. These games typically 
have multiple equihbria reflecting possible inferences from disequihbrium behav- 
iors. The motive for forward induction insists that the conditional probabihties 
vaihdating a sequentiEil equilibrium should be zero for histories dependent on 
strategies that are suboptimal at every equihbrium in the component. Thus, 
one can. prune those branches of the game tree that occm* only when such sub- 
optimeil strategies cire used. See Banks and Sobel (1987) cind Cho and Kreps 
(1987). 




89 



strategy that repeats a particular behavior can attract imitation, leading to 
an equilibrium in which the perturbed strategy is the only one used, except 
in the last stages of the game. These models are relevant to the subsequent 
discussion because typically (as in the repeated chain-store and prisoners’ 
dilemma games) their striking feature is that it is a strategy that is inadmis- 
sible in the stage game that attracts imitation. 



2. A Critique of Admissibility 

We now examine the role of admissibility in constructing equilibrium re- 
finements such as those described above. First I suggest via examples that 
the known deficiencies of stable sets derive primarily from the imposition 
of admissibility criteria. The justification offered for admissibility is that it 
is a cornerstone of single-person decision theory, but this rationale applies 
only to selection among equilibria, not to selection among components (e.g., 
Kohlberg and Mertens, p. 1014; Mertens, 1989, p. 577(c)). Applying admissi- 
bility criteria to component selection is therefore misplaced. This still leaves 
ample latitude to select equilibria within an invariant essential component 
using criteria of admissibility or perfection. I conclude with cautionary re- 
marks: in some contexts admissibility is unduly restrictive even in selecting 
among equilibria within an essential component. 

In the construction of stable sets, admissibility is ensured by allowing 
only strategy perturbations, rather than all payoff perturbations. This for- 
mulation might seem legitimate in view of the fact that differences between 
weak-perfect or sequential (justified by payoff perturbations) and perfect (jus- 
tified by strategy perturbations) equilibria are nongeneric. In fact, however, 
the difference between the existential quantifier (‘there exist perturbations 
yielding nearby equilibria’) used for these equilibrium selections, and the 
universal quantifier (‘all perturbations yield nearby equilibria’) used for com- 
ponent selections can be substantial. In fact, generic games can have stable 
sets (immune to strategy perturbations) inside components that are not es- 
sential (immune to payoff perturbations). For this reason, I see no justifica- 
tion for restricting the allowable payoff perturbations when selecting among 
components. 

In the Appendix, three examples illustrate how the exclusion of some 
payoff perturbations allows implausible components to survive the criterion 
of stability. 

— The first is van Damme ’s (1989) generic example in which there is a stable 
set that fails to induce an equilibrium in a subgame. Mertens’ (1989) re- 
vised definition produces a larger stable set that remedies this deficiency. 
Nevertheless, one sees easily that these stable sets lie in an inessential 
component, as one can see from a simple payoff perturbation. 

— The second example is the nongeneric game Trivial Pursuit in which there 
is a stable set whose outcome for one player is inferior to his stable outcome 
in a continuation that he has the option to elect. Again, a simple payoff 
perturbation suffices to eliminate this stable set lying in an inessential 
component. 

— The third example is Cho and Kreps’ (1987) ingenious example of a stable 
set that motivates their conclusion that “if there is an intuitive story to go 




90 



with the full strength of stability, it is beyond our powers to offer it here’’ 
(p. 220). But once again, a simple payoff perturbation suffices to show that 
this stable set resides in an inessential component. 

I conclude from such examples that in devising criteria for selecting among 
components, it is better to use all payoff perturbations. Of necessity, this 
approach may select essential components with equilibria using inadmissible 
strategies. The second step, therefore, is to use criteria such as admissibility 
to select, say, stable sets (or proper or perfect equilibria) within the selected 
component.^ This two-step procedure was implicit in Kohlberg and Mertens’ 
original work, but unfortunately they abandoned it when they attempted to 
construct a definition that obtains all desirable properties in a single step. 

Besides the above examples, the deficiency of the single-step approach is 
evidenced by Mertens’ (1995) second example (which he calls “pretty damn- 
ing as to the behavior of stable sets in at least some non-generic games” ) of a 
game with perfect information in which the unique stable set includes all ad- 
missible equilibria (which includes a two-dimensional set) , of which only one 
is subgame-perfect. Moreover, this subgame-perfect equilibrium yields the 
only stable outcome derived from all games in the neighborhood of payoff 
perturbations except those in a ^slice’ satisfying a single equality condition. 
Such examples indicate that a second step of selection within an essential 
component is inescapable. 

It is important to realize, however, that a two-step procedure has its own 
deficiencies. To illustrate, the Appendix presents a generic extensive-form 
game Q with an invariant essential component containing two stable sets (in 
the strong sense of Mertens’ definition) and two proper equilibria, reflecting 
the fact that its projection map (from its neighborhood in the equilibrium 
graph to the space of games) has degree 2. Neither of these stable sets is 
immune to payoff perturbations, even though they reside in a component 
that is immune. 

We turn now to the selection of equilibria within an (invariant) essential 
component and ask: Are the strategy perturbations the best set to invoke in 
selecting among equilibria? Clearly, Selten’s pure-strategy perturbations are 
a minimal set sufficient to ensure admissibility and consistency.® On these 
grounds one can say that Selten’s construction is exactly right. On the other 
hand, I think it is prudent to realize that admissibility is justified only if 
one is quite sure about the validity of some particular extensive form of the 
game. As Fudenberg, Kreps, and Levine (1988) prove, every pure-strategy 
equilibrium is the limit of strict equilibria of nearby ‘elaborations’; that is. 



^ An example of the effectiveness of this two-step procedure even when admissi- 
bility is not an issue is van Damme’s (1987, p. 119) generic extensive game with 
a two proper equihbria, one of which he argues is “not sensible.” A simple payoff 
perturbation suffices to show that this equihbri urn’s component is inessential. 

® These comments apply to normal-form pertm*bations; for extensive-form ver- 
sions, one can also use the perturbations used by van Damme (1984) to define 
quasi-perfection. A somewhat larger set that includes mixed-strategy pertur- 
bations is appau-ently a necirly maximal set ensuring admissibility. One subset 
of mixed-strategy perturbations ensuring admissibility is used by Kohlberg and 
Mertens (1986, p. 1025) to construct proper equihbria; however, they show by 
example (p. 1026, Fig. 9) that the full set fails to ensure admissibility. 




91 



games in which players may have differing information about which pertur- 
bation applies. Fudenberg and Maskin (1986), Myerson (1986), van Damme 
(1987) illustrate particular examples. Further, Bagwell (1995) provides an 
example in which perturbations of the observability of actions induce payoff 
perturbations that justify an inadmissible equilibrium in the unique essential 
component containing the subgame-perfect equilibrium. BagwelFs Stackel- 
berg game is described in the Appendix. In all these cases, the inadmissible 
equilibria are justified by payoff perturbations, but not by strategy pertur- 
bations. 



3. Conclusion 

In sum, I see no harm and substantial advantages to selecting invariant essen- 
tial components based on consideration of all payoff perturbations. Doing so 
eliminates anomalous stable sets in the known examples. As a second step, one 
can select a stable set or a proper or perfect equilibrium within the essential 
component, provided there is assurance that admissibility is an appropriate 
criterion. This second step is problematic, however, whenever potentially rel- 
evant perturbations of the extensive form generate payoff perturbations not 
induced by strategy perturbations. 

Acknowledgement: Research support was provided by NSF grant SES9207850. 
Srihari Govindan, John Hillas, and Eric van Damme kindly pointed out errors 
in a previous draft. 



4. Appendix: Examples 

4.1 Example 1: The Beer-Quiche Game 

The distinction between strategy and payoff perturbations is immaterial in 
generic cases. To illustrate, Figure 4.1 shows a version of Kreps’ Beer-Quiche 
game (1990, p. 465; Kohlberg and Mertens, 1986, p. 1031). In this game, 
player I chooses Left or Right and then player II chooses Across or Up; how- 
ever, only I knows whether chance has chosen the payoffs along the Top or 
Bottom (with probabilities p and 1 — p, where 0 < p < 1/2). The stable 
component contains the pure strategy equilibrium in which I surely chooses 
Left, and II chooses Up after Left and Across after Right. The unstable com- 
ponent is reversed: I surely chooses Right, and II chooses Across after Left 
and Up after Right. The latter violates forward induction because Fs unused 
strategy that chooses Left after Top and Right after Bottom is suboptimal 
in that component; if it is deleted then IPs action after Left becomes subop- 
timal, and if that is revised then I surely prefers Left. In terms of strategy 
perturbations this conclusion is evident because IPs strategy cannot be op- 
timal whenever the conditional probability of a deviation to Left is less after 
Top than Bottom. In terms of payoff perturbations the reasoning is altered: 
a perturbation that makes IPs unused optimal strategy (i.e.. Up invariably) 




92 



ro 


CO 

o 


,0 


./ 

\ 

\ 3, 

\ 

\ 

\ 

\ 

\ 


( 

,1 


p 

) 

2, 

1-p 


\ 

\ 

\ 

\ 

\ 

\ 

\ 

\ II 
/ 

/ 

,1 / 

/ 

/ 

/ 

/ 

/ 



I 



Fig. 4.1. The Beer-Quiche game. 



superior requires II to use this strategy exclusively, in which case I prefers 
Left after Bottom.^ 



4.2 Example 2: van Damme’s Game 

Figure 4.2 depicts the generic extensive-form game studied by van Damme 
(1989) in which player I initially chooses either Up or to play a simultaneous- 
move subgame with player II. This game hcis a stable set with the payoff 
(2,2) obtained when player I chooses Up. This outcome cannot arise from an 
essential component: a negative perturbation of IPs payoff from (Up, Middle) 
yields a game in which no equilibrium uses Up. The effect of such a perturba- 
tion on II ’s best-response regions within the simplex of Ps strategies is shown 
in Figure 4.3. 

From his study of this game, van Damme argued that the outcome Up 
violates a plausible interpretation of forward induction; moreover, no equi- 
librium in a minimal stable set induces an equilibrium in the subgame after 
I chooses Right. Mertens’ (1989) definition provides a larger stable set that 
does include subgame equilibria. Even so, it is stable with respect to strategy 
perturbations but not payoff perturbations. 



4.3 Example 3: Trivial Pursuit 

Figure 4.4 depicts the game Trivial Pursuit. In the continuation after player 
II initially chooses Right, were this known to I, Ps strictly dominant strategy 

^ Other than the imphcit corollciry seemingly imphed by the elaborate proof in 
Kreps and Wilson (1982, strengthened by the results of Blume and Zame, 1994) 
of the generic coincidence of perfect cind sequenticd equihbria, I know no direct 
demonstration of the generic equivalence of these two styles of recisoning. 




93 



2.2 



Up 


N 


Left 


Middle 


Right 




Top 


0.0 


3,2 


0,3 




Bottom 


3,3 


0.2 


3,0 



Fig. 4.2. Van Damme’s example of a generic game with a stable set in an inessential 
component. 



I’s Strategies II ’s Strategies 

H’s Best Responses I’s Best Responses 




equilibrium (Bottom, Left). 



Fig. 4.3. Van Damme’s example depicted graphiccilly, showing the effect of per- 
turbing player IPs payoff. 




94 



0,1 1,2 




Fig. 4.4. The game TrivicJ Pursuit with a stable set in an inessential component. 



is Up, ensuring a payoff of 1, and the unique stable payoffs are (1,2). Now 
consider the full game with the parameter satisfying 0 < d < 1 . This adds 
an initial move (Up) by II from which he gets a certain payoff of 1 and 
I gets less — both of which are inferior to their payoffs from the stable 
outcome after Right were it anticipated by I. However, there is a stable set 
in which all equilibria require both players to randomize their initial choices 
— including a positive probability of Up initially by II. This stable set is not 
in the essential component, however, because every negative perturbation of 
either of IPs payoffs from Up yields a game in which II surely chooses Right 
initially, as shown in Figure 4.5. 

This game is nongeneric only to the extent that IPs payoffs from Up 
initially are replicated by a mixture of his payoffs from Right, so Up is a 
redundant strategy for II. Nevertheless, the normal form of this game cannot 
be reduced by deleting Up, because Ps payoffs are affected. 



4.4 Example 4: Cho and Kreps’ Signaling Game 

Figure 4.6 depicts graphically the relevant data for a signaling game like 
Example 1 except that there are three types of player I and player II has three 
actions. They consider an equilibrium outcome in which all types of I choose 
Right (R). In the figure, the left simplex comprises IPs possible beliefs about 
Ps type after observing Ps alternative action Left (L), and shows its partition 
into IPs best response regions. The right simplex shows the partition of IPs 
simplex of behavioral strategies after Left into Ps best response regions; e.g., 
at the top the notation (L,R,R) indicates that types 1, 2, and 3 prefer Left, 
Right, and Right respectively if II is sufficiently likely to choose his Action 
1 after Left. Cho and Kreps (1987) show that this configuration enables 
the outcome from the specified equilibrium [in which I uses the strategy 
(R,R,R)] to be stable. One sees easily, however, that this equilibrium is in 




95 



Vs Strategies 
N’s Best Responses 



Up 

Right-Up I Right-Right 
Up Right 



A negative perturbation of ll’s 
payoff from Up leaves only the 
equilibrium (Up, Right-Up). 



M's Strategies 
I’s Best Responses 




Fig. 4.5. The best-responses for Trivial Pursuit, showing a perturbation of IPs 
payoff from Up that eliminates the inessential component. 



M’s beliefs after I chooses L 
M’s best action after I chooses L 



Type 1 




M’s behavioral strategy after L 
I’s best responses 



Action 1 




Fig. 4.6. The data for Cho and Kreps’ example of a stable set in an inessential 
component. 



96 



an inessential component. A positive payoff perturbation of any one of IPs 
optimal strategies ensures that it must be used exclusively, in which case the 
corresponding type prefers Left. For instance, a positive payoff perturbation 
of Action 1 requires that II uses only Action 1 in response to Left, but then 
I’s Type 1 prefers Left. 

4.5 Example 5: The Game f2 

Figure 4.7 depicts the generic extensive-form game , and Figure 4.8 shows 




Fig. 4.7. The generic extensive-form game Q . 



the players’ best responses. The component in which I uses only strategy a 
is invariant and essential; in fact the projection from its neighborhood has 
degree 2. The degree is reflected in the presence of the two proper equilibria 
where II uses e or f, each of which is contained in a Mertens-stable set: the 
interval [e,de] or the interval [f,df]. Neither of these stable sets is immune to 
payoff perturbations. For instance, those perturbed games with two equilibria 
near f and def have no equilibria near [e,de]. 

4.6 Example 6: Bagwell’s Stackelberg Game 

This example illustrates that even in the games of perfect information that 
motivate Selten’s 1965 criterion of subgame perfection, the sole essential com- 
ponent can contain inadmissible equilibria justified as the limits of equilibria 
of nearby games with imperfect observability. Figure 4.9 displays the exten- 
sive form of Bagwell’s (1995) Stackelberg game. First I chooses Up or Down, 
then chance reveals ‘up’ or ‘down’ to II (with a probability p of an erro- 
neous report), and then II chooses Up or Down. If the error probability is 
zero then this is a game with perfect information: the unique subgame-perfect 




98 



equilibrium has I choosing Up and II choosing Up after up and Down after 
down. This subgame equilibrium lies in the unique essential component that 
requires II to choose Down after down with conditional probability at least 
one-half. (The inessential component has I choosing Down and then II choos- 
ing Down, provided IPs conditional probability of Down after up is at least 
one-half.) However, any perturbation that allows a positive probability of er- 
roneous reports has only one equilibrium close to the essential component, 
and the equilibrium it is close to is not the perfect equilibrium, but rather the 
inadmissible equilibrium where II mixes equally between Up and Down after 
down. This is a robust feature of games with imperfectly observed actions. 

The important feature of the ‘observability’ perturbation above is that it 
induces a payoff perturbation that cannot be mimicked by a strategy pertur- 
bation. Indeed, a perturbation of Ps strategies is capable of excluding only 
the dominated strategies of II that choose Up invariably or Down invariably. 
In contrast, the payoff perturbation induced by a positive error probability is 
one that makes these strategies undominated. Unlike strategy perturbations, 
which envision nonrational trembles, payoff perturbations account for slight 
chances of inaccurate observations. This is similar to the effect of reputational 
considerations in repeated games, where again the relevant perturbation is 
one that converts a dominated strategy into an undominated strategy. 



References 



1. Bagwell, Kyle (1995), ‘Commitment cind Observability in Games,’ Games and 
Economic Behavior^ 8: 271-80. 

2. Banks, Jeffrey, and Joel Sobel (1987), ‘Equilibrium Selection in Signaling 
Games, Econometrica^ 55: 647-61. 

3. Blume, Lawrence, cind William Zame (1994), ‘The Algebraic Geometry of Per- 
fect and Sequential Equilibrium,’ Econometrica^ 62: 783-94. 

4. Cho, In-Koo, and David Kreps (1987), ‘Signaling Games and Stable Equilibria,’ 
Quarterly Journal of Economics^ 102: 179-221. 

5. Fudenberg, Drew, David Kreps, and David Levine (1988), ‘On the Robustness 
of Equihbrium Refinements,’ Journal of Economic Theory^ 44: 354-80. 

6. Fudenberg, Drew, and Eric Mciskin (1986), ‘The Folk Theorem in Repeated 
Games with Discoimting and with Incomplete Information,’ Econometrica^ 54: 
533-54. 

7. Kohlberg, Elon, and Jean-Frangois Mertens (1986), ‘On the Strategic Stabihty 
of Equilibria,’ Econometrica, 54: 1003-38. 

8. Kreps, David (1990), A Course in Economic Theory. Princeton: Princeton Uni- 
versity Press. 

9. Kreps, David, and Robert Wilson (1982), ‘Sequential Equilibria,’ Econometrica^ 
50: 863-94. 

10. Mertens, Jean-Frcingois (1989), ‘Stable Equilibria - A Reformulation,’ Mathe- 
matics of Operations Research^ 14: 575-625. 

11. Mertens, Jean-Frangois (1995), ‘Two Examples of Strategic Equihbrium,’ 
Games and Economic Behavior^ 8: 378-88. 

12. Myerson, Roger (1978), ‘Refinement of the Nash Equihbrium Concept,’ Inter- 
national Journal of Game Theory^ 7: 73-80. 




99 



13. Myerson, Roger (1986), ‘Multistage Games with Communication,’ Economet- 
rica^ 54: 323-58. 

14. Peairce, David (1984), ‘Rationailizable Strategic Behavior and the Problem of 
Perfection,’ Econometrica, 52: 1029-51. 

15. Selten, Reinhard (1965), ‘Spieltheoretische Behandlimg eines Oligopolmodells 
mit Nachfragetragheit,’ Zeitschrift fur die gesamte Staatswissenschaft, 121: 301- 
24. 

16. Selten, Reinhaird (1975), ‘Re-examination of the Perfectness Concept for Equi- 
hbrium Points in Extensive Games,’ International Journal of Game Theory^ 4: 
25-55. 

17. Selten, Reinhcird (1978), ‘The Chciin-Store Paradox,’ Theory and Decision, 9: 
127-59. 

18. van Damme, Eric (1984), ‘A Relation between Perfect Equilibria in Extensive- 
Form Games cind Proper Equilibria in Normal-Form Games,’ International 
Journal of Game Theory, 13: 1-13. 

19. van Damme, Eric (1987), Stability and Perfection of Nash Equilibria. Berlin: 
Springer- Verlag. 

20. van Damme, Eric (1989), ‘Stable Equilibria and Forwcird Induction,’ Journal 
of Economic Theory, 48: 476-96. 

21. van Damme, Eric (1991), ‘Refinements of Nash Equilibrium,’ in J.J. Laffont 
(1992), Advances in Economic Theory: Sixth World Congress. New York: Cam- 
bridge University Press. 




Equilibrium Selection in Team Games 

Eric van Damme 

CentER for Economic Research, Tilburg University, P.O. Box 90153, 5000 LE 
Tilburg, The Netherlands. 



Abstract. It is shown that in team games, i.e. in games in which all players 
have the same payoff function, the risk-dominant equilibrium may differ from 
the Pareto dominant one. 



1. Introduction 

The general theory of equilibrium selection that has been proposed in 
Harsanyi and Selten (1988) invokes two completely different selection criteria: 
risk dominance and payoff dominance. The first is based on individual ratio- 
nality, while the second incorporates collective rationality. The latter criterion 
captures the idea that if one equilibrium E\ yields all players in the game uni- 
formly higher payoffs than the equilibrium E 2 does, then rational players are 
more tempted to play the former. The first criterion captures the idea that, 
in a situation where players do not yet know which of the two equilibria, E\ 
or JE 2 , will be chosen, players will lean towards that equilibrium that appears 
less risky in the situation at hand. For the case of 2 x 2 games, Harsanyi and 
Selten give an axiomatic characterization of their risk-dominance relation. 
In the special case of a symmetric 2x2 game, the risk-dominance relation 
can be easily characterized: E\ risk dominates E 2 if and only if each player 
finds it optimal to play according to E\ if he expects the other to play in 
accordance with E\ with a probability of at least 1/2. 

The criteria of payoff dominance and of risk dominance may yield conflict- 
ing recommendations, and in such cases Harsanyi and Selten give precedence 
to payoff dominance. An example of such a conflict is illustrated in the stag 
hunt game from Figure 1 which has been adapted from Aumann (1990). (Also 
see Harsanyi and Selten (1988, Sect. 10.12)). Each player has two strategies, 
a safe one and a risky one, and if both play their risky strategy, the unique 
Pareto efficient outcome results. However, playing this strategy is very risky: 
If one player chooses it while the other player chooses the safe strategy, then 
the payoff to the first is only zero. In contrast, the safe strategy guarantees a 
payoff of 7, and it might even yield more. In this stag hunt game (i?, R) is the 
payoff dominant equilibrium, while (5,5) is the risk-dominant equilibrium. 
(Indeed each player finds it optimal to play 5 as long as his opponent does 
not choose R with a probability more than 7/8.) 

The payoff dominance requirement is based on collective rationality, i.e. 
on the assumption that rational individuals will cooperate in pursuing their 
common interests if the conditions of the game permit them to do so. Harsanyi 




101 



R 
S 

Figure 1: stag hunt 



R S 



9,9 


0,8 


8,0 


7,7 



and Selten argue that risk dominance is only important in those cases where 
there is some uncertainty about which equilibrium “should” be selected. If 
one equilibrium gives all players a strictly higher payoff than any other equi- 
librium (and if this equilibrium satisfies all other desirable properties that 
the selection theory imposes) such uncertainty will not exist - each player 
can be reasonably certain that all other players will opt for this equilibrium 
- and this makes risk-dominance comparisons irrelevant. It is this argument 
that leads Harsanyi and Selten to give precedence to payoff dominance. 

Yet, relying on collective rationality is somewhat unsatisfactory. For one, 
it implies that the final theory is not ordinal, that is, two games with the same 
best reply structure need not have the same solutions. For example, the game 
from Figure 1 is best-reply-equivalent to one in which the off-diagonal payoffs 
are zero and in which the payoffs to {R,R) and (5,5) are (1,1) and (7,7) 
respectively, and, in the latter, payoff dominance selects (7,7) as the outcome. 
Secondly, one feels that it should be possible to obtain collective rationality 
as an outcome of individual rationality: If one equilibrium is uniformly better 
than another, then the players’ individual deliberations should bring them to 
play this equilibrium. The reason that this does not happen in a game as that 
from Figure 1 - at least if one views the risk-dominant equilibrium as the 
outcome of the individual deliberation process - is that indeed a player is not 
sufficiently certain ex ante that his “partner” will play the risky equilibrium. 
In fact, he cannot be sure of this exactly because of the fact that his partner 
cannot be sure that he will play it: one only needs a “grain of doubt” in order 
for it to be the unique rationalizable strategy to play safe. (See Carlsson and 
Van Damme (1993ab), and, for an informal argument to that extent, Schelling 
(1960, Chapter 9).) 

The above raises the question of whether, in games in which players can 
indeed be quite certain that the opponents will play the payoff dominant 
equilibrium, or at least in games in which this equilibrium is uniquely focal, 
risk-dominance considerations will induce players to play this equilibrium. 
This note aims to address this issue, and it provides a negative answer. We 
consider team games, that is, games in which all players have the same pay- 
off function. Any maximum of this function is trivially an equilibrium and 
one might argue that, in those cases in which the maximum is unique, this 
maximum provides the unique focal equilibrium of the game. In other words, 
under those situations the conditions are most favorable for individual ra- 
tionality to be in agreement with collective rationality. We provide examples 




102 



to illustrate that, even in these cases, risk-dominance considerations do not 
necessarily lead to the playing of the unique payoff dominant equilibrium. 
Specifically, we show that the modified Harsanyi/Selten theory, that does 
not invoke payoff comparisons, may select a Pareto dominated equilibrium in 
a team game.^ 



2. Notation and Definitions 

In this section we introduce notation, and define team games and the risk 
dominance relation. Readers already familiar with these concepts can imme- 
diately turn to Section 3. 

Let r =< Ai^Ui >iei be a strategic form game. I is the player set, Ai is 
the set of pure strategies of player i and U{ : A R is the payoff function 
of this player {A = Ui^iAi). We write Si for the set of mixed strategies of 
player ^, S = Ui^iSi for the set of mixed strategy profiles and Ui{s) for the 
expected payoff to player i for when s G 5 is played. A strategy profile s* is 
a (Nash) equilibrium of F if no player can improve his payoff by a unilateral 
change in strategy, i.e. 



Ui(s*) = max Si), (2.1) 

Si £Si 

where is shorthand notation for (s*, . . . , Sj, . . . , s*). We 

write E{F) for the set of equilibria of F. The (linear) tracing procedure is 
a map T from S into E{F), hence, it converts each mixed strategy profile 
into an equilibrium of the game. Formally the map T is associated with a 
homotopy. For s G 5 and a homotopy parameter t G [0, 1] write for the 
game < Ai,u\'^ >iei which the payoff function of player i is given by 



u^’^(a) = tui{a) -f (1 - t)ui{s\ai). 



( 2 . 2 ) 



Hence, for t = 1 we have the original game, while for ^ = 0 each player faces a 
(trivial) one-person problem. In nondegenerate cases, the game contains 
exactly one equilibrium e(0, s) and this will remain an equilibrium of as 
long as t is sufficiently small. Now it can be shown that the equilibrium graph 



£; = {(i,s):<G[o,i], se£;(r‘-*)} (2.3) 

^ We will not spell out the full details of the Harsanyi and Selten (1988) theory 
The reader is referred to the flowchart on p. 222 of their book for a quick 
overview of the theory. In the examples we take certain shortcuts through this 
flowchart. We leave it to the reader to prove that these shortcuts are justified. 




103 



contains a unique distinguished curve {e{t,s) : ^ G [0, 1]} that connects 
e(0,s) with an equilibrium e(l,s) of F. (See Harsanyi and Selten (1988) 
and Schanuel et al. (1992) for the technical details. In particular, the latter 
paper points out how e(l,5) can be found by applying the logarithmic trac- 
ing procedure.) The endpoint e(l,s) of this path is the linear trace T{s) of 
s. This tracing map T is used to define the risk-dominance relation. 

Imagine that the players are uncertain about which of two equilibria, s* 
or s**, should be considered as the solution of the game. Player i assumes 
that his opponents already know it and he himself attaches probability Z{ to 
the solution being s* and the complementary probability 1 — to the solution 
being s**. Obviously, in this case he will play his best response against the 
correlated strategy Zis!_^4-(1— of his opponents. Assume that if player i 
has multiple best responses, he plays each of them with equal probability and 
denote the resulting centroid best reply by bi{zi]s*,s**). Now, an opponent 
j of i does not know z’s beliefs Z{. Assume that, according to the principle of 
insufficient reason, such a player considers Z{ to be uniformly distributed on 
[0, 1]. Clearly, an opponent will then predict i to play the mixed strategy 



si(s* 



•>=/ 



bi{zi]s*,s**)dzi. 



(2.4) 



Let s{s*, s**) be the mixed strategy vector determined by (2.4), the so called 
bicentric prior associated with s* and s**. We now say that: 

(i) s* risk dominates s** if T{s{s* ,s**)) = 5*, 

(ii) s** risk dominates s* if T(s(s*,s**)) = s**, and 

(iii) there is no dominance relation between s* and s** if T(s(s*,s**)) ^ 

We note that the risk-dominance relation need not be transitive, nor com- 
plete. We also note that a simple characterization of this relation can be given 
for 2-person 2x2 games with two strict equilibria, say 5* and s**. Write z* 
for the critical belief of player i where he is indifferent between both pure 
strategies, that is, bi{z*]s*,s**) = (1/2, 1/2). Then, s* risk dominates s** if 
and only if Zi Z 2 < I- 

The Harsanyi/Selten solution of a game is found by applying an iterative 
elimination procedure. Starting from an initial candidate set (consisting of 
all so called primitive equilibria), candidates that are payoff dominated or 
risk dominated are successively eliminated until exactly one candidate is left. 
We will consider the modification of that theory that only invokes risk domi- 
nance. In both our examples the initial candidate set will simply be the set of 
all pure equilibria of the game. It will be clear from the above definition that 
it can easily happen that there is no risk-dominance relation between pure 
equilibria, say s* and 5 **, and that both are “maximally stable”. In this case 




104 



Harsanyi and Selten propose to replace the pair by one substitute equilib- 
rium, viz. by the equilibrium s*** that results when the tracing procedure is 
applied to the mixed strategy in which each player i chooses l/2s* -f l/2s**. 
(Note that this, so called centroid strategy, typically differs from the bicentric 
prior as determined by (2.4).) Hence, in case of a deadlock with two equally 
strong candidates s* and s**, these equilibria are eliminated from the initial 
candidate set. They are replaced by the equilibrium s*** and the process 
is restarted with this new candidate set. As in both our examples, a single 
equilibrium remains after at most one substitution step has been performed, 
there is no need to go into further details of the process. 

We conclude this section by giving the definition of a team game. F =< 
AijUi >i^i is said to be a team game if U{ = uj for i,j G /, hence, all players 
always have the same payoff. Writing u for this common payoff function, we 
say that the team game is generic if there exists a unique a* € A at which 
u attains its maximum. Attention will be confined to symmetric games, i.e. 
the payoff to a player depends only on which actions are chosen and not on 
the identities of the players choosing them. Because of this symmetry we can 
confine ourselves to analyzing the situation from the standpoint of player 1. 
In the next two sections we investigate risk dominance in generic symmetric 
team games and compute the associated (modified) solutions. 



3. A Two-Person Example 

The discussion in this section is based on the 2-person game from Figure 2. 
(The right hand side of the picture displays the best reply structure associated 
with this game.) 





T 


M 


B 


T 


3 


0 


2 


M 


0 


3-e 


2 


B 


2 


2 


2 4- e 




Figure 2: A Symmetric Two-Person Team Game (0 < e: < 1) 



The game has three (pure) equilibria with payoffs 3, 3-e and 2 4- e. We first 
investigate the risk dominance relationship between T and B. Note that when 
a player is uncertain whether the opponent will play T or B, but is certain 
that this player will not choose M, then this player will never be tempted to 
choose M since M is never a best response against a mixture zT 4- (1 — z)B. 
This shows that M is irrelevant for the risk-dominance relationship between 
T and B and that this relationship can be determined simply in the 2x2 




105 



reduced game spanned by T and B. Now, as is shown by the RHS of Figure 
2, the bicentric prior relevant for this risk-dominance relation assigns almost 
all weight to T, hence, the tracing path starts at T and it stays there: T risk 
dominates B. A similar argument establishes that M risk dominates B. 

It remains to investigate the risk-dominance relation between T and M. 
As is shown by the RHS of Figure 2 the bicentric prior relevant for the com- 
parison of these equilibria, is approximately equal to (1/3, 1/3, 1/3). It follows 
that the unique equilibrium of the game is {B,B). Since {B, B) is a strict 
equilibrium of it is a strict equilibrium of jT^’^ for all t € [0, 1], hence, the 
distinguished curve in the graph E is constantly equal to (B,B). Therefore 



T{s{T,M)) = B 



(3.1) 



and there is no risk- dominance relation between T and M. Hence, there is 
a deadlock: Both T and M dominate B, but T and M are equally strong. 
To resolve the deadlock we apply the substitution step, hence, we start the 
tracing procedure with the prior 1/2T -f 1/2M. Again, as the RHS of Figure 
2 makes clear, the unique best response against this prior is B, so that the 
tracing path starts at B and remains there. The substitution set eliminates 
the pair {T, M} and replaces it with the equilibrium B. Hence, if we modify 
the theory of Harsanyi and Selten, by not imposing the payoflF dominance 
requirement, then the equilibrium B is selected in the game of Figure 2. 
Individual rationality, as incorporated into risk dominance, does not lead to 
collectively efficient outcomes, not even in generic symmetric team games. 

At the intuitive level, one may explain the phenomenon as follows. Risk 
considerations favor the selection of equilibria that give "reasonably good” 
payoffs against a set of diffuse priors: A player does not know what the others 
will do and he investigates what action gives good outcomes no matter what 
the others do. These considerations favor actions that have large stability sets, 
i.e. that are best responses against many mixed strategies of the opponents. 
In Figure 2, B is such a good and safe strategy. In fact, we could make the 
stability set of B to cover almost the entire strategy simplex without losing 
the fact that T is the unique payoff dominant equilibrium: just replace 2 by 
3-2e everywhere in the payoff matrix. By increasing the payoff associated to 
jB, one makes B more attractive, hence, at the same time T is made less 
attractive. What this makes clear is that there is nothing special about team 
games. Either one assumes that the logic of common payoffs and collective 
rationality is so strong that players do not have any doubt to start with about 
what to play, or one allows for prior doubt and then one does not see how 
the common payoff assumption helps to reduce it. 

The doubt concerning what equilibrium to play may, for example, arise 
out of slight payoff uncertainty as in Carlsson and Van Damme (1993a,b). 
One may imagine that it is common knowledge among the players that they 




106 



are playing a team game, but each player may have a tiny bit of private 
information about what the actual payoffs are. If the uncertainty is small, 
then, if the actual game is generic, the strategy combination that attains 
the maximum will be mutually known to a high degree. However, around a 
nongeneric game the latter will not hold. As Carlsson and Van Damme show, 
such nongeneric games exert an influence on “far removed” generic games: 
since players choose safe strategies in non-generic games, they are forced to 
choose safe strategies also in generic games, in order to avoid coordination 
failures. 

To illustrate this argument, consider the modification from Figure 2 as 
in Figure 3. This is a nongeneric game. As Schelling (1960, Appendix C) 
already argued, the unique focal equilibrium in this game is .H: If players 
cannot communicate, then, if they aim to coordinate on T or M, they will 
actually succeed only with 50% probability and, hence, it is better to play 
B (5/2 > 1/2-3 + 1/2-0). Now, consider a game that is close to the one from 
Figure 3, but that is generic and that has a unique maximum associated with 
T. Should a player play T? Well, if he is not exactly sure that he observed 
the correct payoffs the answer is: maybe not. In that case, the actual payoffs 
may be such that the maximum is at M, or that his opponent thinks that 
the maximum is there. In such a case, it might still be better to choose B 
and thereby avoid the coordination problem. 





T 


M 


B 


T 


3 


0 


2 


M 


0 


3 


2 


B 


2 


2 


5/2 



Figure 3: A Coordination Game 



To conclude this section, we give one individualistic rationality argument 
that, for 2-person team games, distinguishes the payoff maximal equilibrium 
from any other pure equilibrium. Is it true that in such games, the pay- 
off maximal equilibrium pairwise risk dominates any other pure equilibrium. 
Specifically, if s* is the Pareto dominant equilibrium and s** is any other pure 
equilibrium, then in the 2x2 game in which the players only have {s*,s**} 
available, s* risk dominates s**. Hence, the Pareto dominant equilibrium may 
be said to be the pairwise risk-dominant one. A proof is simple and uses the 
alternative characterization of risk dominance for 2x2 games given in the 
previous section: Since the sum of the off-diagonal payoffs is smaller than 
the sum of the diagonal (equilibrium) payoffs, the sum of the players’ critical 
probabilities for switching away from the payoff maximal equilibrium is less 
than one. An illustration is provided by the reduced games associated with 




107 



the equilibrium T from Figure 2, which are displayed in Figure 4. In the game 
on the LHS, T dominates M, while in the game on the RHS, T dominates 



B, 



T M T B 



T 


3 


0 


T 


3 


2 


M 


0 


3-e 


B 


0 


2-e 



Figure 4: Reduced Gaines Associated with Figure 2 



It should be noted that, in general, the concept of pairwise risk domi- 
nance captures the overall risk situation rather badly, see Carlsson and Van 
Damme (1993b). This is also evident from the LHS of Figure 4: when a 
player believes that his opponent may play T or M, then he has an incentive 
to play however, B is not present in the reduced game. In this respect 
it is also interesting to refer to the relationship between risk dominance and 
the stochastic stability of equilibria in an evolutionary context (Kandori et 
al. (1993), Young (1993)). In symmetric 2x2 games only the risk-dominant 
equilibrium is stochastically stable, but Peyton Young already provided an 
example of a 3 x 3 game in which the stochastically stable equilibrium differs 
from the pairwise risk-dominant one. Recently, Kandori and Rob (1993) have 
shown that for 2 player symmetric games that satisfy the Total Bandwagon 
Property (TBP) and the Monotone Share Property (MSP) only the pairwise 
risk-dominant equilibrium is stochastically stable. (TBP says that any best 
response against a mixture is an element of the mixture; MSP says that if a 
pure strategy is eliminated, the shares of all other pure strategies increase in 
the completely mixed equilibrium; the game from Figure 2 violates the Total 
Bandwagon Property.) 



4. A Three-Person Example 

The above two-person example is somewhat unsatisfactory since the solution 
process involves using the tie-breaking procedure and the latter might be 
considered ad hoc. The aim of this section is to provide a three-person team 
game that has a non-payoff maximal equilibrium that strictly risk dominates 
any other pure equilibrium. In fact, the example has the additional (desir- 
able) feature that the details of the tracing procedure do not matter: since the 
game is symmetric, the risk-dominant solution is determined directly from 
the bicentric prior (2.4). Finally, the example shows that for 3-player games 
the Pareto dominant equilibrium need not be even pairwise risk dominant. 




108 



The example in question is the game given in Figure 5. (Here x is a real num- 
ber in the interval [0, 1], player 1 chooses a row, 2 a column and 3 a matrix. 
Obviously, the payoff dominant equilibrium is L if x > 1/3, while it is R if 
X < 1/3). 



L R L R 



L 


3x 


2x 


L 


2x 


X 


R 


2x 


X 


R 


X 


1 



L R 



Figure 5: The Three-Person Team Game F{x) (0 < x < 1) 



It is easily seen that F(x) has two strict Nash equilibria, viz. (L, L, L) and 
(ii, i2, i?). To determine the risk-dominance relationship between these two 
equilibria, we compute the bicentric prior as in (2.4). If the players 2 and 3 
play (L, L) with probability 2 : and (i2, R) with probability 1 — z, then player 
1 strictly prefers to play L if and only if 

zx > {1 — z){l — x). 



or, equivalently. 



2 ; > 1 — X. (4.1) 

Applying the principle of insufficient reason, the players 2 and 3 attach a 
probability x to the inequality (4.1) being satisfied. Hence, the prior beliefs 
of the players 2 and 3 are described by player I’s mixed strategy 



xL + (1 — x)R 



(4.2) 



Since the game F(x) is symmetric with respect to the players, the mixed 
strategy (4.2) actually is the prior of each player i G {1,2,3}. To determine 
the risk-dominance relationship between {L,L,L) and (iZ, i?, i?), we have to 
determine the best response of player i when his opponents j and k indepen- 
dently randomize according to (4.2). The reader easily verifies that L is the 
unique best response if and only if 



x(l ~ (1 - x)2) > (1 - x)^ 




109 



or, equivalently, 



— 3a: + 1 < 0 (4.3) 

Hence, if x € then {L,L,L) is the risk-dominant equilibrium of 

the game. Now > 1/3, so that, there exists a range where the risk- 

dominant solution is {R, R, R , ) even though the payoff dominant equilibrium 
is (L, L, L). 

We conclude this section by noting that also the equilibrium selection 
theory that has recently been proposed in Harsanyi (1995) and that involves 
a multilateral risk comparison of equilibria that is not based on the tracing 
procedure, does not always select the payoff dominant equilibrium in team 
games. Harsanyi proposes to select that equilibrium that has the largest sta- 
bility region. As was already indicated in Section 3, the payoff dominant 
equilibrium may have a rather small stability region. Also for the example 
discussed in this section, the reader may verify that the equilibrium with the 
largest stability region, need not be payoff dominant. (Harsanyi defines the 
stability region of a strategy as the set of correlated strategies of the oppo- 
nents against which the strategy is a best response, hence, the stability region 
of L is the set in the three-dimensional unit simplex where p{R, R) < x. He 
does not just take the Lebesgue measure of this set but first applies a trans- 
formation of the simplex.) 



5. Conclusion 

Prom one point of view one might argue that a generic team game is simple to 
play: Firstly, the payoffs of the players coincide so that there is no conflict of 
interest; secondly, the payoff function admits a unique maximum so that there 
is no risk of confusion. Hence, one might say that the unique payoff dominant 
equilibrium is the unique focal point: Players might view the game just as a 
one-person decision problem and solve it accordingly. These arguments are 
even more compelling for symmetric games. Upon closer inspection it turns 
out, however, that the above argument is not entirely convincing: The Pareto 
dominant focal point is not robust and this is reflected in the fact that risk- 
dominance considerations need not select it. Hence, even in symmetric team 
games, collective rationality need not be implied by individual rationality. 
With Harsanyi and Selten (1988, p. 359) we may conclude that “if one feels 
that payoff dominance is an essential aspect of game theoretic rationality, 
then one must explicitly incorporate it into one’s concept of rationality.” 
Nevertheless, there is a sense in which team games are different from 
games in which the players’ payoff functions do not coincide. Aumann (1990) 




110 



argued that, in the stag hunt game of Figure 1, even communication cannot 
help to bring about the equilibrium {R, R) in case players are convinced a 
priori that (5, 5) is the solution of the game without communication. When 
communication is possible each player will always (i.e. no matter how he 
intends to play) suggest the other to play R since he can only benefit by 
having the other do so. Consequently, no new information is revealed by 
communication, hence, communication cannot influence the outcome. One 
may argue that things are somewhat different if it is common knowledge that 
the game is a team game: In this case a player has no incentive whatever to 
suggest an outcome different from the payoff maximal one. It remains to be 
investigated whether risk dominance selects the payoff dominant equilibrium 
if one, or more, rounds of preplay communication are added to the game. 

At a somewhat more abstract level one may raise the question of why 
payoff dominance and risk dominance may be in conflict even in team games. 
I conjecture that this is because of the ordinality property of the risk- 
dominance concept. Hence, the conjecture is that there exist two team games 
with the same best reply correspondence that have different payoff-dominant 
equilibria. Thus far, I have not been able to formally prove this conjecture. 



References 



Aumann, R.J. (1990). “Nash Equilibria are not Self- Enforcing” , in: J.J. Gabszewicz, 
J.-F. Richard and L.A. Wolsey (eds.), Economic Decision-Making: Games, 
Econometrics and Optimisation 201-206. 

Caxlsson, H., and E. van Damme (1993a). “Global Games and Equilibrium Selec- 
tion”, Econometrica 61, 989-1018. 

Carlsson, H., and E. van Damme (1993b). “Equilibrium Selection in Stag Hunt 
Games”, in: K. Binmore, A. Kirman and P. Tani (eds.). Frontiers of Game 
Theory, MIT Press, Cambridge. 

Harsanyi, J. (1995), “A New Theory of Equilibrium Selection for Games With 
Complete Information”, Games and Economic Behavior 8, 91-122. 

Harsanyi, J.C., and R. Selten (1988). A General Theory of Equilibrium Selection 
in Games, MIT Press, Cambridge, MA. 

Kandori, M., G. Mailath, and R. Rob (1993). “Learning, Mutation and Long Run 
Equilibria in Games”, Econometrica 61, 29-56. 

Kandori, M. and R. Rob (1993). “Bandwagon Effects and Long Run Technology 
Choice” . Discussion Paper 93-F-2, Faculty of Economics, University of Tokyo. 

Schanuel, S.H., L.K. Simon and W.R. Zame (1991). “The Algebraic Geometry 
of Games and the Tracing Procedure”, in: R. Selten (ed.). Game Equilibrium 
Models, Vol. 2: Methods, Morals and Markets, Springer Verlag, 9-43. 

Schelling, T. (1960). The Strategy of Conflict, Harvard University Press. 

Young, P. (1993). “The Evolution of Conventions”, Econometrica 61, 57-84. 




Sustainable Equilibria in 
Culturally Familiar Games 

Roger B. Myerson 

J. L. Kellogg Graduate School of Management, Northwestern University, Evanston, 
Illinois 60208, U.S.A. 



It is a pleasure to write for a volume in honor of Reinhard Selten, who has been to 
me both a mentor and a friend. Like many others, I am indebted to him for guidance 
and inspiration in many domains, from the logical study of rational behavior, to the 
pleasures of walking in the German countryside. 

The definition of equilibrium was offered by Nash (1951) as the basic 
characterization of rational behavior in games, but Selten (1965, 1975) showed that 
the set of Nash equilibria may be too weak to characterize rational behavior in many 
games. That is, there may be strategic scenarios which pass the test of being Nash 
equilibria, but which do not fit our intuitive sense of what rational behavior should 
be in a game. So Selten launched the literature on refinements of Nash equilibrium, 
one of the great central strands of the literature of noncooperative game theory. 

To evaluate a refined equilibrium concept, we may apply four criteria. First, the 
definition of the refined equilibrium concept should have some intuitive appeal as a 
characterization of what is rational behavior. Second, in games where we have 
strong intuitions about what are the reasonable equilibria, the refined solution 
concept should coincide with our intuitions, eliminating the unreasonable equilibria 
but not eliminating the reasonable ones. Third, the set of refined equilibria should 
be nonempty for all games to which this solution concept is supposed to be applied. 
Fourth, our refined solution concept should be invariant under transformations of the 
game that seem intuitively irrelevant. No solution concept in the literature has yet 
fully satisfied all of these criteria (even as far as we can reach a consensus about 
what is "intuitively reasonable"), and so the search for new refinements of the 
equilibrium concept continues. In this essay, I want to sketch one problem area in 
which more work on refinements of Nash equilibrium seems particularly needed. 

My intuition about this problem comes from consideration of the well-known 
"Battle of the Sexes" game: 



Player 2 
^2 ^2 

Player 1 ai 3,2 0,0 

bi 0,0 2,3 



Table 1. The Battle of the Sexes Game 






112 



This game has three equilibria: (ai,a2) which gives payoffs (3,2), (bi,b2) which gives 
payoffs (2,3), and a mixed equilibrium (.6[aj ]+ .4[bJ, .4[a2] + 6[b2]) which gives 
expected payoffs (1.2, 1.2). I believe that we need a refinement of Nash equilibrium 
that excludes the mixed equilibrium and includes only the two pure-strategy 
equilibria. 

If this game were played only once, by a pair of subjects in an experimental 
laboratory setting, then I would consider the mixed equilibrium to be a very 
reasonable characterization of how rational players might behave. Indeed, given the 
obvious symmetry of the game, one might plausibly argue that it would be the only 
reasonable characterization of rational behavior in such a situation, where the 
players have no history to guide them. 

But most games that interest us are not played in such isolation. More 
commonly, players are aware of a history of similar games that have been played by 
similar players, and this shared cultural history may crucially influence the players' 
expectations about how the game will be played in the future. For this Battle of the 
Sexes game, if the distinct roles of "player 1" and "player 2" are identified in the 
historical record of past outcomes, then the history of previous play can break the 
symmetry of the situation in favor of either player 1 or player 2, and thus can focus 
the players on one of the two pure-strategy equilibria. 

This point about cultural history is implicit in the story from which this game 
derives its name. We may imagine that this game is played by couples on their first 
date, where player 1 is the man, player 2 is the woman, aj and a2 denote the strategy 
of going somewhere that the man prefers, and bj and b2 denote the strategy of going 
somewhere that the woman prefers; but both want most to be together. In a world 
where such couples really must make their decisions simultaneously and separately, 
I very much doubt that they would persist in the kind of uncoordinated confusion 
that we find in the mixed equilibrium. Surely a social expectation would evolve that 
steers such couples to one of the two pure-strategy equilibria. Cultural expectations 
favoring the woman (bj,b2) might evolve in one society, while expectations favoring 
the man (aj,a2) might equally well evolve in another cultural context. Once cultural 
expectations have focused on either pure-strategy equilibrium, however, Schelling's 
(1960) focal-point effect would operate, leading all future players to reproduce the 
expected pattern of behavior, because each player would understand that deviation 
from the cultural norms would reduce his or her payoff to zero. 

It is hard to see how cultural expectations of the mixed equilibrium could be 
sustained, however. For example, if the outcome (bi,b2) occurred 100 times in a row 
(as must eventually happen in a sufficiently long history of plays of the mixed 
equilibrium), then surely each player would increase his or her subjective probability 
of the other player choosing bi or b2 at the next play of the game. But when this 
subjective probability is increased above the probability in the mixed equilibrium, 
then choosing b2 or b^ would become optimal for each player, and so the repetition 
of the (bi,b2) outcome should continue. Thus any evidence that pushes the players 
slightly away from the mbced equilibrium can ultimately move them all the way to 
one or the other of the pure-strategy equilibria. 




113 



When we say that a game is played in a culturally familiar context, we are not 
assuming that anybody plays this game more than once. We are assuming that the 
game will be played in an infinite sequence of times, and the history of previous 
plays will be known by the players each time the game is repeated. But to guarantee 
that the payoffs in the matrix fiilly describe the effect that each player's strategy can 
have on his own payoff, we must also assume that each individual plays the game 
only once, so there are two new players in each repetition of the game. That is, we 
are assuming that, at each repetition of the game, no player has any incentive to 
develop a reputation or to otherwise affect the expectations that will be held in future 
plays of the game. 

Another version of such cultural evolution has been suggested by Woodford 
(1990). Woodford's model is posed in a price-theoretic framework, but we can 
describe here a simplified game-theoretic version of the same story. Suppose that a 
payoff-irrelevant random variable (say, with finitely many possible values) is drawn 
independently from some fixed distribution before each play of the game, and the 
value of this random variable is announced to the players and is recorded along with 
their strategy choices in the published history. Then it is possible that equilibrium 
expectations could be based on a shared perception that the only relevant part of past 
history is in the cases where the value of the random variable was the same as its 
current value. One possible equilibrium scenario is that, for each possible value of 
the random variable, the players follow the mixed equilibrium until (ai,a 2 ) or (bi,b 2 ) 
occurs in a round with this random value, and thereafter they play the same 
pure-strategy equilibrium every time this random value is observed. So in this story 
the cultural expectations will lead to behavior that might appear random to an 
outside observer who ignored the payoff-irrelevant random variable; but the 
outcomes will eventually all be either (aj,a 2 ) or (bj,b 2 ), which is certainly not the 
pattern of outcomes for the mixed equilibrium ( 6[ai]+.4[b|],.4[a2]+.6[b2]) (where 
(ai,b 2 ) happens 36% of the time, for example). 

Thus, when the Battle of the Sexes game is played as a culturally familiar game 
(with the roles of players 1 and 2 being distinctly identifiable across plays), we may 
expect that the players should have evolved a pattern of expectations which rules out 
the mixed equilibrium but admits either of the pure-strategy equilibria. For now, let 
us say informally that the two pure-strategy equilibria seem to be "sustainable" in a 
way the mixed equilibrium does not. This description begs the question of what 
kind of formal solution concept can we define to identify the "sustainable" equilibria 
that might be expected to persist in any strategic-form game, when it is played in a 
culturally familiar context. That is, how should we extend this intuitive concept of 
"sustainability" from the Battle of the Sexes game to a general refinement of Nash 
equilibrium that can be applied to any strategic-form game? 

One naive approach would be to view this as a question of distinguishing 
between pure-strategy equilibria and randomized-strategy equilibria. However, 
some games have no pure-strategy equilibria, and so the set of pure strategy 
equilibria cannot be the refinement of Nash equilibrium that we are seeking. Even 




114 



the rule "eliminate properly randomized or mixed equilibria when pure-strategy 
equilibria exist" does not seem right, as the following example illustrates. 







Player 2 








^2 


bz 


C 2 


Player 1 




4,0 


0,4 


3,0 




bi 


0,4 


4,0 


3,0 




Cl 


0,3 


0,3 


3,3 



Table 2. A game where the pure-strategy equilibrium seems wrong. 



The unique pure-strategy equilibrium is (Ci,C 2 ), but this equilibrium is in weakly 
dominated strategies and so seems unlikely to persist. If the players perceive that 
there is any positive probability of either player using another strategy, even by 
accident, then neither player can rationally choose Cj or C 2 . 

Another approach would be to focus on the fact that, in the Battle of the Sexes 
game, the two "sustainable" equilibria are both Pareto-superior to the unsustainable 
mixed equilibrium. But the unsustainable (Cj,C 2 ) equilibrium in Table 2 is actually 
Pareto-superior to the more reasonable mixed equilibrium (5[ai]+.5[bi], 
.5[a2]+.5[b2]) for this game. (The game may be viewed as a modified Prisoners’ 
Dilemma game.) Furthermore, consider the following "Stag Hunt" game (adapted 
from a similar game discussed by Harsanyi and Selten (1988) in Section 10. 12). 



Player 2 
^2 ^2 

Player 1 aj 6,6 0,4 

bi 4,0 3,3 

Table 3. The Stag Hunt Game 

This game has three equilibria: a pure-strategy equilibrium (aj,a 2 ) which gives 
payoffs (6,6), a pure-strategy equilibrium (bi,b 2 ) which gives payoffs (3,3), and a 
mixed equilibrium (.6[ai]+.4[bi],.6[a2]+.4[b2]) which gives expected payoffs (3.6, 
3.6). In this game, I would argue that the two pure-strategy equilibria are both 
sustainable, even though one of them (bi,b 2 ) is Pareto-inferior to both of the other 
two equilibria. In a culture where everyone expects others to choose bj or b 2 with 
high probability, the introduction of a small probability of deviation to a, or a 2 
would not cause players to rationally switch away from bj or b 2 . Of course, the 
same could be said about the Pareto-superior (aj,a 2 ) equilibrium: In a culture where 
everyone expects others to choose aj or a 2 with high probability, the introduction of 
a small probability of deviation to bi or b 2 would not cause players to rationally 
switch away from aj or a 2 . On the other hand, the mixed equilibrium does seem 
unstable. In a dynamic environment where people initially expect 60% aj and 40% 
bj choices, any small increase in the perceived rate of choice of either action would 
lead others to further increase their rational selection of that action (because a, is best 







115 



for player 1 if the probability of a 2 is anything more than 60%, whereas bj is best for 
player 1 if the probability of a 2 is anything less than 60%). So any perceived 
deviation from the mixed equilibrium should induce larger deviations in the same 
direction until there was a general convergence to one of the two pure-strategy 
equilibria. Thus, my intuition is that both the (bi,b 2 ) equilibrium and the (a,,a 2 ) are 
"sustainable" equilibria in Table 3, but the mixed equilibrium is not. 

In the terminology of Harsanyi and Selten (1988), the (bi,b 2 ) equilibrium in 
Table 3 risk dominates the (aj,a 2 ) equilibrium. Indeed, each player would prefer to 
choose his b-strategy whenever the other player's probability of choosing the 
b-strategy is anything more than 40%. The model of Kandori, Mailath, and Rob 
(1993) predicts that the (bi,b 2 ) equilibrium will be played infinitely more often than 
the (ai,a 2 ) equilibrium, in the long run of a dynamic society where different pairs 
play this game in an infinite sequence. However, I would argue that our 
"sustainability" solution concept should include both (aj,a 2 ) and (b,,b 2 ) for the game 
in Table 3, without distinguishing either pure-strategy equilibrium as more 
sustainable than the other, but the concept should exclude the mixed equilibrium of 
this game. I believe that, for example, if we are looking for simple game-theoretic 
models to help us better understand the problems of economic development that 
separate rich nations from poor nations, then there may be no better place to start 
than with this Stag Hunt game, and with the recognition that a culture could persist 
in either the Pareto-superior (aj,a 2 ) equilibrium or the Pareto-inferior (bi,b 2 ) 
equilibrium. 

Kohlberg and Mertens's (1986) concept of stability does not exclude any of the 
three equilibria in either the Battle of the Sexes or the Stag Hunt game. For each of 
these two games, any sufficiently small perturbation of the payoffs would still leave 
a game that has a mixed equilibrium close to the mixed equilibrium of the given 
game. Both Selten's (1975) concept of trembling-hand perfect equilibrium and 
Myerson's (1978) concept of proper equilibrium also admit all three equilibria of 
these games. In particular, the mixed equilibrium is perfect and proper for each of 
these two games, because any randomized equilibrium that assigns positive 
probability to all pure strategies is perfect and proper. 

There is at least one well-known refinement of Nash equilibrium in our 
published literature which does what we want for both the Battle of the Sexes and 
the Stag Hunt games: Kalai and Samet's (1984) concept of persistent equilibrium . 
For both of these games, the two pure-strategy equilibria are persistent equilibria but 
the mixed equilibrium is not. Furthermore, Kalai and Samet have proven the 
general existence of persistent equilibria for all finite strategic-form games. Other 
variations on persistence have been discussed in the literature, such as Harsanyi and 
Selten's (1988) concept of equilibria in primitive formations, and Basu and 
Weibull's (1991) concept of equilibrium in minimal curb sets, which (as Van 
Damme, 1994, has remarked) is equivalent to persistence for generic strategic-form 
games. 

Let us denote a general finite strategic form game by 
r = (N, (Ci)i^N, (Ui)ieN) 




116 



where N is the set of players, Cj is the set of pure strategies for player i, and Uj is the 
utility function for player i (mapping strategy profiles in Cj to real numbers). 
Given any finite game in strategic form, let us say that a block is any profile of 
nonempty subsets of all players' pure strategy sets. That is, B = (Bi)jgN is a block if 
each Bi is a nonempty subset of Cj. A block is a curb set if, for every profile of 
randomized strategies that assign zero probability to all strategies outside the block, 
the best responses of each player are all within the block. ("Curb" is an abbreviation 
for "closed under rational behavior.") In the analysis of simple strategic form 
games, a formation is the game restricted to a curb set that is also closed under best 
responses when correlated strategies are taken into account. A block is a retract if 
there exists some positive number e such that, for every profile of randomized 
strategies that assign less than e probability to all strategies outside the block, each 
player has a best response that stays in the block. We may say that a retract or curb 
set or formation is minimal if it is does not contain any subsets that share this 
property. A minimal formation has been called primitive by Harsanyi and Selten 
(1988), and a minimal retract has been called persistent by Kalai and Samet (1984). 

A persistent equilibrium is an equilibrium that assigns positive probability only to 
strategies that are in one minimal retract. 

Our interest in persistent retracts and primitive formations can be motivated by 
the following evolutionary story. When by chance the strategies outside of some 
formation or retract happen to be very rarely used during a long interval, then the 
players should begin to believe that these outside strategies would be unlikely in the 
future, and this assumption (by the definition of a formation or retract) would tend to 
confirm itself. So, during the long history of repetitions, the set of strategies that the 
players consider likely should eventually shrink until it coincides with some minimal 
set that is self-confirming (in some appropriate sense). 

For both the Battle of the Sexes game and the Stag Hunt game, the retracts (and 
the curb sets, and the formations) are ({aijbJ, {a 2 ,b 2 }), ({ai},{a 2 }), and ({bi},{b 2 }). 
So ({ai},{a 2 }) and ({bi},{b 2 }) are the persistent retracts (and the primitive 
formations). Thus, (aj,a 2 ) and (bj,b 2 ) are persistent equilibria. But support of the 
mixed equilibrium is not contained in either persistent retract and so the mixed 
equilibrium is not a persistent equilibrium. 

Following an argument of Kohlberg and Mertens (1986), Van Damme (1994) 
has emphasized that the concept of persistent equilibrium is not invariant under 
substitution of equivalent subgames, however. For example, consider the following 
simple game. 



Player 2 
X2 Jl 

Player 1 x^ 4,-1 -4,1 

yi -4,1 4,-1 



Table 4. A Game with Unique Equilibrium Payoffs (0,0) 






117 



This game has an equilibrium (.5[xi] + .5[y,], .5[x2] + 5[y2]) which gives expected 
payoffs (0,0), and there are no other Nash or correlated equilibria of this game. Now 
consider again the Battle of the Sexes game, but suppose that the (0,0) payoffs in 
that game really represent situations in which the players must go on to play the 
game in Table 4. Then we get an elaborated version of the Battle of the Sexes game 
with can be represented in strategic form as shown in Table 5. 

Player 2 

ax2 ay2 

Player 1 axj 3,2 3,2 

ayi 3,2 3,2 

bxi 4,-1 -4,1 

byi -4,1 4,-1 

Table 5. An Elaborated Battle of the Sexes Game 

In this elaborated Battle of the Sexes game, there are no retracts, formations, or curb 
sets other than the block of all strategies in the game. Thus, all equilibria are 
persistent, including the elaborated version of the original mixed equilibrium 
(.3[axil + .3[ayJ + .2[bxi] + .2[byi], .2[ax2] + .2[ay2] + .3[bx2] + .3[by2]). 

So the substitution of subgames with unique equilibria, a seemingly irrelevant 
transformation of the game, has transformed and trivialized the set of persistent 
equilibria. Nonetheless, I would argue that the elaborated versions of the (former) 
pure-strategy equilibria 

(.5[axi] + .5[ayi], .5[ax2] + .5[ay2]) and 
(.5[bxi] + .5[byi], .5[bx2] + -5[by2]) 

still deserve to be called "sustainable” in a way that the full-support mixed 
equilibrium does not. 

(One possible way of rescuing the primitive-formation or persistent-retract 
approach has just appeared in a new paper on dual reduction; see Myerson, 1995. 
Using this new technique, the game in Table 5 can be reduced to the Battle of the 
Sexes game in Table 1. So the possibility of applying the persistence refinement 
after dual reduction may deserve further attention in future research.) 

Maynard-Smith's (1982) concept of evolutionary stable strategies has also been 
viewed as a way of distinguishing the two pure-strategy equilibria of the Stag Hunt 
game from the mfaced equilibrium. For any symmetric game two-player game, an 
evolutionary strategy is a pure or randomized strategy a such that, for any other pure 
or randomized strategy t, there is some mbcture of a and x against which the strategy 
a is better than the strategy x. In the Stag Hunt game, the strategy a = .6[ai] + .4[bj] 
fails to be an evolutionary stable strategy because, when player 2 randomizes 
between this strategy and x = [a 2 ], player 1 finds that using ai is never worse than 
using a. 



bx2 by2 

4,-1 -4,1 

-4,1 4,-1 

2,3 2,3 

2,3 2,3 




118 



Player 2 

^2 ^2 ^2 



Player 1 a. 


4,4 


5,0 


0,5 


t>i 


0,5 


4,4 


5,0 


Cl 


5,0 


0,5 


4,4 


Table 6. A game with no ESS. 







Evolutionary stable strategies sometimes do not exist, however. For example, 
the game in Table 6 has a unique equilibrium, in which each player randomizes 
uniformly over his or her three strategies, choosing each with probability 1/3. This 
equilibrium gives each player an expected payoff of 3. But when the number of 
players using the strategy is increased above 1/3, then using Uj becomes better than 
randomizing uniformly over all three strategies. 

Recently there has been wide interest in the study of evolution and adaptive 
dynamics in games (see Mailath, 1992, and Crawford, 1993, for example). It may 
seem obvious that the solution being sought in this paper should be definable as the 
set of limits of some dynamic process of learning and adaptation in repeated plays of 
the game. Unfortunately, no simple adaptive-dynamics models are known for which 
convergence to equilibrium can be proven from generic initial conditions. Without 
such a convergence result, we cannot prove general existence for a refinement that is 
defined as the set of equilibria that can be (robustly) achieved as limits of such 
adaptive dynamics. Furthermore, if the limit points of the distribution of outcomes 
are not at least contained within the set of correlated equilibria, then there must exist 
some simple transformations of behavior (of the form "when the adaptive process 
cues you to strategy do strategy bj instead") that would strictly improve some 
players' average payoffs over arbitrarily long periods of time. 

The tracing procedure (see Harsanyi, 1975, and Harsanyi and Selten, 1988) 
offers another possible approach to the characterization of "sustainability." Given 
any strategic-form game T as above, if p = (Pi)ieN is any profile of randomized 
strategies and t is any number between 0 and 1, then F(t,p) may be defined as the 
game in which there is probability t that all the players are playing F together, but 
there is probability 1-t that each player i is playing separately against machines that 
use the strategies p_i. So F(l,p) is the same as the original game T; but F(0,p) is the 
game in which each player separately faces opponents who are behaving according 
to p. We may say that an equilibrium a can be reached from p if there exists a 
continuous function ©(•) such that 0(t) is an equilibrium of F(t,p), for all t, and 0(1) 
= a. 

It is easy to see that any equilibrium of T is reachable from itself However, the 
mixed equilibrium of the Battle of Sexes game can be reached from only a 
one-dimensional set of alternative conjectures p, whereas each of the two 
pure-strategy equilibria can be reached from a full two-dimensional set of alternative 
conjectures p. Thus, the size of the set of alternative conjectures that can reach an 
equilibrium may offer a way to systematically distinguish "sustainable" from 





119 



"unsustainable" equilibria. However, the computational difficulties of the tracing 
procedure have made it difficult for me to check the general properties of this 
approach. 

I can think of another approach to our problem which is more purely 
mathematical. The generic oddness of the number of Nash equilibria (see Harsanyi, 
1973) is derived from the fixed-point theorems of algebraic topology. To 
understand the intuition behind these theorems, it may be helpful to think about the 
following example. Let y:R -> R be any continuous function such that 
y(x) = + 0 O, and lim^^ y(x) = -oo. 

(For example, think of y(x) = x^ - x.) The graph of y(«) must cross the x-axis (where 
y = 0) an odd number of times because, as we move from negative x to positive x, 
there must be one more crossing up from below than crossings down from above. 
That is, each crossing of the x-axis can be assigned an index value, +1 for each 
crossing up from below (where y(x) is locally increasing in x) and -1 for each 
crossing down from above (where y(x) is locally decreasing), and then the sum of 
the indexes of all crossings must be +1. (Tangencies where the graph does not cross 
the x-axis, as for y(x) = x - x at x = 0, are not counted as crossings, or get index 0.) 

This mathematical heuristic suggests that we might look for some way of 
assigning index numbers to Nash equilibria such that, for all generic games, each 
equilibrium can be assigned the index +1 or -1, and the sum of the indexes of all 
equilibria is +1. If any such theory were applied to the Battle of Sexes game, the 
symmetry of the two pure strategy equilibria should guarantee that they must have 
the same index number, and so the pure-strategy equilibria of this game would have 
index +1 and the mixed equilibrium must have index -1. Thus, generalizing this 
heuristic, we may wonder whether the mathematics of differential topology might be 
able to give us some sophisticated index function such that the sustainable equilibria 
have index +1 and the unsustainable equilibria have index -1. 

We can probe this intuition somewhat further, however, without waiting for 
some magic formula or theorem to drop from the realm of higher mathematics. If 
such an index-number theory existed, then we can also predict how it would apply to 
the following game. 







Player 2 










^2 ^2 


^2 


d2 


Player 1 


ai 


0,6 1,4 


0,3 


1,0 




bi 


1,0 0,3 


1,4 


0,6 



Table 7. A game with three equilibria. 



The optimal strategy for player 2 depends on the probability of player 1 choosing aj 
as follows: a2 is optimal if the probability of aj is greater than .6, b2 is optimal if the 
probability is between .5 and .6, C2 is optimal if the probability is between .4 and .5, 
and d2 is optimal if the probability is less than .4. So this game has just three 
randomized equilibria: 




120 



C6[aJ + .4[bi], .5M + .5M), 

C5[aJ + .5[b,],.5[b2] + .5[cd), and 
(.4[ai] + .6[bJ, .5[c2] + .5[dJ). 

There is an obvious symmetry between the first and third of these equilibria, and so 
they must get index +1, and the middle equilibrium must get index -1. Thus we 
should ask, is there any reasonable sense in which the middle equilibrium of Table 7 
is less "sustainable” than the other two equilibria? The approach of looking for 
primitive formations and persistent retracts would not suggest any distinction among 
these three equilibria, but perhaps this game deserves some further consideration. 

In this paper, I have discussed the importance of trying to define some formal 
concept of "sustainability" of equilibria, to distinguish equilibria that are likely to be 
played in culturally familiar settings. I have not offered any precise formal definition 
of "sustainability," however. Instead, I have tried to describe an open problem, 
which I have pondered for long time. I have occasionally found promising new 
approaches to this problem, but so far none have led to the kind of simple and 
appealing formulation that I feel intuitively must be out there somewhere. Still I 
have hope that tomorrow someone may have the conceptual breakthrough that can 
help to fill this important gap in the game-theory literature. 



References 

Basu, K. and J. W. Weibull, "Strategy Subsets Closed under Rational Behavior," Economics 
Letters 36:141-146 (1991). 

Crawford, V. P., "Introduction to a Special Issue on Adaptive Dynamics," Games and 
Economic Behavior 5:315-319 (1993). 

Harsanyi, J. C., "Oddness of the Number of Equilibrium Points: A New Proof," International 
Journal of Game Theory 2:235-250 (1973). 

Harsanyi, J. C., "The Tracing Procedure," International Journal of Game Theory 4:61-94 
(1975). 

Harsanyi, J. C., and R. Selten, A General Theory of Equilibrium Selection in Games . 
Cambridge, MA: M.I.T. Press (1988). 

Kalai, E., and D. Samet, "Persistent Equilibria," International Journal of Game Theory 
13:129-144(1984). 

Kandori, M., G. J. Mailath, and R. Rob, "Learning, Mutation, and Long Run Equilibria in 
Games." Econometrica 6 1:29-56 (1993). 

Kohlberg, E., and J.-F. Mertens, "On the Strategic Stability of Equilibria," Econometrica 
54:1003-1037(1986). 

Mailath, G. J., "Introduction: Symposium on Evolutionary Game Theory," Journal of 
Economic Theoty 57:259-277 (1992). 

Maynard-Smith, J., Evolution and the Theory of Games Cambridge: Cambridge University 
Press (1982). 

Myerson, R. B., "Refinements of the Nash Equilibrium Concept," International Journal of 
Game Theory 7:73-80 (1978). 

Myerson, R. B., "Dual Reduction and Elementary Games," Northwestern University 
discussion paper (1995). 




121 



Nash, J. F., "Noncooperative Games," Annals of Mathematics 54:289-295 (1951). 

Schelling, T. C., Strategy of Conflict . Cambridge, MA: Harvard University Press (1960). 
Selten, R., "Spieltheoretische Behandlung eines Oligopolmodells mit Nachfragetragheit," 
Zeitschrift fiir die gesamte Staatswissenschaft 121:301-324 and 667-689 (1965). 

Selten, R., "Reexamination of the Perfectness Concept for Equilibrium Points in Extensive 
Games," International Journal of Game Theory 4:25-55 (1975). 

Van Damme, E., "Strategic Equilibrium," (1994). To appear in Handbook of Game Theory, 
Volume 3. edited by R. J. Aumann and S. Hart, Amsterdam: North-Holland. 

Woodford, M., "Learning to Believe in Sunspots," Econometrica 58:277-307 (1990). 




Evolutionary Conflict and the Design of Life 

Peter Hammerstein 

Max-Planck-Institut fur Verhaltensphysiologie, Abteilung Wickler, 82319 Seewiesen, 
Germany 



1. Introduction 

During biological evolution there have been many circumstances under which 
game-like conflict must have occurred among entities that could reproduce, such 
as self-replicating molecules, genes, chromosomes, and organisms. These „games 
of real life“ resulted from the Darwinian struggle for long-term existence. They 
had a major influence on the course of evolution and left their impressive traces in 
the design of organisms. For example, the existence of two different sexes appears 
to be an evolutionary consequence of the severe Darwinian conflict among those 
parts of the genetic material that are not located in the cell’s nucleus (cytoplasmic 
DNA). Without identifying this „cytoplasmic war“, we would not have reached a 
deep understanding of why there are two sexes and why cytoplasmic DNA is 
usually transmitted by one of the sexes only. Similarly, it is impossible to 
understand the behavioural and morphological differences between males and 
females without studying their intra- and intersexual „games‘". Conflict analysis 
turns out to be generally one of the most powerful tools for revealing the 
evolutionary logic of organismic design. This explains the similarity in how 
evolutionary biologists and game-theorists approach their problems. 

In the following I describe some cases of evolutionary conflict with intriguing 
consequences for the organisation of genetic material, cells, and organisms. All 
examples can be understood intuitively and no formal mathematical models are 
presented. I do not even claim that models from classical game theory can be used 
for analysing these cases of conflict. However, it is the „flavour of a game“ which 
certainly can be felt when looking at the biological problems discussed in this 
essay. 



2. Conflict in the Early Days of Life 

We all remember Miller’s famous experiments in which he created artificial 
„lightning“ that struck his laboratory version of the „primitive soup“. He 
demonstrated the formation of proteinogenic amino acids in an experimental 
setting which mimicked conditions on the primitive Earth that might have caused 
the origin of life. However, the relevant context for the origin of life is not well 
enough described by the picture of a „soup exposed to thunderstorms“. Many 
important molecules - especially those needed for membrane formation - have not 




123 



been synthesized in such electric discharge reactions. It seems more convincing 
that the first important steps towards life were made close to hypothermal vents in 
the deep sea, where they probably took place on mineral surfaces, such as that of 
pyrite (Wachtershauser, 1988). The origin of membranogenic lipids can be 
explained for a two-dimensional chemical world on such a surface. An 
appropriate metaphor would be to consider the first forms of life not as the 
„gamish of the primitive soup“ but as the „topping of the primitive pizza“. The 
expression „primitive pizza“ was recently suggested by two biologists (Maynard 
Smith & Szathmary, 1995) with strong culinary inclinations. In their delicate 
description of the origin of life, the pizza’s cheese layer stands for the first lipid 
membrane and bubbles of this layer can perhaps be viewed as primitive 
predecessors of what later in evolution became the cell. 

After this first stage of primitive life, many steps must have occurred before 
the cheese bubbles could evolve into a true cell with all its sophisticated features. 
Szathmdry & Demeter (1987) discuss an intermediate stage in cell evolution with 
the following properties. Their „protocells“ consist of a membrane that forms a 
compartment in which a simple metabolism can take place. In the early protocell 
there are several kinds of molecules, say A, B, and C, which replicate separately 
within the compartment. These replicators are not linked and chromosomes have 
not yet evolved in Szathmary & Demeter’s view of this stage of cell evolution. A 
protocell produces daughter cells by fission, whereby random samples of the 
existing molecules A, B, and C are passed on to each member of the new cell 
generation. The rate of division of a protocell depends on the types and 
proportions of replicating molecules contained. 

Under this assumption, there is conflict among replicators within a protocell 
regarding the speed of their replication. By replicating faster than others, a 
molecule of a given type can increase the average proportion of its own type in 
the daughter cells. This would be true even if the faster replicator gained its speed 
by deteriorating chemical resources of the protocell. Such a replicator would be 
more successful than others at the expense of the overall protocell performance. 
This really looks like a conflict situation known as the „tragedy of the commons^. 
In view of this potential conflict, the protocell stage of evolution must have been 
very fragile. It is also interesting to observe that the tragedy of the commons 
seems to have threatened living matter already at such an early stage. 

Why is it the case that the protocells have not become extinct before more 
advanced forms of life could evolve? Due to the assumed absence of horizontal 
replicator transmission and due to the lack of recombination and of cell fusion we 
find in the protocell world an ideal situation for group selection to operate. A line 
of descendants containing a very fast replicator will simply die out without 
„infecting“ any other lineage by transmission of the non-cooperative molecule. 
But one problem remains: due to the basic properties of chemistry there will 
always be small differences in replication rates between different types of 
replicators. This could have lead to a dramatic long-term change in all replicator 
proportions and it could have entailed the extinction of all protocells. 




124 



Their complete extinction apparently did not happen. Had it happened, this 
might have prevented further evolution of life and, perhaps, this article could not 
have been written. In order to understand the long-term survival of protocells, 
note that the sampling effect during their fission produces daughter cells with new 
proportions of replicators. Due to this sampling effect, some of them will not have 
an excessive amount of the faster replicators. As Szathm^ has shown, the joint 
effect of this „stochastic correction^ and of group selection can be strong enough 
to keep a population of protocells alive and to favour the evolution of some 
cooperation between replicators of the same protocell. However, life on Earth 
would not have gone very far with this basic design of the cell. The reason is that 
stochastic correction only works efficiently if the number of molecules per 
compartment is very small. Under such conditions evolution would have taken 
place in a chemical world with relatively few dimensions. 



3. Chromosomes and the Drunken Walk of the Diploid 

As we saw in the previous section, early protocell evolution could not generate 
cells containing many replicators. Undoubtedly, this limitation has been finally 
overcome by the evolution of linkage between replicators, that is by the origin of 
chromosomes. Once there is linkage between two or more of the replicators, their 
replication rates are equal and this enhances the scope for efficient stochastic 
correction. Furthermore, since linkage ensures that a group of cooperating 
replicators will not be separated during protocell fission, there is also an enhanced 
potential for the evolution of cooperation within the cell. 

All these advantages of chromosomes do not immediately explain why they 
have evolved in the first place. One reason is that chromosome replication is slow 
as compared to the separate replication of its constituents. For the case of two 
linked replicators, Maynard Smith & Szathm^ry (1993) consider a twofold 
disadvantage in replication speed as a conservative estimate of the "cost of 
linkage". They are able to show that such a disadvantage can be offset by selective 
advantages if the number of "genes" in the original protocell is small and if the 
particular combination of the linked genes is needed for efficient cell 
reproduction. Under these circumstances, linkage evolves because genes with 
complementary effects now benefit strongly from their cooperation and gene 
competition due to different replication rates is reduced. 

Although the evolution of chromosomes was perhaps the most important step 
towards a limitation of reproductive competition in the genome, this has not been 
sufficient to overcome all intragenomic conflict and to completely „tame the 
genes". As soon as there is more than one chromosome in the same cell, genes can 
achieve a replication advantage if they cause the chromosome on which they 
"travel" to gain a disproportional share in the next generation of chromosomes. 
Such potential gains become a particular threat to intragenomic cooperation when 
the additional features of cell fusion and genetic recombination also come into 
play. Both new features strongly undermine the strength of group selection. This 




125 



means that little power is left to the selective force that seems to have maintained 
"genetic order" in early protocell evolution. Other mechanisms must have 
replaced group selection as the "policeman^ that holds the genes in check and 
ensures their cooperation. 

To cut a long story short, evolution has finally produced very elaborate 
patterns of how chromosomes are passed on to new generations. In order to 
discuss the sophistication of genetic transmission at later stages of evolution, let us 
now turn our attention to sexually reproducing organisms with diploid genetic 
material in the cell’s nucleus. The nuclear genetic material then consists of pairs 
of homologous chromosomes. A parent transmits only one of its two homologous 
chromosomes via gametes (eggs or sperm) to an offspring. This requires the 
segregation of chromosomes during gamete production. In principle, such a 
segregation could be achieved by a single „reduction division“ which transforms 
one diploid cell into two haploid daughter cells. But evolution did not favour this 
simple design of reproductive physiology. Instead of producing an elegant 
solution, evolution has invented a complicated process, called meiosis, during 
which chromosome numbers are first doubled and the cell then undergoes two 
consecutive reduction divisions. 

Hurst (1993) calls this puzzling aspect of meiosis "the drunken walk of the 
diploid". In order to make one step forward, the diploid makes first one step 
backward and then two steps forward. How can we explain this strange design 
found in many forms of life? Remember that gametes are the devices that carry 
chromosomes into the next generation. In a thought experiment, suppose that only 
a single reduction division takes place which directly produces two haploid "sister 
cells". Consider now a mutant gene that disturbs the process of reduction division 
by killing the sister cell as the reduction division takes place. Chromosomes with 
this mutant gene would end up in more than half of the gametes. Such an effect is 
called meiotic drive and the driving mutant may initially invade the population 
even if it has otherwise negative effects on organismic performance. 

This resembles vaguely the reproductive conflict among replicators in early 
protocells. But we can no longer invoke group selection as the ordering force that 
reduces the damaging consequences of this conflict. It may please all alcoholics to 
learn that it is the „drunken behaviour of the diploid“ which acts as an institution 
against meiotic drive (Haigh and Grafen, 1991). The step backward, i.e. the 
apparently useless first duplication of chromosomes, creates a copy of the meiotic 
drive gene and therefore opens the possibility that this gene "shoots itself in the 
foot" when it kills a sister cell. This is an advanced way of forcing the nuclear 
genes into cooperation. We do not know, however, whether this effect of the 
drunken walk has really caused its long-term maintenance in evolution. The origin 
of this phenomenon probably relates to the properties of mitotic cell division 
which seems to be the historical predecessor of meiosis. 




126 



4. Cytoplasmic Conflict and its Resolution: 

The Origin of the Sexes 

The story of intragenomic conflict continues outside the cell’s nucleus. After the 
nucleus had been tamed, a new evolutionary battlefield probably arose in the 
world of cytoplasmic DNA. It is very likely that reproductive competition now 
occurred among organelles, such as mitochondria or chloroplasts. There seems to 
be no obvious evolutionary force that would have restricted this competition with 
high efficiency. However, evolution has resolved most of the cytoplasmic conflict 
by improving the design of life in an „ingenious“ way. This improvement was 
achieved by the origin of two different mating types, only one of which 
transmitted its organelles to the next generation. The Darwinian creation of these 
mating types initiated the evolution of the two sexes. Who would have thought 
that the sexes were „born“ in a cytoplasmic war? 

This idea about the origin of binary sexes was first published by two 
psychologists (Cosmides & Tooby, 1981) and later elaborated by several 
biologists (Hoekstra, 1987; Hurst & Hamilton, 1992). In order to understand this, 
suppose that there are no mating types yet. Gametes are then all alike and there is 
no egg-sperm dimorphism. When two such isogametes fuse, both contribute 
organelles to the new cell that originates from their fusion. During this fusion , the 
organelles of the two gametes find themselves in a competitive situation regarding 
their prevalence in the new cell. Hurst and Hamilton argue that this probably has 
led to the evolution of active attempts to harm organelles of the other gamete. 
Such evolved warfare must have had adverse effects on cell performance after 
fusion. This in turn should have created a selective advantage for those nuclear 
genes that are capable of undermining cytoplasmic warfare. Hamilton and Hurst 
imagine, for example, a mutation in the nuclear DNA which suppresses the 
harmful activities of all those organelles with which it travels in the gamete. Even 
if the fusion occurs with another gamete that does not carry the suppressor gene, 
the cytoplasmic war will be short in view of the unilateral disarmement caused by 
the suppressor gene. At the population level, a polymorphic equilibrium can be 
reached containing both the suppressor and the non-suppressor gene. 

In order to complete the evolution of the sexes, two more steps are required. 
First of all, we need the origin of two mating types, each of which fuses only with 
the other. Taking the equilibium of suppressor and non-suppressor as the new 
evolutionary starting point, assortative mating between suppressor and non- 
suppresor can easily evolve. By this process, the two mating types „plus“ and 
„minus“ come into existence. The symmetry of gametes is now entirely broken 
and, in a second step, evolution can attach further attributes to plus and minus. 
This must have largely facilitated the differentiation into egg and sperm, i.e. the 
evolution of the „true“ sexes. Once males and females existed, their reproductive 




127 



competition at the organismic level must have generated various kinds of intra- 
and intersexual conflict. For example, there is quite a potential for evolutionary 
conflict between the sexes regarding mating, paternity, and parental investment. 
At this point, the Darwinian tale begins to describe phenomena that we are all 
very familiar with. 



References 

Cosmides, L.M. & Tooby, J. (1981). Cytoplasmic inheritance and intragenomic 
conflict. J. theor. Biol 89, 83-129. 

Haigh, D. & Grafen, A. (1991). Genetic scrambling as a defense against meiotic 
drive. J. theor. Biol 153, 531-558. 

Hoekstra, R.F. (1987). The evolution of sexes. In The evolution of sex and its 
consequences, ed. S.C. Stearns, pp. 59-91. Basel: Birkhauser Verlag. 

Hurst, L.D. (1993). Drunken walk of the diploid. Nature, 365, 206-207. 

Hurst, L.D. & Hamilton, W.D. (1992). Cytoplasmic fusion and the nature of 
sexes. Proc. R. Soc. Lond. B, 247. 189-194. 

Maynard Smith, J. & Szathmary, E. (1993). The origin of chromosomes. I. 
Selection for linkage. J. theor. Biol. 164, 437-466. 

Maynard Smith, J. & Szathmary, E. (1995). The major transitions in evolution. 
Oxford: Freeman. 

Szathmary, E. & Demeter, L. (1987). Group selection of early replicators and the 
origin of life. J. theor. Biol 128, 463-486. 

Wachtershauser, G. (1988). Before enzymes and templates: theory of surface 
metabolism. Microbiological Reviews, 52, 452-484. 




Evolutionary Selection Dynamics 
and Irrational Survivors* 



Jonas Bjornerstedt^, Martin Dufwenberg^, Peter Norman^, 
and Jorgen W. Weibull'* 



^ Department of Economics, Stockholm University. 

^ CentER for Economic Research, Tilburg University, and Department of Eco- 
nomics, Uppsala University. 

^ Department of Economics and Institute for International Economic Studies, 
Stockholm University. 

^ Department of Economics, Stockholm School of Economics. 



Abstract o We consider certain classes of evolutionary selection dynam- 
ics in discrete and continuous time and investigate whether strictly domi- 
nated strategies can survive in the long run. Two types of results are pre- 
sented in connection with a class of games containing the game introduced 
by Dekel and Scotchmer [5]. First, we use an overlapping-generations version 
of the discrete-time replicator dynamics and establish conditions on the de- 
gree of generational overlap for the survival and extinction, respectively, of 
the strictly dominated strategy in such games. We illustrate these results by 
means of computer simulations. Second, we show that the strictly dominated 
strategy may survive in certain evolutionary selection dynamics in continuous 
time. Journal of Economic Literature Classification Numbers: C72, C73. 

1. Introduction 

A basic rationality postulate underlying much of non-cooperative game theory 
is that players never use strategies that are strictly dominated. Hence, in so 
far as one is interested in establishing evolutionary foundations for solutions 
concepts in non-cooperative game theory, an important question is whether 
evolutionary selection processes do eliminate such strategies. 

We here consider a few classes of evolutionary selection processes in dis- 
crete and continuous time, and investigate conditions under which strictly 
dominated strategies are eliminated, or, alternatively, can siurvive in the long 
run. The set-up is standard for evolutionary game theory: we imagine a 
large (infinite) population of individuals who are randomly drawn to play a 
symmetric and finite two-player game in normal form. The classes of selec- 
tion dynamics here considered include the discrete-time and continuous-time 
versions of the replicator dynamics used in biology, but are wide enough to 

’^We wish to thank Josef Hofbauer, Klaus Ritzberger and Suzanne Scotchmer for helpful 
comments to earlier drafts of this manuscript. Bjornerstedt and Norman thank the Jan 
Wallander Foundation for financial support. 




129 



also include a range of social and cultural transmission processes based on 
replication by way of imitation (cf. Bjornerstedt and Weibull [2], Weibull 
[12]). In the case of biological evolution, the payoffs in the game in question 
represent reproductive fitness while in the case of social evolution the payoffs 
represent preferences - shared by all individuals. 

Our study relates to the following established results: 

Fact 1 The continuous-time replicator dynamics eliminates any strictly dom- 
inated pure strategy (Akin [1]). 

Fact 2 The discrete- time replicator dynamics eliminates any pure strategy 
strictly dominated by a pure strategy (Samuelson and Zhang [10]), but 
may fail to eliminate a pure strategy strictly dominated only by a mixed 
strategy (Dekel and Scotchmer [5]). 

Fact 3 All continuous-time payoff-monotone evolutionary selection dynam- 
ics eliminate any pure strategy strictly dominated by a pure strategy 
(Samuelson and Zhang [10]). 

Below we present two groups of results. Firsts we devise a class of over- 
lapping generations models in discrete time, a class that contains the usual 
discrete- time and continuous- time replicator dynamics as a special and limit- 
ing case, respectively. The model we construct is different from that devised 
by Cabrales and Sobel [4], see Section 3.6. The discrepancy between facts 1 
and 2 above is explained theoretically in this framework with reference to the 
degree of intergenerational overlap: Akin’s [1] result is extended to a range 
of high degrees of overlap and Dekel’s and Scotchmer ’s [5] result is extended 
to a range of low degrees of overlap. We also report a series of computer 
simulations that indicate how the intergenerational overlap affects the nature 
of the dynamic evolutionary solution paths in a game studied by Dekel and 
Scotchmer. For low degrees of overlap the paths converge to the unique Nash 
equilibrium of the game, for an intermediate range of overlaps the paths form 
an interior limit cycle, and for low degrees of overlap the paths diverge to- 
ward the boundary. In the last two cases the game’s strictly dominated pure 
strategy (that is, however, not dominated by any pure strategy) survives in 
the long run. 

Second^ we investigate to what extent the extinction of strictly dominated 
strategies can be guaranteed in a wide range of evolutionary selection pro- 
cesses modelled in continuous time. More exactly, we consider two such 
classes: the class of ’’payoff monotone dynamics” (Samuelson and Zhang [10]) 
with the property that pure strategies with higher payoffs have higher growth 
rates, and the wider class of ’’weakly payoff monotone dynamics” (Friedman 
[6]) which only requires that the population composition moves in the gen- 
eral direction of increasing payoffs. We provide a game and a weakly payoff 
monotone dynamics such that a pure strategy that is strictly dominated by 




130 



all other pure strategies in the game survives in the long run, in a sizable pop- 
ulation share and for a wide range of initial conditions. Dekel and Scotchmer 
[5] construct a game that has a strictly dominated strategy that is not strictly 
dominated by any pure strategy. We provide a payoff monotone dynamics 
under which this strategy survives, again in a sizable population share and 
for a wide range of initial conditions.^ 

The rest of the paper is organized as follows. Section 2 introduces basic 
notation and terminology. Section 3 contains our discrete-time analysis, cen- 
tred on the mentioned overlapping-generations replicator dynamics. Section 
4 contains our continuous-time analysis, and Section 5 concludes. 

2. Notation and Preliminaries 

We consider finite symmetric two-player games in normal form. Let / = 
{1, 2..., A:} be the set of pure strategies (the same for both players). A mixed 

strategy is then a point x on the unit simplex A = G • 

Let A denote the k x k payoff matrix, so that aij is the payoff to strategy 
i when played against strategy j. We represent a pure strategy i as the 
(degenerate) mixed strategy which assigns probability one to the i’th pure 
strategy. Geometrically, this is the i’th vertex of the mixed strategy space, 
which we will denote G A. Then u (e*, a;) = e^Ax is the expected payoff of 
a pure strategy i when played against a mixed strategy a: G A. Accordingly, 
u{x^y) is the expected payoff of a mixed strategy a: G A when played against 
a mixed strategy y G A. 

A strategy a: G A is strictly dominated if there exists a strategy y £ A 
which always earns a higher payoff, i.e. if u (t/, z) > u{x,z) for all strategies 
z G A. A strategy pair (a:,y) is a Nash equilibrium if a; is a best reply to y 
and y is a best reply to x. 

Much of the subsequent analysis will refer to modifications and extensions 
of the so-called ROCK-SCISSORS-PAPER game, the payoff matrix of which is 
the special case a = 0 of 

/ 1 24-a 0 \ 

A — \ 0 1 2-f-a j (for a > — 1). (1) 

\ 2 + a 0 1 / 

Strategy 1 (rock) ’’beats” strategy 2 (SCISSORS), which ’’beats” strategy 3 
(paper), which in its turn beats strategy 1. The mixed strategy x = (3)3)3) 
is clearly a best reply to itself, so (x, x) constitutes a Nash equilibrium. In- 
deed, this is the only Nash equilibrium (granted a > —1). A game with payoff 
matrix 1 will be referred to as a RSP{a) game, 

^Nachbar [ 9 ] shows that if an interior solution trajectory to a payoff monotone selection 
dynamics converges, then the limiting state is a Nash equilibrium strategy. Hence, survival 
of a strictly dominated strategy requires that the solution in question diverges. 




131 



We here consider a large population of individuals who every now and then 
are randomly drawn to play some finite and symmetric two-person game as 
defined above, each individual using some pure strategy whenever playing the 
game, and each individual having the same probability to be drawn to play 
the game. In this context, a point x G A represents a population state, i.e., a 
vector of population shares € [0, 1] of individuals playing the different pure 
strategies i € I. Accordingly, u (x, x) is the average payoff in the population. 

The most well-known evolutionary selection dynamics for this setting is 
the replicator dynamics. Modelled in continuous time it is defined by the 
following system of ordinary first-order differential equations: 

x\ = [u(e\ Xt) — u{xt, Xt)] x\ (for all 2 € / and t G R). (2) 

Here the growth rate of the population share associated with a pure strategy 
i equals the difference between its own payoff u (e^, x^) and the average payoff 
u{xt,Xt)? 

Applied to an RSP{a) game with positive a, this dynamics is known to 
induce solution orbits all of which converge in the interior of the simplex to 
the unique Nash equilibrium strategy x. For negative a, all interior solution 
orbits instead diverge toward the boundary of the simplex, while in the bench- 
mark case a = 0 all interior solutions form periodic orbits around x. 

Modelled in discrete time, the replicator dynamics takes the form of the 
system of first-order difference equations: 

^ ^ '^^\ x\ (for alH G / and t G N). (3) 

U \Xi, Xi) 

Here the population share associated with a pure strategy i increases from one 
period to the next by a factor which equals the ratio between its own payoff 
tx(e%xt) and the average payoff u{xt,Xt).^ Applied to an RSP{a) game, 
this dynamics induces solution orbits all of which (except when starting at 
the Nash equilibrium strategy x) diverge toward the boundary of the simplex, 
irrespective of the parameter value a (Weissing [13], see Theorem 1 below). 

For references to the continuous and discrete replicator dynamics, and for 
discussions and analyses of these, see Hofbauer and Sigmund [8] . 

3. Evolutionary Selection in Discrete Time 

In this section we develop an overlapping-generations replicator dynamics 
which subsumes the above discrete and continuous replicator dynamics as a 
special and limiting case, respectively. The present model is used to illustrate 
how three qualitatively different long run outcome patterns are possible in 

^Here and in the sequel all random variables are approximated by their mean values. A 
careful investigation of the appropriateness of this approximation is called for. 

^Unlike in the continuous-time version, the discrete-time version presumes that the 
average payoff u(x, x) is always positive. 




132 



Dekel’s and Scotchmer’s game, dependent on the overlap between generations. 
It turns out that a certain result by Weissing [13] for the discrete replicator 
dynamics in a generalized class of ROCK-SCISSORS-PAPER games is useful for 
our theoretical investigation. The results are illustrated by means of computer 
simulations. 



3ol An Overlapping-Generations Dynamics 



One may derive the discrete replicator dynamics (3) as follows. Let the time 
unit be the lifetime of a generation, suppose pure strategies ’’breed true,” i.e., 
that each offspring inherits its single parent’s pure strategy. Let the payoff 
u (e^ , x) to pure strategy i against the mixed strategy x represent the number 
of (surviving) offspring to an individual using strategy i in population state 
a; € A. Let Ut denote the number of individuals in generation t, and let n\ 
be the number of these who use strategy i. Then 

nl+i=u(e\xt)nl (for t = 0, 1, 2, (4) 
and hence, by linearity of u in its first argument. 



nt+i='^n\^^=u{xt,xt)nt (for « = 0, 1, 2, ..)• (5) 

t€/ 



Dividing (4) by (5) gives (3). 

In this scenario, all individuals are simultaneously replaced by their off- 
spring, at times t = 1,2,3.... Suppose instead that generations overlap so 
that replacements take place r times per time unit and only involve the frac- 
tion 5 = 1/r of the total population. If each individual in the population has 
the same probability of being drawn each time, and u (e% x) is the number 
of offspring to an individual using strategy i in population state then the 
resulting dynamics is 

= {1 - 6) n\ + 6u {e\ xt) nl {for t = 0,S,26, (6) 

Summing across all strategies i G I and dividing through (analogously with 
the derivation of (3)) one obtains the following overlapping generations ( OLG) 
dynamics : 



. _ 1 - g + gu {e\xt) i 

^t+s I — S + 6u{xt,xt)^*' 



(for t = 0, 6,26, 



(7) 



As expected, the discrete replicator dynamics (3) appears as the special case 

5 = 1 . 

Manipulating (7), and taking limits as 5 — ♦ 0, one obtains the continuous 
replicator dynamics (2): 



= li„, ;i±«^ 

5-^0 6 s^o 1 — 6 6u {xt, Xt) 

= [uie^xt) -u{xt,xt)]x\ 



( 8 ) 




133 



T 




Figure 1: Divergence/convergence in RSP{0.3b) depending on the number of 
overlapping generations (6 = 0.05, 6 = 0.6 and 5=1). 



The OLG dynamics (7) thus indeed spans the whole range between the dis- 
crete and the continuous replicator dynamics. (For a comparison with the 
discrete-time dynamics in Cabrales and Sobel see section . 

Figure 1 shows how the solutions to the OLG dynamics (7) may, depending 
on the share 6 of simultaneously reproducing individuals, converge or diverge 
from the unique Nash equilibrium state x in a RSP{a) game with a > 0. The 
smooth and convergent path in the diagram approximates the limiting case 
when 5 0 - for which we have already noted that the continuous replicator 

dynamics (2) converges from all interior initial states. 

If we rewrite equation (7), for an arbitrary r = |, as 

4+s ^ r-l + u{e\xt) 
x\ r - 1 -\-u{xt,Xt) 

then it is apparent that having r > 1 generations is equivalent with adding a 
positive constant c = r— 1 to each entry of the payojff-matrix A in the discrete- 
time dynamics (3). Note that c increases with r from zero to plus infinity as 
r increases from one to plus infinity. Letting c — > oo is thus equivalent with 
letting r — 4 oo (which in its turn is equivalent with letting 5 — > 0), suggesting 
that the discrete replicator dynamics (7) approaches the continuous replicator 
dynamics (2) as c increases, an observation made by Hofbauer and Sigmund 
[8] concerning the discrete replicator dynamics (3). 




134 



A general property of OLG dynamics (7) is that a strategy’s population 
share increases (decreases) over time whenever the strategy earns above (be- 
low) the population average: x\_^^ > x\ if u{e^,Xt) > u[xt,Xt), and likewise 
for the reversed inequality. 



3o2 Circulant Rock-Scissors-Paper Games 

Weissing [13] calls a symmetric 3x3 game with payoff matrix 

( a b c \ 

c a b \ (for c < a < b) (10) 

b c a J 

a circulant ROCK-SCISSORS-PAPER game. Any such game has a unique interior 
Nash equilibrium strategy x (Shapley [11], Weissing [13]). Observe that when 
a=l, 6 = 2-fa and c = 0 the payoff matrix (10) corresponds to a RSP{a) 
game (for a > — 1), and then x is the mid-point Games with payoff 

matrix A as in (10) will here be termed RSP{a,b, c) games. 

Weissing [13] shows that under the discrete replicator dynamics (3), as 
applied to a RSP{aj 6, c) game, the unique interior Nash equilibrium strategy 
X is either a global attractor, a global repellor, or a global center (Weissing 
[13] Theorem 6.8). It turns out that part of his result and proof is important 
for the subsequent analysis. More exactly, we will use: 



Theorem 1 (Weissing ) Let x be the unique Nash equilibrium strategy in 
a RSP{a^ 6, c) game, for c < a < b. Then the following holds under the 
dynamics (3): 

(i) X is a global attractor if a? < be. 

(ii) X is a global repellor if a‘^ > be. 

To prove this result Weissing [13] uses the function W defined on the 
interior of the unit simplex A by 



W{x) 



xAx 

X1X2XZ 



( 11 ) 



He shows that x is the unique critical point of this function (which tends to 
plus infinity near the boundary of A), and, moreover, that along any solution 
to (3) starting from any other interior initial state xq, W (xt_|-i) — W (xt) has 
the same sign as — be. Hence W is monotonically decreasing (increasing) 
over time if < be (a^ > be). Consequently, all interior solution trajectories 
to (3) converge to x if < be and diverge from x toward the boundary of 
the simplex if > be. 

Applied to an RSP{a) game, this result says that the discrete replicator 
dynamics diverges from the unique Nash equilibrium strategy (|,^,|) to- 
wards the boundary of the simplex, for any parameter value a > — 1. More 
generally, we noted above that having r > 1 overlapping generations in the 




135 



OLG dynamics (7) results in the same orbits in state space as when adding the 
constant r — 1 to each payoff in the discrete- time dynamics (3). Consequently, 
the OLG dynamics with r overlapping generations in an RSF{a) game in- 
duces the same orbits as the discrete replicator dynamics in an RSP{a, 6, c) 
game with a = r, 6=lH-a-f-r and c = r — 1. By Theorem 1, this tells us 
that the OLG orbits in a RSP{a) game converge (diverge) if a(r — 1) > 1 
(a(r — 1) < 1). Hence, the orbits diverge for negative a, while for a positive 
a they converge if there is sufficient overlap, r > ^ + 1, and diverge if the 
overlap is small, r < L -f 



3o3 Rock-Scissors-Paper-Dumb Games 



To show the possibility for strictly dominated strategies to survive the discrete 
replicator dynamics (3), Dekel and Scotchmer [5] added a fourth pure strategy, 
baptized ’’dumb” by Cabrales and Sobel [4] to a particular RSP{a) game. 
The game they used corresponds to the special case a = 0.35 and = 0.1 of 
the following 4x4 payoff matrix: 



1 


2 + a 


0 




0 


1 


2 + a 


13 


2 + a 


0 


1 


(3 


1+/? 


1 + 13 


1+/? 


0 / 



(for 0 < 3/3 < a < 4/3). 



( 12 ) 

We will refer to any such game as a RSPD{a, /3) game. The unique Nash 

equilibrium of such a game is (x;x), where x = (^,0) = (i? 

fourth pure strategy, DUMB, is strictly dominated by x since u (x, = 

^ > 1 -f /3 = u (e^, e^) for all z < 4, and u (x, = (3 > 0 = u e^) . 



3o4 Irrational Survivors in OLG Dynamics: Theory 

Recall that under any OLG dynamics (7) a strategy increases its population 
share wherever it earns a payoff above average. By straightforward manipu- 
lations the relevant inequality u (x, x) < u (e"^, x) can be re-written as 



(1 - x“) ((1 - X-) - M < (,1)» + (^2)^ + ,13, 

It is easy to see that this inequality holds along the three edges of the unit 
simplex where DUMB is extinct. For instance, if = 0, then the left- 

hand side equals 1 — 2/3 fa while the right-hand side is not less than 1/2 for 
any combination of x^ and x^. Since a < 4/3 by assumption, (13) is met. The 
region where condition (13) is met is shown in Figure 2. This region consists 
of the points in the unit simplex which lie outside the egg-shaped set. Inside 

^Expressed in terms of 6, the condition for convergence (divergence) is ^ {6 > 

i-a 




136 




Figure 2: Region of simplex where is less than average 



the egg-shaped set the reversed inequality holds. Thus DUMB decreases over 
time inside the egg and grows outside it. For a RSPD{a, P) game, in any 
population state x G A with < 1, let y = (2/^, 2/^, 2/^) denote the relative 
population shares of the non-DUMB strategies: y'^ = for i = 1,2, 3. Such 
a vector y is formally identical with a population state in the corresponding 
RSP{a) game (1). Let v (e^y) denote the payoff to strategy i in population 
state y in the latter game. Inspection of the payoff matrix (12) shows that 
the payoffs in the RSPD{a, P) game in any population state x are related 
to their payoff in the associated population state y in the RSP{a) game as 
follows: 

u (e\ a:) = (l — x^^ v (e\ y) H- x^/3 (14) 

for all strategies i < 4, and for strategy DUMB 

u (e^, x) = (l — x^) (1 -f /3) . (15) 



After some algebraic manipulations of these expressions one finds that the 
OLG dynamics (7), as applied to the full RSDP{a, P) game, induces the 
following dynamics for the vector y: 

^ lt + v{e ,y) (fori = l,2,3andt = 0,5,25,...) (16) 

yl lt+v{y,y) 



where 7 t 



^ ’ Thus the dynamics of y is identical with the discrete 

replicator dynamics (3) in a circulant ROCK-SCISSORS-PAPER game (10) with 




137 



time-dependent parameters at = 1 + 7t, 6t = 2 4- a 4- 7t , and q = 7t- 

Recall that ^ = (x, 0) = O) is the unique Nash equilibrium strategy 

in any RSPD{a, P) game, and let L denote the line segment in A where all 
non-DUMB strategies appear in equal shares: L = (x G A : = x^}. 

Using Weissing’s [13] technique for the discrete replicator dynamics (3) in 
circulant ROCK-SCISSORS-PAPER games, we obtain 

Proposition 2 Consider the OLG dynamics (7) applied to a RSPD{a, p) 
game (for 0 < 3/3 < a < 4/3^. 

(a) The solution through any interior initial state xq converges to x if 
6 < ■ 

(b ) The solution through any interior initial state xq ^ L diverges toward 

the boundary of A if 6 > • In this case the population share x^ does 

not converge to zero. 

Proofo As mentioned in the connection of Theorem 1 above, Weissing 
[13] showed that the function W defined in (11) satisfies the condition that 
W — W (xt) has the same sign as a^ — be along any interior solution 

(starting in a state xq ^ x) to the discrete replicator dynamics (3) in a 
RSP{a, b, c) game. Generalized to such a game with time-dependent param- 
eters, the condition becomes that W (xt+i) — W (xt) has the same sign as 
a? — bfCt. 

In the y-dynamics (16) we have - btCt = 1 — ajt, so in that dynamics 
W (yt-fi) — W {yt) has the same sign as 1 — 07 t. 

First suppose 6 < It is easy to show that then a^t > 1 for all 

x^ € (0, 1). Hence W{yt) is strictly decreasing along the interior solutions to 
the y-dynamics (16) (with initial state yo ^ ^), so yt x = (|, ^, ^) . In 
the full state space of the RSPD{a,P) game, this means that the population 
state Xt converges to the line segment L. But all states x near L lie inside 
the egg-shaped region where DUMB earns below average, so x^ — ► 0 and thus 

Xt X. 

Second, suppose 6 > One can show that x^_^^ < x^ whenever 

Xt > |, and x^_|_^ < | if a:t < along any interior solution to the OLG 
dynamics (7). Hence, we may without loss of generality presume x^ < |. It 
is easy to show that a^t < 1 for all such x| . Hence (for t sufficiently large) 
W{yt) is strictly increasing along any interior solution to the y-dynamics 
(16) (with initial state yo ^ ^), so yt diverges towards the boundary of the 
unit simplex in y-space. In the full state space of the RSPD{a, p) game 
this means that the population state xt diverges toward the boundary D = 
{x G A : X* = 0 for some i < 4}, from any interior initial state xq ^ L. All 
states X G A near the edge D fl {x G A : x^ = 0} lie outside the egg-shaped 
region where xf earns below average, so xf increases over time near D. In 
particular, x^ does not converge to zero along any interior solution trajectory 
starting at a state xq ^ L.M 

The proof of the second claim in this proposition establishes that for suf- 




138 



ficiently little overlap in the OLG dynamics, its solution trajectories diverge 
toward the boundary of the simplex where at least one of the non-DUMB 
strategies is extinct. The overlap condition is payoff dependent, 6 > 
where the right-hand side expression is approximately 0.43 in the game stud- 
ied in Dekel and Scotchmer [5]. They considered the discrete replicator dy- 
namics (3), so 5 = 1. Hence, the dynamic path will sooner or later reach 
outside the egg-shaped set, and DUMB then increases. 

In fact, as the population state approaches this boundary its movement 
slows down and it spends longer and longer time intervals close to the edges 
of the simplex where two of the three non-DUMB strategies are close to ex- 
tinction. If one such strategy, say PAPER, would be completely extinct, ROCK 
performs better than SCISSORS, and SCISSORS will converge to zero. With 
only DUMB and ROCK left it is easy to see that the dynamics converges to- 
wards (1/2, 0,0, 1/2). One may thus conjecture that for all 6 sufficiently large 
the set T = {(^,0,0, , (O, ^,0, , (0,0, ^)} of ’’subsimplex Nash equi- 

librium” points constitutes an attractor in the statistical sense of the time 
spent close to points in T.^ 



3o5 Irrational Survivors in OLG Dynamics: Computer Simulations 

We have used the OLG dynamics (7) for numerical computer simulations of 
RSPD{a^ /?) games. The simulations support the above results and provide 
some insights into the nature of the dynamics for intermediate degrees of 
overlap, i.e., for the range of 5- values for which Proposition 1 neither ensures 
convergence nor divergence. In most of the simulations reported here we have 
used the parameter values of Dekel and Scotchmer [5]: ct = 0.35 and = 0.1. 
Below we refer to these as the DS-payoffs. 

The Convergent Case: As expected from Proposition 1, the population 
state converges to x = the unique Nash equilibrium strategy, 

for S sufficiently small, more exactly for 6 < With DS-payoffs, ^ 

0.26. Our simulations indicate convergence for all 6 below some critical value 
somewhat below 0.28, see Figure 3. 

The Divergent Case: As expected from Proposition 1, the population state 
diverges toward the boundary where at least one non-DUMB strategy is ex- 
tinct for 8 sufficiently large, more exactly for 8 > With DS-payoffs, 

^ 0-43. The computer simulations exhibit such divergence for all 8 
exceeding approximately 0.40, see Figure 4. As indicated above, the set T of 
’’subsimplex Nash equilibrium points” seems to be a global attractor in terms 
of sojourn times. 

Limit Cycles: With DS-payoffs and values of 8 between 0.28 and 0.4 our 
computer simulations indicate the existence of a limit cycle, see Figure 5. 
Note that initial conditions are irrelevant for the limiting outcome also in 



®In the topological sense, the attractor is larger: it will also contain some connecting 
curves between these three subsimplex Nash equilibria. 




39 




Figure 3: Simulated replicator evolution in the RSPD game: Convergence 
with 8 = 0.2, initial state (0.3,0.01,0.01,0.49). 




Dumb 



Rock 



Figure 4: Simulated RSPD replicator evolution: Divergent case with 5 = 0.5 
and initial state (0.34,0.33,0.32,0.01). 





140 




Figure 5: Simulated replicator evolution in the RSPD game with limit cycle 
when S = 0.3 and initial state (0.49,0.01,0.01,0.49). 

this intermediate case. To check the numerical robustness of the limit cycle 
performance we have run one thousand simulations with randomly generated, 
uniformly distributed, initial population states for a variety of parameter 
values. Figure 6 shows the population state after 2000 iterations, each point 
corresponding to a different initial state, and all runs were made with the 
DS-payoffs and the overlap 6 = 0.3. 

In terms of irrational survivors, our computer simulation results suggest 
that the strictly dominated strategy 4 in RSPD{a,P) games may survive in 
the long run for a wide range of initial conditions and in a sizable population 
share (about 0.5). This survival in fact occurs for a wider range of payoflF 
values (a,/3) and overlaps (5) than the theoretical prediction in Proposition!: 
The dominated strategy appears to survive also in the intermediate range of 
5- values. 

3o6 The Cabrales-Sobel Dynamics 

The following discrete- time dynamics was suggested in Cabrales and Sobel 
[4] (op.cit. p.414): 

i /•. CN t , i 



{Wi € I a,nd t — 0,S,26, (17) 





141 




Figure 6: RSPD population states after 2000 iterations with ^ = 0.3. 



One can give an interpretation of this dynamics in terms of boundedly rational 
dynamic adaptation. For this purpose, consider a population consisting of rit 
individuals, n\ of whom use strategy i at time t. The payoff earned by strategy 
i in this population then is u (e^,a;t) nj, and the strategy’s share of the total 
payoff earned in the population is (with the time index dropped) 

n(e%x)n* _ u (e\x) x' 
u(x,x) 

Assume that at each time t = 0,5,25,... the share 5 of individuals review 
their strategy choice in the light of ciurent payoffs. If reviewing individuals 
switch to each strategy in proportion to its share of the total payoff earned, 
one obtains: 

nj . ^ = (1 - 5) nj H ^ ’ *\* 5nt (Vi G I and t = 0, 5, 25, ..). (19) 

u{xt,xt) 

Dividing by the size of the total population rit one obtains (17). 

Just as in the OLG model, if 5 equals unity, (19) reduces to the discrete- 
time dynamics (3). However, taking the limit as 5 — > 0 gives 

_ u{e\xt) -u{xt,xt)_i 

u{xt,xt) 



(20) 





142 



a system of differential equations that is not the same as in the continuous 
replicator dynamics (2). Nevertheless, the orbits of (20) and (2) coincide since 
the denominator in (20) only influences the speed at which the solutions to 
(20) travel along these orbits. (In multi-population dynamics this makes a 
difference also in terms of orbits.) 

4. Evolutionary Selection in Continuous Time 

4.1 Classes of Selection Dynamics 

All continuous-time selection dynamics to be studied here can be written as 
a system of differential equations in the form 

x\ = {xt) x\, (21) 

where for each pure strategy z, is a Lipschitz continuous ^’growth rate” 
function such that the orthogonality condition (^) = 0 is met in 

all states x. The system (21) then has a unique solution through any initial 
population state and this solution stays forever in the simplex. The most 
well-known evolutionary selection dynamics in this form is the continuous 
replicator dynamics (2), where each growth rate (x) equals the strategy’s 
’’excess” payoff u (e^, x) — u (x, x). 

Following Weibull [12] we call a continuous-time dynamics (21) pay- 
off monotone {relative monotone in Nachbar [9] and order compatible pre- 
dynamics in Friedman [6]) if 

{x) > {x) u (e\x) > u {e^ jx) . (22) 

Payoff monotonicity hence requires that the population share associated with 
a pure strategy with a higher payoff increases at a higher rate than the popu- 
lation share associated with a pure strategy with a lower payoff. Clearly the 
replicator dynamics (2) is payoff monotone in this sense. 

Likewise, a dynamics (21) will be called weakly payoff monotone {weakly 
compatible in Friedman [6]) if it moves in the general direction of increasing 
payoffs. More exactly, the requirement is that, unless all non-extinct pure 
strategies earn the same payoff, 

(x) x^u{e\x) > 0, (23) 

iei 

and, if all non-extinct pure strategies do earn the same payoff their growth 
rates are zero. 

Let PM and WPM refer to these two classes of evolutionary selection 
dynamics. They are related as follows: PM C WPM. To see this, consider 
any payoff monotone dynamics, and first note that if all non-extinct strategies 
(those with > 0) earn the same payoff, they must all have the same growth 
rate by mono tonicity, and, by orthogonality, this has to be zero. Secondly, 




143 



suppose that the payoffs of non-extinct pure strategies are not all equal. Note 
that the left hand side of (23) is unchanged if some constant c is added to all 
payoffs in the game (using orthogonality). So is the payoff ranking between 
pure strategies. Hence, without loss of generality we may assume that each 
payoff u (e^,a:) has the same sign as the corresponding growth rate {x). 
Then all terms on the left hand side of (23) are non-negative and at least one 
is positive, hence their sum is positive. 

Samuelson and Zhang [10] show that if a pure strategy is strictly dominated 
by another pure strategy then the population share of the dominated strategy 
will go to zero along any interior solution trajectory to any payoff monotone 
dynamics. One may thus wonder: (i) Is extinction of such strategies guar- 
anteed also under weakly payoff monotone dynamics? (ii) Is extinction of 
pure strategies that are strictly dominated by some mixed strategy guaran- 
teed in any of these two classes of evolutionary selection dynamics? By way 
of counter-examples we will here show that the answer to both questions is 
negative. 



4o2 Irrational Survivors in Weakly Payoff Monotone DynEunics 

Proposition 3 A pure strategy that is strictly dominated by a pure strategy 
may survive under weakly payoff monotone dynamics. 



In order to prove this claim, consider the symmetric 4x4 game given by 
the payoff matrix 



/ 1 
0 
2 

-/5 



2 0 

1 2 

0 1 

-/? -/3 



0 \ 
0 
0 






(for /? > 0) 



(24) 



The first three pure strategies together constitute the bench-mark RSP{G) 
game. Recall that the solutions to the continuous replicator dynamics (2) in 
this game constitute closed orbits around the unique Nash equilibrium point 
(see Section 2). The fourth pure strategy, DUMB, is strictly dominated by all 
the other three pure strategies. 

Consider first the dynamics 



f (rr) x'’ for i < 4 

^ 0 for z = 4 



(25) 



where (pn is the replicator growth-rate function of the RSP{G) ’’subgame,” 
formally defined by = e^Bx — x'Bx, where is as in (24) with (3 = 

0. Since the replicator dynamics is weakly payoff monotone (indeed payoff 
monotone), so is the dynamics (25), except at the line segment L where all 
three strategies z < 4 appear in equal shares. All population states on this 
line segment are stationary in (25) while weak payoff monotonicity requires 




144 



a negative growth rate for strategy DUMB (whenever > 0). Other solution 
trajectories only rotate in closed orbits around L, so the population share of 
DUMB remains constant over time along all solution trajectories. 

In order to establish the claim in Proposition 2 it suffices to modify the 
growth rates in (25) in a small region containing L so that the dynamics 
becomes weakly payoff monotone. For then all initial states sufficiently far 
outside this small region will have closed orbit solutions lying outside the 
region, with constant, forever. 

One way to obtain this is to let strategy 4 decline in a cylinder of arbitrary 
radius 6 > 0 containing L. For this purpose, let d{x) be the Euclidean distance 
from any population state x £ A to L, and define 



/ (^) + {0, (5 — d (x)} for z < 4 

\ — (1 — max {0, 5 — d (x)} for z = 4 



(26) 



The resulting dynamics (21) is Lipschitz continuous and meets the orthogo- 
nality condition. At all population states at distance 6 or more from L, (p 
coincides with ip, while at closer population states cp adds a slight drift up- 
wards in all strategies z < 4 and a slight drift downwards in strategy 4. To 
see that the resulting dynamics is weakly payoff monotone, just note that 



{x)x'u{e\x) = 
i&I 



3 

= (x) x^u(e\ x) -f (1 -f /?)(! — x"^)x^ max {0, 6 — d (x)} , (27) 

i—l 

where, in the 5-cylinder, the first term is non-negative, actually positive when 
X ^ L, by weak payoff monotonicity of the replicator dynamics, and the second 
term is positive except where x"^ = 0 and x"^ = 1. 

4o3 Irrational Survivors in Payoff Monotone Dynamics 

We finally establish that a pure strategy that is strictly dominated by a mixed 
strategy may survive a payoff monotone dynamics, again for a large region of 
initial states. This result is due to Bjornerstedt [3], and we here only sketch 
his proof. 

Proposition 4 A pure strategy that is strictly dominated by a mixed strategy 
may survive under payoff monotone dynamics. 

In order to prove this claim, we re-consider a RSPD{a, p) game and con- 
struct a particular payoff monotone dynamics that comes with a story. First 
imagine a large but finite population of individuals spending their time play- 
ing this game, each individual using some pure strategy all the time. How- 
ever, every now and then an individual randomly reviews his strategy choice. 




145 



with equal probability for each individual. If the reviewing individual finds 
that his strategy is among the currently worst performing, i.e., earns payoff 
miuj x), then the individual imitates ’’the first man in the street,” i.e., 
selects a pure strategy in proportion to the current population shares. Now 
add some noise to the individual’s observation of the other pure strategies. 

Formally, suppose that the observed payoff to strategy j equals its expected 
payoff u (e^, x) plus some noise term Sj. Assume that all noise terms are 
i.i.d. with c.d.f. F. Thus the probability that the individual’s own pure 
state, 2 , is perceived as worse performing than j is F \u (e*^ , x) — (e\x)]. 

The probability Pi (x) that i is perceived to be among the very worst is 
accordingly 

= • ( 28 ) 

In terms of expected values, the dynamics that arises when the review rate 
is constant over time is (up to a constant rescaling of time) 

X* = x*’Pj (x) Pi (x) x^ = 

j^i j^i 




Pi{x)] x^ 



X 



jei 



x“ 



(29) 



If F has everywhere positive density, this dynamics is payoff monotone: The 
’’inflow” sum Pj{x)x^ is the same for all strategies i and the ’’outflow” 
term. Pi (x), is by (28) a decreasing function of i6(e%x). There is a smaller 
chance that a better performing strategy is weeded out by mistake than that 
a worse performing strategy is weeded out by mistake, and hence the growth 
rate of the former will exceed that of the latter. 

We can study the qualitative properties of this dynamics for small obser- 
vational errors by reducing the variance of the distribution F toward zero. In 
the limit we have that Pi (x) = 1 if z is among the worst-performing strategies 
and otherwise Pi (x) = 0.® In a RSPD{a, /3) game there are four geometri- 
cally well-shaped regions within each of which exactly one of the four pure 
strategies is worst, see Figure 7. 

For instance, near the vertex of pure strategy 1 (rock) pure strategy 2 
(scissors) is worst. Accordingly, in the absence of observational errors only 
SCISSORS is decreasing: 



^ = x^ — 1 and ^ for all z ^ 2, (30) 

X‘^ x^ 

and likewise for the vertices i = 2,3. Not surprisingly, pure strategy 4 (dumb) 
is worst when the other three strategies appear in approximately their Nash 

®The resulting limiting vector field in (29) is thus discontinuous. However, the discon- 
tinuities only appear on certain hyperplanes, see Figure 8. 




146 




Figure 7: Region of simplex where is the worst strategy. 




Figure 8: Direction of movement in (29) without observational error. 






147 



equilibrium shares (|). This region is the indicated ’’conical cylinder” C 
around the line segment L where all three strategies z < 4 appear in equal 
shares (see Figme 7). 

In the absence of observational errors, the solutions to (29) are well-defined 
straight line segments inside each of these four regions. Inside the set C the 
movement makes a ray from the vertex where = 1 toward the boundary 
face where — 0. Having left the set (7, the solution orbit will not enter 
again, as the vector field points away from C. Outside the set C the movement 
is directed linearly out toward one of the three boundary faces where each of 
the other pure strategies z < 4 is extinct, see Figure 8 which describes the 
dynamics outside the set C on the boundary face where DUMB is extinct. 

It is not difficult to find a cylinder D (parallel to the a;^-axis in R^) con- 
taining the above mentioned set C (in its relative interior) such that the 
vector field of the dynamics (29) in the limiting case with no observational 
errors points outward on the surface of D. Then DUMB increases outside D. 
By continuity and compactness the set D has these two properties also for 
sufficiently small observational errors. Hence, for initial states outside D and 
for sufficiently small observational errors, the population state will remain 
forever in D and DUMB will increase forever. Computer simulations suggest 
that all interior solutions outside D swirl around forever, spending longer 
and longer times near the set T of ’’subsimplex Nash equilibrium” points, cf. 
Section and see Figure 4.^ 

5. Concluding Remarks 

We have studied the possibility that ’’irrational behavior,” in the sense of 
strictly dominated pure strategies, may survive in the long run under evo- 
lutionary selection dynamics. For this purpose we constructed an overlap- 
ping generations replicator model in discrete time and showed how a strictly 
dominated strategy necessarily dies out when the degree of intergenerational 
overlap is large but survives when this overlap is small. We then investigated 
whether strictly dominated strategies can survive in continuous- time evolu- 
tionary dynamics. We showed that strictly dominated strategies that are not 
strictly dominated by any pure strategy may survive in payoff monotone evo- 
lutionary dynamics, and that even a pure strategy that is strictly dominated 
by all other pure strategies in a game may survive in weakly payoff monotone 
evolutionary dynamics. 

^We know, however, that the solutions do not converge to any one of the points in 
T, since the limit point to an interior solution under any payoff monotone dynamics is 
necessarily a Nash equilibrium strategy (Nachbar [9]). 




148 



Since elimination of strictly dominated strategies is a relatively weak ra- 
tionality assumption underlying much of non-cooperative game theory, these 
findings appear potentially disturbing for the attempts that are currently 
being made to provide evolutionary foundations for non-cooperative game 
theory. 

References 

[1] E. Akin, ’’Domination or Equilibrium”, Mathematical Biosciences 50 (1980), 
239-250. 

[2] J. Bjornerstedt and J.W. Weibull, ’’Nash equilibrium and evolution by imi- 
tation”, DELTA WP 93-23, 1993, forthcoming in K. Arrow et al. (eds ). The 
Rational Foundations of Economic Behavior, Macmillan, 1995. 

[3] J. Bjornerstedt, ’’Experimentation, Imitation and Evolutionary Dynamics”, 
mimeo.. Department of Economics, Stockholm University, 1995. 

[4] A. Cabrales and J. Sobel, ”On the Limit Points of Discrete Selection Dynam- 
ics”, J. Econ. Theory 57 (1992), 407-419. 

[5] E. Dekel and S. Scotchmer, ”On the Evolution of Optimizing Behavior”, J. 
Econ. Theory 57 (1992), 392-406. 

[6] D. Friedman, 1991, ’’Evolutionary Games in Economics”, Econometrica 59, 
637-666. 

[7] M. Hirsch and S. Smale, Differential Equations, Dynamic Systems, and Linear 
Algebra, Academic Press, 1974. 

[8] J. Hofbauer and K. Sigmund, The Theory of Evolution and Dynamical Systems, 
Cambridge University Press, 1988. 

[9] J.H. Nachbar, ’’Evolutionary Selection Dynamics in Games: Convergence and 
Limit Properties”, Int. J. Game Theory 19 (1990), 59-89. 

[10] L. Samuelson and J. Zhang, ’’Evolutionary stability in Asymmetric games”, J. 
Econ. Theory 57 (1992), 363-391. 

[11] L.S. Shapley, ’’Some Topics in Two-Person Games”, Ann. Math Stud. 5 (1964), 
1-28. 

[12] J.W. Weibull, Evolutionary Game Theory, MIT Press, Cambridge (USA), 1995. 

[13] F. Weissing, ’’Evolutionary Stability and Dynamic Stability in a Class of Evo- 
lutionary Normal Form Games” , in R. Selten (ed.) Game Equilibrium Models: 
I. Evolution and Game Dynamics, Springer Verlag, Berlin/New York, 1991. 




strict and Symmetric Correlated Equilibria 
are the Distributions of the ESS’s of Biological 
Conflicts with Asymmetric Roles* 

Avi Shmida^’^ and Bezalel Peleg^’^ 

^ Center for Rationality and Interactive Decision Theory, The Hebrew University, 
Jerusalem 91904, Israel 

^ Department of Evolutionary Ecology, The Hebrew University of Jerusalem, 
Jerusalem 91904, Israel 

^ Department of Mathematics, The Hebrew University of Jerusalem, Jerusalem 
91904, Israel 



Abstract. We investigate the ESS’s of payoff-irrelevant asymmetric animal 
conflicts in Selten’s (1980) model. We show that these are determined by the 
symmetric and strict correlated equilibria of the underlying (symmetric) two- 
person game. More precisely, the set of distributions (on the strategy space) 
of ESS’s coincides with the set of strict and symmetric correlated equilibria 
(described as distributions). Our result enables us to predict all possible sta- 
ble payoffs in payoff-irrelevant asymmetric animal conflicts using Aumann’s 
correlated equilibria. It also enables us to interpret correlated euqilibria as 
solutions to biological conflicts: Nature supplies the correlation device as a 
phenotypic conditional behavior. 



1. Introduction 

Evolutionary game-theory studies conflict situations in which individuals 
adopt actions (strategies) in order to maximize their Darwinian fitness 
(Maynard-Smith, 1982; Reichert and Hammerstein, 1983). In contrast to 
game theory which assumes rationality in human behavior, evolutionary 
ecology bases its “rationality” on the natural selection mechanism of the 
Darwinian process: Nature selects genotypes (which represent morphological 
and/or behavioral characters) that give higher fitness (Parker and Hammer- 
stein, 1985; Hammerstein and Selten, 1991). Thus, in evolutionary ecology 
we assume that animals’ and plants’ traits are ‘winning strategies’ that have 
been selected through evolution because of their higher fitness. 

This work was supported by Volkswagen Foundation Grant 1/63691 to ECO- 
RATIO group. We would like to thank the Institute of Advanced Studies of 
The Hebrew University of Jerusalem for its hospitality which contributed to 
our “coordination”. We acknowledge helpful conversations with R. Aumann, S. 
Ellner, U. Motro, R. Selten, J. Sobel, and P. Young. We thank S. Zamir and D. 
Goldberg for comments on the manuscript. A.S. would like to thank especially 
R. Selten, A. Neyman and M. Maschler for introducing him to the intriguing 
world of game theory and competitive interactions. 




150 



Player II 
D H 



-- D 

<L) 

P-* 

crj 

^ H 



b > s 
d < a < b 

s > d s < d 

V ^ V ^ 

game of chicken prisoners dilemma 



a , a 


(S) 




d.d* 



D - Dove 
H- Hawk 

o - An equilibrium of 
the chicken game 

^ - An equilibrium of 

the Prisoner ’s dilemma 



Figure 1. The “Hawk-Dove” game: An example of a symmetric 2x2 bimatrix 
game. Each player has two pure strategies: “Hawk” and “Dove”. If s > then we 
obtain the “game of chicken” . It has two asymmetric equilibria in pure strategies. 
(It also has a symmetric mixed equilibrium). When d > s we have the “prisoner’s 
dilemma” . It has exactly one equilibrium which is symmetric and pure. 



On this background Maynard-Smith established the concept of Evolution- 
arily Stable Strategy (ESS). This is a strategy such that if all the members of 
a population adopt it, then no mutant strategy can invade (Maynard-Smith, 
1982, p.204). It can be proved that an ESS is a Nash equilibrium which 
satisfies an additional stability condition (see, e.g.. Van Damme, 1987). 

Only symmetrical equilibria in symmetric games are relevant to biologi- 
cal conflicts of animals and plants (Reichert and Hammerstein, 1983; Parker 
and Hammerstein, 1985; Hammerstein and Selten, 1991). Fig. 2.a. repre- 
sents an example of the game of chicken with Hawk and Dove strategies^. 
This game has three Nash equilibrium points. Two equilibrium points with 
pure strategies exist: [{H,D)] (D,H)]. (If player one chooses strategy H and 
player two chooses strategy D, then we have the equilibrium {H^D). (D,H) 
is interpreted similarly) . These two equilibria are not symmetric because the 
players use different strategies. The third equilibrium point which is mixed 
[p (D) -h (1 —p){H) where p = 1/2] is symmetric, and it is chosen by the two 
players. 

Contests between animals within the same species, in which one oppo- 
nent behaves aggressively (“Hawk” strategy) and the other opponent behaves 
peacefully (“Dove” strategy) have attracted ethologists for many years: The 
animal which chooses “Dove” behaves as if it was an altruist. 

Maynard-Smith solved this debate in a game theoretical approach by 
proving that playing “Dove” against “Hawk” — is an equilibrium. That is, 
“Dove” is the best reply to “Hawk” . The problem is that the two pure equi- 
libria are not symmetrical and therefore cannot be realized without further 




151 



Fig 2.a. 



Fig 2.b. 



Player II 
D H 



Player II 
D H 



2,2 


(Q) 


<S) 


0,0 



-1 ,-i 


CD 


CD 


1 , 1 



D - Dove 
H - Hawk 

- An equilibrium 



Figure 2. Two examples of the “game of chicken”. 

Figure 2. a. The two asymmetric pure equilibria cannot be realized in any bio- 
logical contest (based on this game). In a bimorphic population with two distinct 
phenotypes evolution may lead to an ESS in behavior strategies which yields the 
payoff (3,3) (= 1/2(4, 2) + 1/2(2, 4)). This is the largest symmetric payoff; hence, 
it is the maximum payoff in any (biolgoical) contest based on this game. Also no- 
tice that (3, 3) is the payoff of the (symmetric and strict) correlated equilibrium 
l/2(H,D)-\-l/2{D,H). 

Figure 2.b. The original example of Maynard-Smith of the “Hawk-Dove” game 
(1982, p. 17). 

Each arrow points in the direction of the strategy that yields a higher payoff against 
a given opposing strategy. 



structure which was introduced by Selten (1980). Maynard-Smith (1982, p. 
17,18,22) suggests at least three different biological situations which make 
possible equilibrium behavior in the “Hawk-Dove game” : 

a) A normal form game as in Fig. 2. a. which has only a mixed symmetric 
equilibrium {p = 1/2). Some ecologists do not consider the mixed strategy 
as a completely specified biological situation (Selten per comm.). 

b) A game with an additional strategy named “Retaliator” which is executed 
as follows: play “Dove” if your opponent plays “Hawk” and “Hawk” if 
your opponent plays “Dove”. In such a situation the retaliation is an ESS. 
This situation is not consistent with the basic assumption of game theory 
that strategies are chosen simultaneously. Prom the biological setup, the 
retaliator strategy is unacceptable because player one can play retaliator 
only if he plays after player two; similarly for player two. Therefore, the 
two players cannot both be instructed to play retaliator. 

c) A game with an additional strategy — “Bourgeois” which says: If you are 
an “owner” play “Hawk” and if you are an “intruder” play “Dove” . In 
such situations the bourgeois strategy is an ESS. 

Situation c) can be developed to a semi-extensive^ game that corresponds 
to a correlated equilibrium distribution (Aumann 1974, 1987) — this is the goal 
of this study. Parker (1982, 1984) studied further situation c) and coined the 
term “phenotype-limited ESS”. Unlike others (see, for example, Dawkins’ 
conditional behavior (1976, P-74)) he correctly explained the equilibrium sit- 




152 



uation of the “conditional pure strategies” and suggested that it be called 
“Conditional ESS”. 

Selten (1980) built a game model for situation c) and showed that for 
asymmetric animal conflicts^ with asymmetric information, all the ESS’s are 
pure. He allows only contests between animals with non-identical roles, which 
he obtains by choosing appropriately the probabilities of the possible encoun- 
ters. Given a biological game by a fitness matrix, our theorem completely 
specifies all the payoffs that are achievable by some uncorrelated asymmetry 
of nature. 

The biological bases of understanding “Hawk-Dove encounters” in na- 
ture were presented by the fundamental work of Maynard-Smith and Parker 
(1976). They emphasized that (p. 171) “in asymmetric contests some asym- 
metric features will be taken as a cue by which the contest will be settled 
conventionally”. They also emphasize the importance of “roZes” (e.g. — owner 
vs. intruder; juvenile vs. adult; different phenotyphic individuals of the same 
genotype) and the fact that they may be uncorrelated'^ with the outcome of 
the game {payoff irrelevant — Hammerstein, 1981). Their models already con- 
tain analysis of contests with payoff irrelevant asymmetry and incomplete in- 
formation. A basically similar framework is the model of Hammerstein (1981) 
who deals only with the complete information situation. 

Selten (1980) observed, in particular, that the foregoing intraspecific bio- 
logical conflict cannot be solved in pure strategies (in the normal form game) , 
because in equilibrium the two opponents should have the same strategies 
(and in a “game of chicken” each of the two pure equilibria is asymmetri- 
cal (see Fig. 2)). He built a semi-extensive model with incomplete informa- 
tion and “Role asymmetry”^. Selten’s model allows only encounters between 
non-identical roles. We will show that this requirement can be relaxed and 
pure Nash equilibrium in the semi-extensive game (which corresponds to Au- 
mann’s correlated equilibrium) can be achieved also when we allow identical 
roles to meet. We will also show that the pure equilibrium strategies might 
be derived, at least formally, from Aumann’s correlated equilibrium when it 
is strict and symmetric. 

The reader is advised to look into figures 3,4,6, and 7 for the graphical 
demonstration of the foregoing arguments. 

A concrete example with which the reader may be familiar is the follow- 
ing one: In intraspecific biological conflicts with a payoff matrix of a “game 
of chicken” (Fig. 1) the pure equilibria cannot be realized if the biological 
situation cannot be considered as a semi-extensive game with “role asymme- 
try” . Aumann’s lottery device of correlated equilibria is realized in nature by 
evolution of phenotypic roles which reflect both role asymmetry and incom- 
plete information. In such situations not only pure equilibria which are totally 
correlated® can be achieved (Fig. 3), but it is also possible to obtain partially 
correlated equilibria® which yield very large payoffs for both opponents (see 




153 



Nature 

chooses 




1 - Player I 

2 - Player H 

0 - Owner "type" 

1 - Intruder "type" 

D - Dove strategy 
H - Hawk strategy 

O - An equilibrium 



D H 



2,2 


2.4 


o 


0.0 



D H 



2,2 


( 5 ) 


4,2 


0,0 



Figure 3 and Figure 4. An ESS in behavior strategies which yields the payoff of 
a (symmetric and strict) totally correlated equilibrium. 

Figure 3. A representation of an asymmetric conflict (bimorphic population with 
owner and intruder phenotypes), in a semi-extensive form. Each of the two players 
may be of any one of these two phenotypes (with equal probability). The phenotype 
“owner” always plays H and “intruder” always plays D. (0, 1) means that nature 
has chosen Player I as owner and Player II as intruder. (/,0) is interpreted in a 
similar way. The lottery (of nature) 1/2(0, J)-hl/2(J, 0) is identical to Selten’s basic 
probability distribution. The possible roles of animals (i.e., owner or intruder) are 
chosen randomly with equal probability. 



Fig. 6). Such efficient equilibria, in situations of incomplete information, have 
not been suggested before in evolutionary ecology. 



2. Correlated Equilibria 

Let A he an m X m fitness matrix, let = {1, . . • ,m}, and let S = 

X S"^. Thus, the payoff functions of the game specified by A are given by; 

= (0‘jk,akj), all (j,k) 6 S. 

A correlated strategy pair (c.s.p.) is a function / from a finite probability 
space r into S (see Aumann, 1987). Let / be a c.s.p. . The two components 
of / are denoted by and p. Thus, / = (/^ /'■^). If i G {1, 2} then we denote 
—i = {1, 2} \ {i}. Also, throughout the sequel, E denotes “expectation”. 




154 



Nature 



chooses 




1 - Player I 

2 - Player II 

0 - Owner "type" 

1 - Intruder "type” 



D - Dove strategy 
H - Hawk strategy 



c.e.d. - Correlated equilibrium 
distribution 



O 



- Information set 



Player 2 
D H 



2,2 


2,4 


4.2 


0,0 



0 


1 

2 


1 

2 





Figure 4. A representation of an asymmetric conflict by an extensive game. The 
c.e.d. matrix is the distribution of the correlated equilibrium l/2(if, D)-hl/2(Z>, H). 
An information set of a player is a set of decision points with the following proper- 
ties: a) In all decision points, the player has the same information. (According to 
his information, the player cannot distinguish between the different points in the 
set.) b) In all decision points, the player has the same number of alternatives. 
Figures 3 and 4 represent the same (asymmetric) conflict which is payoff- irrelevant. 



Definition 2.1. A c.s.p. / is a correlated equilibrium (c.e.) if for every i G 
{1, 2} and every 5* 

Eh\f)>Eh\r\g^of) 

This definition is a simple expression of “incentive compatibility” (Bayesian 
rationality sensu Aumann, 1987); if ‘nature’ tells a player to play a certain 
strategy, then it is to the benefit of the player to follow nature’s instruction. 

Definition 2.2. A c.e. / is strict if for every i g {1,2} and for every g^ : 
S'^ such that Pr{g'’ o /* ^ /^} > 0 

Eh\f)>Eh\r\g^op) 

Let / be a c.s.p. and let {j,k) G S. We denote qjk — Thus, 

{qjk) is the distribution of /. If / is a c.e. then its distribution is called a c.e. 
distribution (c.e.d.). 





155 




Payoff to player I 

Figure 5. Various regions in the set of feasible payoffs of the “game of chicken” of 
Figures 2. a, 3, and 4. 

(1) The feasible set itself is the convex hull of (0,0), (2,4), and (4, 2). 

(2) The hatched area is the convex hull of the three Nash equilibrium payoffs (2, 2), 
(2, 4), and (4, 2). (2, 2) is the outcome of the mixed equilibrium and (2, 4) and 
(4, 2) correspond to pure equilibria. 

(3) (2|, 2|) is the outcome of a partially correlated equilibrium. 

(4) The interval [(0, 0), (3, 3)] on the 45° line is the set of possible payoffs in biolog- 
ical contests (based on this game). The half-open interval ((2,2), (3,3)] is the 
set of payoffs to strict and symmetric correlated equilibria. By the equivalence 
theorem each point in this interval is an outcome of an ESS of some asymmetric 
conflict based on our game. The payoff vector (3,3), which is Pareto optimal, 
is therefore an outcome of an ESS. 



Definition 2.3. A c.s.p. is symmetric if its distribution (qjk) is symmetric, 
that is Qjk = Qkj for {j,k) G S. 

Example 2.4. Let h be given by the following bimatrix: 



D 

H 



D 



H 



6,6 



2,7 



7,2 



0,0 



This is the “game of chicken” (see Figures 6 and 7). 
Then 




156 



Nature 



chooses 




6.6 


2.7 


7.2 


0.0 



1 

3 


1 

3 


1 

3 


0 



1 - Player I 

2 - Player D 

0 - Owner "lypc” 

1 - Intruder "lypc" 

D - Dove strategy 
H - Hawk strategy 
c.e.d. - Correlated equilibrium 
distribution 

C ^ - Information set 



Figure 6. A representation of an asymmetric conflict with incomplete informa- 
tion (and without “role asymmetry”), by an extensive game — the case of Davies’ 
butterflies (a detailed exposition of the example is given in Section 5). 



1/3 


1/3 


1/3 


0 



is a symmetric and strict c.e.d. which cannot be obtained by mixed strategies 
(see Aumann, 1987). 



3. Asymmetric Animal Conflicts with Payoff- Irrelevant 
Types and Role Asymmetry^ 

An asymmetric animal conflict is a system {U,p, a) where U = {ui , . . . , Un}, 
n > 2, is the set of possible types of the animals; p is a probability distribution 
on U X U that satisfies p{u,v) = p{v,u) iov u,v G U] and a is a function 
which assigns to each pair of types {u,v) a payoff matrix a{u^v) (of the first 
player). Formally, {U,p, a) is a two-person game with incomplete information. 
U is the common set of types of the players; p determines the (consistent) 
beliefs of the two players; and a is the payoff function. (U,p,a) satisfies role 
asymmetry if p{u,u) = 0 for all u e U. (C/,p, a) has pay off -irrelevant types 
if there exists an m x m (fitness) matrix A such that a(n,u) = A for all 




157 




Figure 7. Various regions within the set of feasible payoffs of the “game of chicken” 
of Davies’ butterflies (see the game of Figure 6 ). 

( 1 ) The feasible set itself is the convex hull of the vectors (0,0), (2, 7), ( 6 , 6 ), and 



(7, 2 ). 

(2) The hatched area is the convex hull of the three Nash equilibrium payoffs 

(2,7), (7,2) and ( 43 , 43 ). r i j *ru • 

(3) The dotted region is the set of payoff vectors of correlated equilibria, which 

are “above” the convex hull of payoffs of Nash equilibria. 

(4) The interval [(0,0, ( 6 , 6 )] on the 45° line is the set of possible payoffs in bi- 
ological contests (based on this game). The set of payoffs of strict and sym- 
metric correlated equilibria that dominate the Nash payoff is the open interval 
(( 4 |^ 4 |)j (5|,5|)). Each of these payoff vectors may be obtained as the out- 
come of an ESS (by the equivalence theorem). 



Finally, we remark that payoff vectors of correlated equilibria also exist below the 
line through (7,2) and (2,7). For example, each point in the interval [(3|,3|), 
( 4 |, 4 |)j is the outcome of a correlated equilibrium. 





158 




Figure 8. A graphic representation of the “game of chicken” of Figure 1. We assume 
here that a > {h-{- s)/2 and 6 > s (as in the figure). 

(1) The feasible region is the convex hull of the vectors (d, d), (a, a), and 

{b,s). 

(2) ((6-4-s)/2, (6 + s)/2) is the outcome of a strict symmetric and totally correlated 
equilibrium. Hence, it is also the payoff vector of a (pure) ESS of an asymmetric 
conflict with two (complementary) roles and “role asymmetry”. 

(3) The open interval (((6+ s)/2,(6 -f s)/2),(c, c)) on the 45° line is the set of 
payoff vectors to strict, symmetric, and partially correlated equilibria, which 
(Pareto) dominate the vector ((6 -f- s)/2, (6 + s)/2). c is given by the following 
formula. 

^ _ 2(6 — a)(6 + s)/2 (s — d)a 

2(6 — a) + s — d 2(6 — a) s — d 

Thus, c = a(6 + s)/2 + (1 - a)a where a = 2 (b-a)+Ld ■ 

The details of the computation of c are given in the Appendix B. 




159 



u^v G U . An asymmetric conflict with payoff-irrelevant types is denoted in 
the sequel by (C/,p, A), where A is the payoff matrix. 

Let (U,p, A) be an asymmetric conflict. A (behavior) strategy is a function 
(f : U Sm, where Sm is the (standard) (m — l)-dimensional simplex. Thus, 
ip is a, function which assigns to each type u G U a mixed strategy in the 
(matrix) game A. If (j% i = 1,2, is a strategy of i, then the payoff to i in 
(U,p,A) is Eh^{a ^ i = 1,2. a is a symmetric Bayesian equilibrium if a 
is a best reply to cr. cr is an evolutionarily stable strategy (ESS) if: 

(j is a symmetric Bayesian equilibrium; (3-1) 

if /X is a best reply to <t, that is, Eh^ {a, a) = Eh^{p,a), 

and p ^ (Jy then Eh^{a,p) > Eh^{p,p). (3.2) 

A behavior strategy a is pure if a{u) is a pure strategy (in the matrix game 
A), for every u eU. The following result is a corollary of Selten (1980). 

Proposition 3.1. Let {U,PyA) be an asymmetric animal conflict with payoff- 
irrelevant types and role asymmetry. If a is an ESS then 

a is pure; (3.3) 

if p ^ a then Eh^{p,a) < Eh^{a,a). (3.4) 



4. The Main Result 

4. a. An Equivalence Theorem 

Theorem 4.1: Let A be an m x m fitness matrix. Then (qjk) is the proba- 
bility distribution of a strict and symmetric c.e. iff there exist an asymmetric 
animal conflict with role asymmetry and payoff-irrelevant roles (?7,p, A), and 
an ESS a, such that (qjk) is the distribution of (cr, cr). 

Proof: Sufficiency. Let {Uyp,A) be an asymmetric conflict (consid- 
ered as a 2-person game with incomplete information), and let <j be an ESS. 
Because cr is a (symmetric) pure Nash Equilibrium in ([/,p, A), the distri- 
bution of (cr, (j) : U X U 5 is a c.e.d. (see Aumann, 1987, Sec.3). As p 
is symmetric, (qjk), distribution of (cr, cr) is symmetric. Also because of 
(3.4), (cr, cr) is strict ((cr, cr) itself may be considered as a c.e.; see Aumann, 
1987, Sec. 4(g)). 

Necessity. Let (qjk) be a strict and symmetric c.e.d. . Because (qjk) is 
strict, it satisfies the following inequalities: 

m m 

k) - h}{l, k))qjk > 0 if / j and ^ qjk > 0 (4.1) 

k=l k=l 




160 



Now we have to find an asymmetric animal conflict (f/, p, A) and an ESS a 
with the foregoing properties. Let W = {ei,. . . , em} be a set with m members 
and let 

U = Wx{I,0} = {(ei, /), . . . , (e„, I), ){ei,0), (e„, O)} 

(ei, . . Cm may be considered, e.g., as degrees of fighting experience, O — as 
ownership, and I as intrusion.) Define a probability distribution p onU xU 
by p{{ej,I),{ek,0)) = p{{ek,0){ej,I)) = qjk/ 2 , and p(u,v) = O otherwise, 
for all u, V e U. Finally, define a{ej,I) = a{ej,0) = j for j = 1, . . ., m. 
Then {U,p,A) is an asymmetric animal conflict with role asymmetry. Also, 
because (qjk) is a symmetric c.e.d, a is a symmetric Nash equilibrium. By 
(4.1) a is the unique best reply to a. Hence, a is an ESS. Q.E.D. 

4.b. An Example 

Consider again Example 2.4. We now show how to apply Theorem 4.1 in 
this particular example in order to construct an asymmetric animal conflict 
(U,p,A) and an ESS a whose distribution coincides with the given c.e.d. 
Accordng to the proof of the necessity part, let U = {(ei,/), (e 2 ,/), (ei,0), 
(^ 2 , 0 )} and let p be given by the matrix 







(e2,I) 


(ei,0) 


(£2,0) 




0 


0 


1/6 


1/6 


(e2,/) 


0 


0 


1/6 


0 


(ei,0) 


1/6 


1/6 


0 


0 


(e2,0) 


1/6 


0 


0 


0 



Then (U,p, A), where A is the fitness matrix of Example 2.4, is an asymmetric 
animal conflict with role asymmetry. The behavior strategy a which is defined 
by (j(ei,/) = a(ei,0) = D and a{e 2 ,I) = cr(e2,0) = is an ESS with the 
distribution of Example 2.4. 

Remark 4.2. Let A be a fitness matrix and let {qjk) be a strict and symmet- 
ric c.e.d. By Theorem 4.1 we may construct an asymmetric animal conflict 
(C/,p, A) with role asymmetry and an ESS a with the distribution (qjk)’ How- 
ever, the construction is not unique. Indeed, our particular method is quite 
arbitrary (except that it seems to minimize the number of roles). Therefore, 
we cannot predict the structure of asymmetric animal conflicts that might 
arise in nature. 




161 



5. A Generalization and Examples 

5. a. Asymmetric Animal Conflicts When Contestants Can Have 
Identical Roles 

Let A be an m X m fitness matrix and let {qjk) be a symmetric and strict 
c.e.d. We can always find an asymmetric animal conflict {U,p^A) with m 
roles, and perhaps without role asymmetry, that has a pure ESS a such that 
the distribution of (cr, cr) is {qjk)- Indeed, let U = {ri,...,rm} be a set of 
m possible roles. Define a probability p on U x U hy p{rj,rk) = qjk- Then 
{U,p,A) is an asymmetric animal conflict with payoff-irrelevant types (but 
perhaps without role asymmetry). Define now a (pure) behavior strategy a 
in (U,p,A) by (r{rj) = j, j = l,...,m. Because (qjk) is a symmetric and 
strict c.e.d., a is an ESS. Thus, we can reduce the number of roles to m if we 
do not insist on role asymmetry. 

In 5.b.2 we shall show how to construct an asymmetric animal conflict, 
with two roles and without role asymmetry, and an ESS a that represent 
Example 2.4 (see also Figure 6). 

5.b. Degrees of Seniority as a means to obtain correlated equilibria 
in the “game of chicken”: The field case of Davies’ Butterflies. 

5.b.l. A case of total correlation 

One of the most interesting field studies investigating the “Dove-Hawk con- 
flict” in biology is that of Davies (1978) (for other examples see Maynard- 
Smith, 1982). Male butterflies (speckled wood species) search for sunspots 
which axe clearings in forests. Males compete for sunspots because these are 
the best place to find females. When two males meet in the same sunspot a 
spiral upward flight takes place that ends always in the following fashion: One 
butterfly leaves the sunspot and the other becomes a resident. Davies found 
that in such encounters, the “owner” of the sunspot, that is, the first to find 
it, always wins the clearing. To test whether ownership rather than strength 
or other payoff-relevant traits determine the outcome of the contest, Davies 
caught “owners” and introduced them as “intruders” into other sunspots. In 
all the experiments in which “role asymmetry” was created, Davies observed 
short spiral flights with the same result, confirming Tinbergen’s statement 
“the resident always wins” . 

When Davies simultaneously introduced into the same sunspot two male 
butterflies, a spiral flight occurred which was 10 times longer than typical 
ones. Davies interpreted these long spiral flights as escalated encounters, 
where both contestants play “Hawk”. Maynard-Smith (1982; p.99) empha- 
sizes that Davies’ result “fits beautifully with the predictions of the Hawk- 
Dove Bourgeois model”. Figures 3, 4, and 5 describe in detail how the Davies’ 
butterflies reach correlated equilibria. 

This case, as well as all other biological contests in which only comple- 
mentary roles meet, are called totally correlated^ cases (Forges, 1986). This 




162 



is exactly the result of the seminal model of Selten (1980) for the “game 
of chicken” . In such biological games nature selects through evolution such 
contests in which only non-identical roles meet. 

Fig. 1 and Fig. 4. demonstrate how one can solve pairwise interaction of 
“games of chicken” by getting only pure strategies encounters of {H, D) or 
{D^H) with payoffs’ pairs (6, s) (5,6), and avoiding encounters of the pair 
{H^H) with low payoff (d, d). 

5.b.2. A case of partial correlation 

Totally correlated® cases are very special examples of correlation (Forges 
1986, 1990). In the formal model (Sec. 4) partial correlation is achieved by 
having in a row j of the probability distribution matrix, more than one entry 
with Qjk > O, where qjk is the probability of the encounter between strat- 
egy j and strategy k, Aumann (1974, 1987) emphasized that such partially 
correlated® strategies may yield equilibria which have higher payoffs than 
other Nash equilibria and totally correlated equilibria (see the examples of 
Figures 6, 7 and, for the general case, see Fig. 8). Can we find a biological 
situation in which such partially correlated cases can be materialized? Selten 
(1980) already considered situations in which a row can have more than one 
entry with basic probability greater than zero (in Selten ’s model for more 
than one entry, u, Wuv > 0). Such a mathematical condition corresponds to 
a situation of incomplete information. Below, we use the Davies’ butterflies 
example as the basis for a model in which incomplete information leads to 
partially correlated equilibrium. The resident (or owner) role will be called a 
senior role and the intruder role will be called a junior role. 

Male butterflies compete for sunspot clearings in a forest in order to 
fertilize females. Females mate only with males that occupy a clearing. The 
first male to reach a clearing checks the opening and realizes that no other 
males are there. Before time T is reached, the first male in the clearing has 
a “junior” role. If no other male has detected the clearing during the time 
interval T, the first male switches to the “senior” role. 

The second male reaching the same opening immediately will observe the 
first male and will assume a “junior role” . 

Our hypothetical “Junior-senior example” does not allow the possibility 
of hidden butterflies and assumes that males have excellent ability to detect 
other males. In such situations the probability that two males arriving at 
the same opening will acquire the same “senior role” is zero^. We assume 
that the density of male butterflies in the forest is high enough so that the 
probability of a pairwise encounter of two butterflies which reach the same 
clearing before the time T is one-third. Then we will get a situation in which 
2/3 of the encounters are between different roles (half Senior- Junior and 
half Junior-Senior) and one-third of the encounters are between “Junior- 
Junior”. There are no encounters of “Senior-Senior”. In such circumstances, 
if the “Senior role” will play the “Hawk” strategy and the “Junior role” will 
play the “Dove” strategy, the payoff of a strict and symmetrical, correlated 




163 



equilibrium can be obtained (Figures 6 and 7). This is to say that none of the 
individuals which have such “conditional pure strategy” (sensu Parker, 1984) 
“can choose a strategy” by which he can get a higher fitness. We have shown 
formally (Sections 4, 5) that the distribution of a strict symmetric correlated 
equilibrium is always the distribution of an ESS. 

Have such partially correlated biological situations been observed in Na- 
ture? In the study of the speckled wood butterfly Davies did not report the 
percentage of pairwise contests between intruders (the analog to our pairwise 
Junior encounters). Jacques and Dill (1980) studied contests of spiders and 
report on many pairwise contests between wandering individuals (analog in 
our model to intruder and Junior roles), that end by one of the contestants 
winning without escalation. We hope that the presentation of the idea of cor- 
related equilibrium to ecologists will open the investigation of the question: 
Are contests with partially correlated strategies (which implies incomplete in- 
formation) which settle “game of chicken situations” common in plants and 
animals? 

What is the internal logic in the base of the “Junior-Senior example”? 
There is an asymmetry in the information. Our model analyses the biological 
case “a^ if” the Darwinian logic underlying such behavior is the following: 

There is asymmetry in the knowledge of the two role types: the Senior type 
knows both his type and the type of his opponent. The Junior type knows 
his type but he is not sure whether his opponent is Junior or Senior. In such 
circumstances there is a range of payoffs in which it is best for the Junior 
type to always play Dove and not Hawk. Such an example is demonstrated 
in Figures 6 and 7. 



6. Discussion 

Evolutionary game theory studies only symmetrical games (Hammerstein 
and Selten, 1991). Fig. 1 represents a group of biological games that we call 
“monomorphic games”: These are games with a genetically monomorphic 
population of potential players. In a monomorhic game equilibrium all the 
individuals in the population must play the same strategy. Thus, the possible 
payoff pairs for pure strategies in Fig. 1 are (a, a) and (d, d). Indeed, in the 
prisoner’s dilemma game when b > a and s < d, (d, d) is an equilibrium 
payoff. But if s > d, the bimatrix game represents the “game of chicken”, 
which has two pure equilibria that are asymmetric and therefore cannot be 
realized in intraspeciflc competition with a monomorhic population. We show 
that by adding further structure in the form of a lottery device, the two pure 
equilibria of the normal form game can be combined to produce a Nash 
equilibrium in a semiextensive game^. The chance device in nature turns out 
to be a situation which assigns to the two opponents with the same genotype 
different information situations, which are known in biology as “roles”. In 
Aumann’s model, the lottery between the two pairs of strategies is part of 




164 



the solution concept, and indeed in nature the biological correlation device of 
the roles is part of the solution of the game. (Both the roles and the strategies 
are chosen simultaneously and the temporal order in Figures 4 and 6 is only 
to help in understanding the model). This is in contrast with Selten (1980) 
and Hammerstein and Selten (1991), who interpreted the chance device as 
part of the model. Overall, this semi-extensive biological game comprises a 
symmetrical game with asymmetric conflict (Hammerstein and Selten 1991, 
p. 48). It is worth noting that the chance device generates role types which do 
not influence the payoff of the game. Both Aumann (1974) in game theory and 
Maynard-Smith and Parker (1976) in animal behavior have independently 
dealt with this case. 

The competitive interaction among individuals in a natural population 
can drive the opponents into situations where both will receive small payoff. 
Such an inefficient situation for both players can result in the following games: 

a) In the prisoner’s dilemma, at equilibrium, both players get very low payoff 
(see Fig. 1). 

b) In the “game of chicken” (Fig. 1) the only symmetrical equilibrium is that 
of mixed strategies in which the players receive lower payoff compared with 
what they could get in the pure equilibria (Fig. 5). 

The following question can be raised. Is it possible in nature to achieve 
more efficient equilibria by natural selections of biological traits? The for- 
mal solution to this question was suggested by Aumann with the correlated 
equilibrium model. To implement Aumann’s correlation machine, evolution 
should develop and invent behavioral or morphological traits with the follow- 
ing characteristics: 

a) The pairwise meeting of phenotypes (carriers of traits) are totally or par- 
tially correlated. 

b) Traits that develop information asymmetry between the opponents. 

c) Traits that create and develop a situation of incomplete information for 
at least one of the opponents. 

Indeed, we find in nature a number of behavioral traits that lead to semi- 
extensive games with correlated equilibrium. Behavioral conventions such 
as seniority, territoriality, and sexuality in parental care may be viewed as 
correlation devices which have been selected by evolution because of their 
highly efficient equilibria. The following characteristics are typical of such 
traits: 

1. Encounters occur mainly among different roles (in totally correlated 
cases). 

2. In cases of encounters between the same roles , evolution will select be- 
havior leading to incomplete information that will result in an efficient 
equilibrium with the appropriate “correlated equilibrium distribution” . 

3. The roles do not have an influence on the payoffs. 




165 



We suggest that natural situations in which different individuals in the 
population can behave differently (i.e., choose complementary roles) and 
thereby render the genotype payoflF more efficient, can evolve only if nature 
builds semi-extensive games in which strict symmetric correlated equilib- 
ria (which correspond to ESS’s) can be materialized Thus, asymmetric 
pairwise interactions are the natural lottery device that maintain correlated 
strategies. 

Thus, the evolution of conventions such as seniority, territoriality and 
unisogamy (in conflicts of parental care) in nature as well as in human cul- 
ture, does not have to be interpreted by “altruistic means” or by “species 
good” (Parker, 1978; Parker and Hammerstein, 1985), but by individualistic 
behavior (i.e., maximization of fitness) which selects those natural correlation 
devices which produce the greatest fitness. The pure strategies we observed 
in nature in such examples are the result of the existence of strict symmetric 
correlated equilibria^. Several human cultural ceremonies can be interpreted 
along this line (Selten, 1991). 

We now try to explain the adaptive value of sex in animals with two sex 
types, by means of our model. Many explanations have been suggested to the 
evolution of sex (Michod and Levin (1988)). Our model is complementary 
and explains how the specialization of the parents in different roles, when 
raising the offspring, leads to a biological equilibrium, that is, to an ESS. 

Aumann’s notion of correlated equilibrium explains the equilibrium be- 
tween male and female in raising their offspring, although they play different 
roles and, hence, receive different payoffs. Indeed, the male - female game, 
for animals, has two asymmetric roles, namely, the two sexes. We may as- 
sume that each player has two strategies: D:= feed the offspring; and H:= 
forage in order to get food for the family. In this way we obtain the situation 
depicted in Fig 3 when we replace “owner” by “male” , and “intruder” by “fe- 
male” . Notice that the equilibrium in Figure 3 is totally correlated and payoff 
dominates the symmetric Nash equilibrium. Also, the roles of the players are 
chosen by Nature. 

Independently of our work, Cripps (1991) has proved a somewhat more 
general result than our Theorem 4.1. Whereas our theorem applies to bima- 
trix games of the special form (A, A^), Cripps proves an equivalence theorem 
for general bimatrix games. We do not know of any biological model which 
is solved by Cripps ’s result and for which our result does not apply. Further- 
more, Cripps does not present any biological applications in his work. Finally, 
our Theorem A in Appendix A is more general than the Sufficiency Part of 
Cripps’s Theorem (Shmida and Peleg, 1991). 

From the theoretical point of view, this study makes a connection be- 
tween Selten’s model and Aumann’s “correlated equilibrium distribution”. 
The main theorem shows that the distribution of every ESS in Selten ’s model 
is the distribution of a strict symmetric correlated equilibrium, and vice versa. 
From the biological point of view, our main result may help the biologist in 




166 



the following way: Suppose he knows the fitness matrix of a biological con- 
flict. Then, he may ask: Which payoffs are obtainable with the help of some 
payoff irrelevant asymmetric conflict baaed on this matrix? The answer to this 
question is given directly by our theorem using Aumann’s definition of corre- 
lated equilibrium, without the need to consider all possible payoff irrelevant 
asymmetric conflicts in Selten’s extensive form. In particular, it supplies a 
computational procedure to obtain all attainable payoffs in “payoff irrelevant 
asymmetric conflicts”. 

Compared with Aumann’s paper (1974), Selten’s theorem (1980) represents a 
different approach to modelling nature. While Selten identifies the roles and 
the “baisic probabilities” as part of the game model, Aumann represents the 
correlated equilibrium distributions as part of the solution concept. We fully 
agree with Professor Selten (per. comm.) that natural situations should be 
explored in game theory by explicit models and one should explain explicitly 
Aumann’s “correlation device” . 

But, on the other hand, as we are concerned with biological evolutionary 
games we prefer to approach the correlation device (i.e., the behavioral games 
which control the conditional phenotype behavior) and the probabilities of 
encounters of different roles as part of the solution concept. Because, in bi- 
ology, there is no temporal order in the semi-extensive “Hawk-Dove game” 
and evolution has selected both the strategies and the roles “as if” it were a 
simultaneous rational decision. 



Footnotes 

^ The Hawk-Dove conflict corresponds to the “game of chicken” situation 
(see Aumann 1974, 1987). 

^ A semi-extensive game in our paper is an extensive game in which a chance 
move chooses different roles (types) which play the same matrix game. In 
usual extensive games, the payoffs may be arbitrary and are attached to 
the endpoints of the game tree. The term ‘semi-extensive’ was suggested 
to us by R. Selten. 

^ By asymmetric animal conflicts the ethologists (people who study animal 
behavior) mean that the contestants in the game may have different “roles” , 
e.g., female vs. male; owner vs. intruder; first vs. second (see Maynard- 
Smith and Parker, 1976 and Hammerstein, 1981). 

Anecdotally, what makes the correlation in the sense of Aumann possible is 
the phenomenon of “uncorrelated asymmetry” (i.e., “payoff irrelevance” — 
see Sec. 3) in animal conflict. Indeed, we wanted to call our paper “Un- 
correlated asymmetry as a mechanism to obtain correlated equilibrium in 
biological conflicts” . 




167 



^ What was considered in Selten (1980) as “information asymmetry” (no 
contests between opponents with the same information) is named by Ham- 
merstein and Selten (1991) “role asymmetry”. We will use only the latter 
term in the present paper. 

® A correlated strategy for the “game of chicken” is totally correlated if the 
pure strategies that it dictates for the players are fully correlated, that is, 
the pure strategy used by Player I determines the pure strategy used by 
Player II, and vice versa. A correlated strategy is partially correlated if 
it is not totally correlated. In biological contests based on the “game of 
chicken” the totally correlated strategies are: 



p 


0 


0 


1-p 



0 < p < 1 and 



0 


1/2 


1/2 


0 



^ For the formal model to be complete and consistent with the example in 
Fig. 4, we must add the following assumptions: 

1. The probability of finding an empty clearing (being first) is 1/2, and 
the probability of being second is 1/2. 

2. The probability of the arrival of a second male in the interval [0,T] is 
1/3. The probability of the arrival of the second in [T, 3T] is 2/3. 

3. After time 3T the first butterfiy leaves. 

4. Three or more butterflies never share the same clearance. 

5. The probability of simultaneous arrival of two males is 0. 

6. If a “senior” butterfly has expelled a “junior”, then no new male will 
enter the clearing before the “senior” leaves. 

® In interspecific competition there is no need to require symmetric equilib- 
ria between the opponents. And, indeed, it is common in such situations 
in nature to obtain equilibria with payoff pairs (6, s) and (s, b) (see Fig. 1). 
Such equilibrium points correspond to an evolutionary phenomenon of spe- 
cialization and differentiation of two different species. 



A. Some remarks on Theorem 4.1 

The Sufficiency Part of Theorem 4.1 is a special case of the following general 
result. Let F = (Kj P, U, Cjp,r) be a finite n-person game in extensive form 
(see van Damme (1987, p.lOO)), let Z be the set of endpoints of P, and let 
G = (A^, . . . , . . . ,u"') be a finite n-person game in strategic form. 

Denote by F(G) = (K, P,U,CjP,G) the game obtained from F by replacing 
r(z) by G at every z ^ Z (i.e., if a play of F arrives dX z ^ Z then the players 
have to play G in F{G)). A pure strategy of player i, 1 < z < n, in F{G) is a 
pair (d% 5*), where d* is a pure strategy (for i) in F and 5'' : Z ^ Clearly, 
if a = (cr^, . . . ,(j”) is an n-tuple of mixed strategies in F{G), then a induces 
a correlated strategy in G. 




168 



Theorem A: If cr = (cr^, . . . ,cr”) is a Nash equilibrium in mixed strategies 
in r{G), then is a correlated equilibrium in G, 

Proof: F{G) is equivalent to the following game Gc (in extensive form): 
Stage 1: Each player 1 < i < n reports a pure strategy d* in P to a media- 
tor. We denote by . . . ,(P) the probability distribution on Z which is 

induced by Stage 2: The mediator chooses z e Z according to 

the distribution . . ,d”) and informs each player on Stage 3: Each 

1 ^ ^ ^ chooses S^{z) G A*. 

Clearly, r{G) and Gc have the same strategic form. Hence, cr is a Nash 
equilibrium in Gc* Therefore, by the revelation principle (see Myerson (1991), 
Chapter 6) , is a correlated equilibrium of G. 



B. The computation of c in Figure 8 



We consider the symmetric bimatrix game 



CX, CL 


s, b 


b,s 


d, d 



where b>a>d^b>s>d and a > (6 -f s)/2. 

We are interested in payoffs of symmetric and strict correlated equilibria 
that Pareto dominate the vector ((6 + 5)/2, {b -h s)/2). 



Let /X = 



a 


P 


P 





be a symmetric correlated strategy, that is: 



a + /3 + 2p = 1 and a, > 0. 


(1) 


/X is a strict correlated equilibrium iff 




aa -f ps > a6 -1- pd; 


(2) 


and 




pb ^ Pd > pa -{■ /3s. 


( 3 ) 


Clearly, (1), (2), and (3) define a convex set. 




The payoff to p is (h^(p), /i^(p)) where 




(p) = h^(p) = aa -f p{b + s) -f /3d. 


( 4 ) 



Obviously, 

c = sup{/i^(/x)|^ satisfies (1), (2), and (3)}. 
Because d < (6 *f 5)/2 we otain from (l)-(5). 



( 5 ) 




169 



c = sup{aa + p{h + s)\aa ps > ab + pd and a 4- 2p = 1} 



= sup {a + 2p 



{b-\- s — 2a) 



b — a 



2(b-g)(b4-g)/2 

2(6 — a) 4- 5 — d 2(6 — a) 4- 5 — d 



^ 2(6 - a) + s 
{s — d)a 



4 



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Avi Shmida and Bezalel Peleg 



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Recurring Bullies, Trembling and Learning^ 



Matthew Jackson and Ehud Kalai 

Kellogg Graduate School of Management, Northwestern University, Evanston, IL 
60208-2009 



Abstract. In a recurring game, a stage game is played consecutively by 
different groups of players, with each group receiving information about the 
play of earlier groups. Starting with uncertainty about the distribution of 
types in the population, late groups may learn to play a correct Bayesian 
equilibrium, as if they know the type distribution. 

This paper concentrates on Selten’s Chain Store game and the Kreps, 
Milgrom, Roberts, Wilson phenomenon, where a small perceived inaccuracy 
about the type distribution can drastically alter the equilibrium behavior. It 
presents sufficient conditions that prevent this phenomenon from persisting 
in a recurring setting. 

Keywords. Recurring Game, Social Learning, Chain Store Paradox 

1 Introduction 

In a recurring game, a stage game is sequentially played by different groups 
of players. Each group, before its turn, receives information about the social 
history consisting of past plays of earlier groups. Game theorists have stud- 
ied such recurring situations using various dynamics like fictitious play, last 
period best response, and random matching, since Nash in his dissertation 
(1950). 

Of interest here is a Bayesian version of a recurring game, where each 
stage game player is uncertain about the types of his or her opponents. Ob- 
served social histories axe used by players to update beliefs about the un- 
known distribution of types in the population. With time, players’ behavior 
converges to that of the Bayesian equilibrium of the stage game, as if the 
true distribution of types in the population were known. 

In a recent paper (Jackson and Kalai, 1995a) we present a general model of 
recurring games and sufficient conditions that yield such social convergence. 
Our purpose in this chapter is to apply this approach to Selten’s (1978) 
chain-store games and study the implications on the phenomenon illustrated 
by Kreps and Wilson (1982) and Milgrom and Roberts (1982) (KMRW for 

^ The authors thank the California Institute of Technology and the Sherman 
Fairchild Foundation for their generous hospitality while doing this research. 
They also thank Eric van Damme for comments on an early draft and the 
National Science Foundation for financial support under greints SBR-9223156 
and SBR-9223338. 




172 



short, also sometimes called the “gang of four”). A specific question is whether 
this phenomenon (described in more detail below) can persist in a recurring 
setting. To better fit the recurring setting we replace Selten’s chain-store 
story with a strategically equivalent story about a bully terrorizing a finite 
group of individuals, the challengers. The rest of the introduction presents 
a verbal description of our model and conclusions. A broader survey of the 
related literature can be found in Jackson and Kalai (1995a). 

The “challenge the bully” stage game (the bully game, for short) is played 
by a bully and L challengers. It consists of L preordered episodes each played 
by the bully and one challenger. In each episode the designated challenger 
chooses whether to challenge the bully or not, and if challenged, the bully 
chooses whether to fight or not. The bully prefers not to be challenged, but 
if challenged he would rather not fight. The challenger prefers challenging a 
nonfighting bully, but if the bully fights he would rather not challenge. (The 
single episode payoffs are represented in Figure 1 of the next section.) Both 
bully and challenger know the outcome of all previous episodes before making 
their decisions. 

Recall that Selten’s paradox is that backward induction applied to this 
game dictates that all challengers challenge and the bully never fights. Yet, 
when L is large, common sense suggests that the bully may choose to fight 
in early episodes in order to build a reputation so that challengers will not 
challenge.^ KMRW point out that the backward induction result depends 
on the complete information assumption. The resolution offered by KMRW 
is to replace the above game by a Harsanyi (1967-68) Bayesian game in 
which there is a small commonly known prior probability that the bully is 
“irrational,” e.g., he prefers fighting to any other episode outcome. In such 
a modified game the more reasonable behavior is obtained as the unique 
sequential equilibrium outcome, even when the realized bully is rational. Thus 
the phenomenon illustrated by KMRW is that in a world with a rational bully, 
a small uncertainty about the rationality of the bully can drastically change 
the equilibrium outcomes, and in particular, induce the rational bully to fight. 

It is not obvious whether this phenomenon will persist if this situation 
recurs. In other words, if new bullies are always born rational, but challengers 
are uncertain about this fact, would bullying behavior persist? On the one 
hand, it seems that statistical updating will lead observers (i.e., players in 
later rounds) through backward induction to the recognition that the bullies 
are rational and with it will come challenging and not fighting. On the other 
hand, a violation of a rational act by a bully that wishes to appear irrational, 
is more convincing when done in view of such learning. Thus, one is left 
with the question of whether such learning will lead the eventual discovery 
of the rationality of the population and thus the unraveling of the KMRW 



^ There axe, of course, Nash equilibria of this type; however, they do not survive 
backward induction. 




173 



phenomenon, or whether the actions of the players will slow the learning so 
that the rationality of the population is never learned. 

To meaningfully address this question, we extend the KMRW game to a 
“doubly Bayesian” recurring game with two “priors.” One prior, r, is a fixed 
probablity distribution according to which a bully-type is drawn before each 
stage game. We assume, however, that this type-generating distribution is 
not known to the players. They assume that it was selected before the start 
of the entire recurring game according to a known probability distribution F 
over a set of possible type generating distributions.^ 

This gives rise to what we call an uncertain recurring game whose ex- 
tensive form game can be described as follows. In period 0 nature randomly 
selects a type generating probability distribution r from a set of possible 
ones, according to the prior probabilities F(t). Players know that a selection 
was made according to T, but no information is given about the realized r. 
For period 1, one bully and L challengers are selected. The bully is randomly 
assigned a type according to the fixed (unknown) type-generating distribu- 
tion r. Only the bully is informed of its realized type and the L -h 1 players 
proceed to play the L episodes of the bully game. Their realized play path 
becomes publicly known. The game recurs, following the same procedure. In 
each period t > \ a new set of challengers and a bully are selected. Again, 
the bully is privately assigned its type according to the same unknown r and 
the players proceed to play the L episodes of the bully game, with the play 
path publicly revealed. 

To study the recurring version of the KMRW phenomenon we will focus 
the above described game, with a realiztion of r that selects rational bullies 
with probability one, indicating a world in which bullies are really rational. 

Note that the above separation to two distributions is necessary since 
a restriction to a single prior r, representing both actual type-generation 
and social beliefs, will force one of the following misrepresentations of the 
phenomenon. If the single r assigns probability one to a rational bully, then 
there would be no social uncertainty about it. On the other hand, if the single 
T assigns positive probability to irrational bully types, then in the recurring 
setting with probability one some irrational types will be realized, leading to 
the KMRW equilibrium, but in a world which truly has irrational bullies. In 
either situation, there would be no social learning. That is, observing past 

^ This approach may be viewed as the multi-axm-bandit “payoff” learning model 
adapted to the Bayesian repeated game “type” learning literature. [See Aumann 
and Maschler (1967) and followers, and the more closely related Jordan (1991).] 
The utility maximizing bandit player starts with a prior distribution over his set 
of possible payoff generating distributions. Ours is a multi-person version, with 
uncertainty over the distribution generating types, modeled through a prior 
over a set of possible distributions. A major difference between our model of 
learning in recurring games and the literature on learning in repeated games 
is that in our setting players are attempting to learn about the distribution of 
types that they will face in their stage game, while in a repeated game players 
learn about the actual opponents they repeatedly face. 




174 



stage games would tell a player nothing new about the distribution of types 
in the population. 

Our recuring game definition of a type is also significantly generalized. 
A type’s preferences can depend on the entire social history and not just on 
his own stage game. So, for example, we can model a “macho bully” whose 
utility of fighting increases after social histories with many earlier fighting 
bullies. Or in a multi-generational setting, a bully may prefer to behave like 
his ancestors. 

We refer to the Nash equilibria of the above uncertain recurring game as 
uncertain Bayesian equilibria. In playing an uncertain Bayesian equilibrium, 
expected utility maximizers perform Bayesian updating of the prior F to 
obtain updated posteriors over the set of type generating distributions. As 
a result, the strategies of each stage game constitute a Bayesian equilibrium 
of the stage game relative to the perceived distribution over types induced 
by the updated prior, but not necessarily relative to the true (realized) type 
generating distribution. 

This discrepancy partly disappears however, after long social histories. 
First, as a consequence of the martingale convergence theorem, the updated 
prior will converge (almost everywhere). This means that players’ learning 
disappears in the limit. Second, the updated distribution will be empirically 
correct in the limit. This means that players’ learning induced predictions 
concerning the play path will match the distribution over play paths induced 
by the true (realized) distribution over types. This second result can be ob- 
tained almost directly from a learning result in Kalai and Lehrer (1993). It 
means that in late games, any discrepancy between the true and the learning 
induced type generating distributions cannot be detected, even with sophis- 
ticated statistical tests. 

The fact that players learn to predict the play path with arbitrary pre- 
cision, does not necessarily imply that they have learned the true type gen- 
erating distribution r or the true distribution of strategies (including off 
the equilibrium path behavior).^ We present an example of challengers who 
never challenge, because their initial beliefs assign high probability to fighting 
bullies, even though the realized bullies would never fight if challenged. No 
learning occurs because no challenging ever occurs. Moreover, in the example 
a single tremble by a single challenger will reveal the non fighting nature of 
the bullies and would cause the entire equilibrium to collapse. 

In order to overcome this difficulty, we introduce trembles a la Selten 
(1975, 1983). The effect of introducing the trembles is to ensure that the 



^ This phenomenon also arises in non-recurring situations, such as those where 
players play a long extensive form or a repeated game. For detailed analyses of 
this in such settings, see Battigalli, Gilli and Molinari (1992), Pudenberg and 
Kreps (1988), Fudenberg and Levine (1993), and Kalai and Lehrer (1993). 




175 



social history leads to informative learning.^ For an exogenously given small 
positive probability we assume that every player chooses strategies that 
assign probabilities of at least q to each one of his or her actions at every 
information set. 

Assuming that players play an uncertain Bayesian equilibrium with trem- 
bling, we obtain the following conclusions for the case where the realized dis- 
tribution results in only rational bullies: With probability one, in late stage 
games, the challengers always challenge, and the bullies never fight. These 
rules are followed with the exception of occasional trembles. This result is 
obtained regardless of the initial beliefs of the society, provided those initial 
beliefs assign some positive probability to the distribution which selects only 
rational bullies. 

Thus the introduction of the trembles in the recurring model leads one to 
eventually play the trembling hand perfect equilibrium (defined with respect 
to the agent normal form). It is important to remark, however, that the 
trembles are not working directly (as in the definition of trembling hand 
perfection), but rather indirectly through the learning that they ensure. This 
is evident since in early stages, the equilibrium outcomes with trembles can 
still be that rational bullies fight and challengers not challenge. 

We wish to remark that the general message of our results is not simply 
the eventual decay of the KMRW equilibrium. For instance, if the realized 
type generating distribution is truly selecting some “irrational” fighting bul- 
lies, then this would also eventually be learned and the eventual convergence 
would be to the KMRW equilibrium - even if initial beliefs placed an ar- 
bitrarily high (but not exclusive) weight on rational bullies. Thus the more 
general understanding of our results is that in a recurring game, given some 
randomness to induce learning, equilibrium behavior will eventually converge 
to that £ts if players were playing an equilibrium knowing the underlying type 
generating distribution.® If players are truly rational, and the game is an ex- 
tensive form, then this will lead to eventual approximate play of a perfect 
equilibrium. 

2 The Recurring Bully Game 

The stage game is a bully game consisting of one bully player, b, and L 
challengers, (ci)i=i,...,L- In sequential episodes i = 1,...,L, challenger C{ 
decides whether to challenge (C) the bully or refrain (R) from doing so. If the 
challenger does challenge, then the bully has to decide whether to acquiesce 
(A) or fight (F). The corresponding episode payoffs, to the challenger and 
bully, are given by the extensive form game pictured in Figure 1. 

® See the concluding remarks of Pudenberg and Kreps (1995) for the suggestion 
of a similar approach to ensure learning and convergence to Nash equilibrium 
in a model with myopic long lived players. 

® See Jackson and Kalai (1995ab) for general results along these lines. 




176 



Ct 



Challenges 



Fights 



Refrains 



- 1,-1 



0,1 



Acquiesces 

1,0 

Figure 1 



Each player is informed of the outcomes of earlier episodes before making a 
decision. The payoffs of challengers are determined according to their episode 
payoffs while the payoff to the bully is the sum of his or her L episode payoffs. 

Formally, a play path of such a bully game is described by a vector 
p = (Xi, . . . ,Xl) with each X*, being in the set {(i?), (C, A), (C, F)}, de- 
scribing the outcome of the episode. A partial play path is such a vector 
(Xi, . . . ,X/), but with 0 < I < L. A (behavioral) strategy ai of challenger 
Ci consists of a probability distribution over the actions R and C for every 
i — 1 long partial play path (Xi, . . . , Xi_i). A bully strategy, 77 , chooses a 
probability distribution over the actions A and F for every partial play path 
of every length. A vector of strategies {{ai),T}) induces a probability distri- 
bution over the set of play paths and defines expected payoffs for all players 
in the obvious unique way. 

The recurring bully game is played in periods t = 1,2, — In each period 
t a new group of players (bully and challengers) are selected to play the stage 
game. Moreover, before choosing their strategies, they are informed of the 
play paths of all earlier groups. 

A social history h of length t is a vector of play paths h = (p^, . . . ,p^). 
The empty history 0 is the only history of length 0, and H denotes the set of 
all finite length histories. The players of the recurring bully game are denoted 
by and cf , where for every fixed h, and describe the bully and the 
challenger to play the z-th episode of the stage bully game after history of h. 
Thus, as the notation suggests, the players b^ and c^ know the social history h 
that led up to their game. Strategies in the recurring bully game are denoted 
by T]^ and erf, with the obvious interpretations. 

3 Games with Unknown Bully Types 

Alternative bully types describe different utility functions that a bully may 
have in playing a bully game. There is a countable set 6 of possible bully 
types. For each type 0 e 0 there is a function ue{h,p) describing the payoff 
to bully b^ of the play path p, when he or she is of type 6. 




177 



An example of a type is the one already defined whose payoff from any 
path (regardless of the social history) is the sum of the episode payoffs de- 
scribed earlier (in Figure 1). We refer to these payoffs as the “rational” ones 
and to this type as the rational type, denoted by r. 

An example of an “irrational” bully, motivated by KMRW, is one with 
payoff k > 0 for every “strong path” p in which he or she never acquiesces, and 
—k for weak paths p, exhibiting some acquiescing actions. We keep the terms 
“rational” and “irrational” in quotes, since “irrational” bullies are supposed 
to be maximizing their expected payoffs. It is simply that the “irrational” 
payoffs do not match those of the game in Figure 1. Of course, this is simply 
a modeling convenience as any behvior can always be made to be expected 
utility maximizing, if one can arbitrarily manipulate the payoff function. 

More complex “irrational” types can condition their payoffs on the ob- 
served social history. For example in the type just described we can replace k 
by k^, taking on large values after social histories h consisting only of tough 
play paths and small values for social histories h with weak bullies. This may 
represent an ego maniac bully who compares himself to earlier ones. 

Probability distributions over the countable set of possible bully types, 
0, are referred to as type-generating distributions. Every such distribution r 
defines a recurring Bayesian bully game as follows. After every social history 
h, nature chooses a bully type 9^, according to the distribution r (indepen- 
dently of all earlier choices in the game) . The bully is informed of his or 
her realized type 6 ^, before choosing his strategy. All other players are only 
aware of the process and know the distribution r by which 9^ was selected. 

Strategies in the Bayesian recurring bully game are denoted by {rj^) and 
(c7^), with rj 0 denoting the strategy of the bully when he or she is of type 
9 , and is a challenger strategy as before. 

Notice that a Bayesian recurring bully game, with a type generating dis- 
tribution r, induces after every history h a Bayesian bully game of reputation, 
similar to KMRW, with bully types payoffs given by ue(p,h) and a prior r 
on possible types 9. 

We wish to model, however, Bayesian recurring bully games with uncer- 
tainty about the type generating distribution. It is such uncertainty that 
introduces learning from the social history. An uncertain Bayesian recurring 
bully game (uncertain recurring game for short) is played as follows. In a 
0-time move, nature selects a type generating distribution r according to a 
prior probability distribution F. Without being informed of the realized r, 
the players proceed to play the r Bayesian recurring bully game. We assume 
that the exogenously given distribution F has a countable support (F{r) > 0 
for at most countable many r’s) and that it and the structure of the game 
are commonly known to all players.^ Players update F based on observed 
histories and the strategies that they believe to be governing play. This up- 

see Jackson and Kalai (1995a) for discussion of extensions to allow for players’ 

prior beliefs to be type dependent. 




178 



dated distribution induces a distribution over types, which is the basis for 
the Bayesian stage game that they play. 

Notice that an uncertain recurring bully game may be thought of as a 
“doubly Bayesian” game since we use a prior to select a distribution that 
serves as the priors of the stage games to come. The strategies of the uncertain 
recurring game are the same as the ones of the Bayesian game since the 
information transmitted and feasible actions are identical in both games. 
However, as will be seen in the sequel, expected utility computations and 
Bayesian updating are more substantial in the uncertain recurring game. 

An uncertain Bayesian equilibrium is a vector of strategies ((^0)), (erf )) 
which are best reply to each other in the extensive form description of the 
uncertain recurring game defined by the prior F. 

4 Social Learning 

In this section, we consider a given uncertain Bayesian recurring bully game 
with a prior F and fixed strategies ((ryf ), (erf)). We first clarify some proba- 
bilistic issues. 

A fully described outcome of the uncertain recurring game is a sequence of 
the type , — Such an outcome generates progressive socially- 

observed histories . . . with = (p^, . . . ,p^). To describe the probability 
distribution on the set of all outcomes it suffices to define consistent proba- 
bilities for all initial segments of outcomes. We do so inductively by defining 
P(r) = F(t) and 

P(t,0\p\. . . = P(t,0\p\. . . 

where is the probability ofp^"^^ under the strategies ((pf), (erf)) with 

h — and 6 — . 

Players, observing only the buildup of histories . . ., do not know 

the chosen type generating distribution r, but they can generate posterior 
probability distribution over it using the initial distribution and conditioning 
on current histories. Thus, their posteriors P^, P^, . . . , are defined for every 
type generating distribution f by P^(t) = P(t) and P^(f) = P{f \h^). 

Each such distribution over type generating distributions induces a direct 
distribution over types. Thus we have updated posterior distributions on the 
next period type denoted by 7^,7^ . . with j^{6) = Although 

we supress the notation, both P* and 7^ are history dependent. 

After every history the bully, , and the challengers, (cf ), play a 
Bayesian bully game. The bully knows his or her realized type, and it 
is commonly assumed that he or she was drawn according to the distribution 
7^. If the original strategies ((pf),(crf)) constitute an uncertain Bayesian 
equilibrium of the recurring game, then ((pf ), (erf)) with h = and 0 = 6^'^^ 
constitute a Bayesian equilibrium of the stage Bayesian game with the prior 

7 ‘. 




179 



Of course, the assumed prior, 7 ^, against which individual challenger op- 
timal strategies are chosen, is “wrong”, since the real prior, by which is 
chosen, is the unknown r. In Harsanyi’s (1967-68) definition of a Bayesian 
equilibrium this presents no problem. The strategies, with the commonly 
known assumption that the prior is 7 ^, are formally a Bayesian equilibrium. 
But in a recurring setting, when types are repeatedly drawn, a discrepancy 
between a real prior and an assumed prior may lead to statistical contra- 
dictions. These statistical discrepencies will disappear as players learn from 
observed histories. 

To describe the effects of learning we proceed in two steps. Our first propo- 
sition states that players’ updated distributions (over distributions) converge 
almost surely. Effectively, after some random time, players stop learning. 

Proposition 1. For almost every outcome, j^{0) converges to a limit 7 ( 0 ) 
uniformly for all types 6 E 0. 

Proposition 1 follows from the martingale convergence theorem. We refer 
readers to Jackson and Kalai (1995a) for a proof. 

Our next proposition states that when players stop learning they have in 
fact learned all that they could, and so they are arbitrarily correctly predict- 
ing the play path after some random time. Proposition 2 can be proven using 
Proposition 1, or can be seen more directly as a consequence of Theorem 3 
in Kalai and Lehrer (1993).^ 

With the fixed stage game strategies, let p be the probability distribution 
induced on the stage game play paths by the real prior, r, and let p be the one 
induced by the assumed prior, 7 *. If for every play path p{p^'^^) = p{p^^^) 
then no observable contradictions, even statistical ones can arise. We refer to 
the strategies and the assumed prior 7 ^ as an empirically correct Bayesian 
equilibrium whenever p = p. If for some 6 > 0 we have | p{p^'^^ ) — ^(p*+^) |< e 
for all play paths we refer to it as an empirically e-correct Bayesian 

equilibrium. 

Proposition 2. In an uncertain Bayesian equilibrium of an uncertain recur- 
ring bully game, for almost every outcome and for every e > 0 , there exists a 
time T, such that 7 ^, together with the induced t-\-l period strategies, consti- 
tute an empirically e-correct Bayesian equilibrium of the stage game for all 
t>T. 

The fact that late period stage-game players play an empirically e-correct 
Bayesian equilibrium relative to their updated type generating distribution 7 ^ 
does not mean that they play a Bayesian equilibrium relative to the correct 

® Theorem 3 in Kalai and Lehrer (1993) states that Bayesian updating of beliefs 
containing a “grain of truth” must eventually lead to correct predictions. The 
restriction in the current model, to a countable set of type generating distribu- 
tions, implies that for almost every outcome jT(r) > 0, which implies that the 
beliefs contain a grain of truth. 




180 



distribution r. It only means that players’ predictions concerning the play 
path are approximately correct. Players may be mistaken concerning oflF the 
equilibrium path behavior. This is illustrated in the following example. 

Example 

Consider three types of a bully, r, /, and a, described by their episode 
payoffs as follows. The rational type, r, has the original extensive game payoffs 
described earlier. The fighting type, /, has a payoff 1 for fighting and 0 for 
any other outcome (replacing the payoffs in Figure 1 ), while the acquiescing 
type, a, has a payoff of 1 for acquiescing and 0 for any other outcome. Let 
p, ijj and a, be the type generating distributions that select respectively with 
probability one the types r, / and a. 

Suppose the prior F assigns high probability to a world with fighting bul- 
lies, i.e., F{xIj) is high, and only a small probability to a and to p. Suppose 
the real type generating distribution was selected to be a, i.e. only acquiesc- 
ing bullies are generated. The strategies with all challengers refrain, a and 
r type bullies always acquiesce, and / type bullies always fight, constitute 
an uncertain Bayesian equilibrium. This is so because the initial stage game 
prior 7 ® assigns high probability to a fighting bully and to refrain from chal- 
lenging is therefore rational. But since there is no challenge in the first period 
the updated posterior on bully types remains unchanged, i.e. 7 ^ = 7 ^, the 
same logic applies to the second period, and so on. Moreover, the induced 
stage game Bayesian equilibria are empirically accurate. It is clear, however, 
that these strategies do not induce a Bayesian equilibrium of the stage game 
relative to the real prior a. In other words, if the challengers knew that their 
bully is drawn with probability one to be of the a type, then they would 
challenge. 

5 Trembling Rational Players 

The uncertain Bayesian equilibrium in the above example is highly unstable. 
If at any time, in the infinite play of the recurring game, a challenger trembles 
and challenges, the whole equilibrium will collapse after observing that the 
bully does not fight. The equilibrium is able to survive only because society 
never learns what bullies would do if challenged. 

If a small amount of noise, in the form of trembles by players, is intro- 
duced, then more learning will occur and equilibria such as the one in the 
example above will eventually be overturned. In short, a small amount of im- 
perfection will lead players to learn behavior at all nodes in the tree. While 
there are several ways to model such trembles, we follow Selten’s (1975) ap- 
proach by restricting the set of strategies that a player can choose. 

Let g be a positive small number describing the probability of minimal 
trembles. The uncertain Bayesian recurring bully game with q-trembling is 
obtained from the usual uncertain recurring bully game by restricting the 




181 



players to the choice of behavior strategies that assign probability greater or 
equal to q to every action available in each information set. 

Our next observation is that under g-trembling, if a Bayesian equilibrium 
is empirically e-correct, then it must also be approximately correct for all 
conditional probabilities in the game. For example, the probability that the 
bully fights the challenger, conditional on the event that the chal- 
lenger challenges, must be similar when computed by the learning induced 
distribution 7 and by the true (realized) distribution r. 

To see this point, let /x be the distribution induced on play paths through 
the learning induced distribution 7 , and let /x be the one obtained from the 
true (realized) distribution r. The fact that | /x(p) — /x(p) |< e for any play 
path p implies that this can be made (by starting with a smaller e) true for 
any event (i.e. a set of play paths) in the game. The g-trembling property 
implies that there is a small positive number s such that the probability of 
any non empty event in the game is at least s. The conditional probability of 
event Ei given E 2 is P{Ei and E 2 )/P{E 2 ). Since we have all denominators 
(over all possible £? 2 ’s) being uniformly bounded from zero, by making e 
sufficiently small we can make all such ratios when computed by /x or by /x 
be within any given 8 of each other. Thus we obtain the following. 

Define /x to be strongly empirically e-correct if for any two events Ei , and 
E 2 the conditional probabilities of Ei given E 2 , computed by /x and by /x are 
within e of each other. 

Lemma 3. Consider any Bayesian equilibrium {{rje) , {cri)) of a q-tremhling 
Bayesian bully game with an assumed type generating distribution 7 and a 
true type generating distribution r. For every e > 0 there is a 5 > 0 such 
that if the equilibrium is empirically 8 -correct it must be strongly empirically 
e-correct. 

Notice that by combining the result of Proposition 2 with the above 
lemma, we can conclude that for almost every outcome of an uncertain 
Bayesian equilibrium of the recurring game, after a sufficiently long time, 
stage game players must be playing a strongly empirically e-correct Bayesian 
equilibrium. 

We consider now the special case, of an uncertain Bayesian equilibrium 
of the recurring game, where the true (realized) type generating distribution, 
p, generates the rational type r with probability one. The prior P over type 
generating distributions can be arbitrary, as long as P{p) > 0 . 

For a game with g-trembling, a bully stage strategy q is essentially acqui- 
escing if its probability of fighting at every information set is the minimally 
possible level g, i.e. is only due to trembling. Similarly, a challenger i strat- 
egy is essentially challenging if the probability of refraining is q. We can now 
state the following result. 

Proposition 4. Consider an uncertain Bayesian equilibrium of an uncertain 
recurring bully game with q-trembles {q > 0). If the realized type generating 




182 



distribution places weight one on rational players, then for small enough q 
(1/2 > q) and for almost every outcome there is a finite time T such that the 
t-period stage game strategies must he essentially acquiescing and essentially 
challenging for all t>T. 

Given the previous propositions and lemma, the conclusion of this propo- 
sition is straightforward. For every e > 0 there is a sufficiently large T so 
that all the later stage games are played by strongly empirically e-correct 
Bayesian equilibrium. So all we have to observe is that for sufficiently small 
e, a strongly empirically e-correct Bayesian equilibrium must consist of es- 
sentially acquiescing and essentially challenging strategies. This is done by a 
(backward) induction argument, outlined as follows. 

In the last episode, following any play path, a rational bully must ac- 
quiesce with the highest possible conditional probability given the (positive 
probability) event that the last challenger challenges. Therefore the rational 
bully must acquiesce with probability 1 — q in the last episode. Thus, after 
time T (as defined in Lemma 3) the last episode challenger’s assessed proba- 
bility of the bully acquiescing in that episode is at least 1 — q — e. Therefore, 
for sufficiently small e and ^ < 1/2 the last challenger’s unique best response 
is to challenge, and so he or she challenges with probability 1 — q. Since this is 
true for any play path leading to the final episode, the assessed probability of 
challenge in the final episode is at least 1 — q — e, indepedent of the play path 
leading to that episode. It follows that a rational bully in episode L — 1, will 
have a unique best response of acquiescing in that episode. The L — 1 period 
challenger, assessing this to be the case with probability at least 1 — q — e, 
will essentially challenge, and so on. 

6 Concluding Remarks 

The KMRW phenomenon extends to a general folk theorem for finitely re- 
peated games with incomplete information, as shown by Fudenberg and 
Maskin (1986). The failing of the phenomenon in a recurring setting has 
parallel implications for this folk theorem. Jackson and Kalai (1995ab) con- 
tains general results that have direct implications on this question. Again, we 
should emphasize that our results imply that under certain conditions players 
will learn to play as if they knew the realized type generating distribution. To 
the extent that there exists a true diversity of types in the population, this 
is learned and players will play accordingly. Thus, the variety of equilibrium 
outcomes allowed for by the folk theorem can still be realized, but will require 
a true diversity of types, rather than just a perceived one, in order to survive 
in a recurring setting. 

Stronger, and even more striking, versions of the KMRW phenomenon 
might be possible in the manner described by Aumann (1992). This would 
involve higher order misconceptions on the part of players. For example, 




183 



replace the KMRW situation, where challengers are uncertain about whether 
the bully is rational or fight loving, by a situation where all challengers know 
that the bully is rational, but are uncertain about whether other challengers 
also know it. If, for instance, challengers believe that other challengers believe 
that there may be a fighting bully, then the KMRW results might be extended. 
This situation is incorporated into our model by using the type space to allow 
an explicit description of the beliefs a player holds about the beliefs of other 
players (and choosing r’s and F to refiect this uncertainty). Proposition 4 
covers these cases and thus, if bullies are born rational, then this extended 
version of the KMRW equilibrium would also unravel in a recurring setting. 

Getting back to Selten’s paradox, it seems to become more severe in the 
recurring setting. A bully in later stages of the game who fights the first 
challenger can be explained as having trembled, and thus is not perceived 
as being irrational. Moreover, in the equilibrium play in late enough stages, 
even a bully who fights the first few challengers will be explained as hav- 
ing trembled several times, as this likelihood is larger than the alternative 
explanation of the bully being irrational. Although in late enough play it is 
relatively more likely that this behavior is due to trembles rather than irra- 
tionality, both of these events were very unlikely to start with. Thus, it may 
be that the challenger would prefer to doubt the model altogether (or believe 
that the bully has done so), rather than to ascribe probabilities according to 
it. Thus we are back at Selten’s paradox. 

Let us close with two comments relating to “technical” assumptions that 
we have maintained in our analysis. The restriction in this chapter to count- 
ably many types is convenient for mathematical exposition. The generaliza- 
tion to an uncountably infinite set (and an uncoutably infinite set of possible 
type generating distributions) requires additional conditions, in particular 
to assure that Proposition 2 extends. Lehrer and Smorodinsky (1994) offer 
general conditions which are useful in this direction. 

One strong assumption we have made is that players start from a com- 
mon prior over priors. This is not necessary for the results. The content of 
Propositions 1 and 2 can be applied to situations where each type of player 
has their own beliefs. The equilibrium convergence result then needs to be 
modified, since players’ beliefs may converge at different rates. When these 
convergence rates are not uniform, then the conclusions are stated relative 
to a set of types which receives a probability arbitrarily close to one (under 
the realized distribution). 



References 



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Markets and Games: Essays in Honor of Frank Hahn, edited by P. Dasgupta, 
D. Gale, O. Hart, and E. Maskin, MIT Press, Cambridge MA, 214-227. 




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Aumann, R. J. and M. Maschler (1967), “Repeated Games with Incomplete Infor- 
mation: A Survey of Recent Results,” Mathematical ST - 116, Ch. Ill, 287-403. 

Battigalli, P., M. Gilli, and M.C. Molinari (1992) “Learning and Convergence 
to Equilibrium in Repeated Strategic Interactions: An Introductory Survey,” 
Richerche Economiche^ 96, 335-378. 

Fudenberg, D. and E. Maskin (1986), “The Folk Theorem in Repeated Games with 
Discounting and Incomplete Information,” Econometrica^ 54, 533-554. 

FHidenberg, D. and D. Kreps (1988), “A Theory of Learning, Experimentation and 
Equilibrium in Games,” mimeo: Stanford University. 

Fudenberg, D. and D. Kreps (1995), “Learning in Extensive Form Games: Self 
Confirming Equilibrium,” Games and Economic Behavior j 8, 20-55. 

Fudenberg, D. and D. Levine (1993), “Steady State Learning and Nash Equilib- 
rium,” Econometrica, 61, 547-573. 

Harsanyi, J. (1967-68), “Games of Incomplete Information Played by Bayesian 
Players (Parts I, II, and III)”, Management Science^ 14, 159-182; 320-334; 486- 
503. 

Jackson, M. and E. Kalai (1995a), “Social Leaxning in Recurring Games”, mimeo, 
California Institute of Technology. 

Jackson, M. and E. Kalai (1995b), “Learning to Play Perfectly in Recurring Exten- 
sive Form Games” , mimeo, California Institute of Technology. 

Jordan, J. (1991), “Bayesian Learning in Normal Form Games,” Games and Eco- 
nomic Behavior J 3, 60-81. 

Kalai, E. and E. Lehrer (1993), “Rational Learning Leads to Nash Equilibrium”, 
Econometrica, 61, 1019-1045. 

Kreps, D. and R. Wilson (1982), “Reputation and Imperfect Information”, Journal 
of Economic Theory, 27, 253-279. 

Lehrer, E. and R. Smorodinsky (1994), “Repeated Large Games with Incomplete 
Information” , Tel- Aviv University. 

Milgrom, P. and J. Roberts (1982), “Predation, Reputation, and Entry Deterrence”, 
Journal of Economic Theory, 27, 280-312. 

Nash, J. (1950), “Non-Cooperative Games”, PhD Thesis, Mathematics Department, 
Princeton University. 

Selten, R. (1975), “Reexamination of the Perfectness Concept for Equilibrium 
Points in Extensive Games”, International Journal of Game Theory, 4, 25-55. 

Selten, R. (1978), “The Chain-Store Paradox”, Theory and Decision, 9, 127-159. 

Selten, R. (1983), “Evolutionary Stability in Extensive 2-Person Games,” Mathe- 
matical Social Sciences, 5, 269-363. 




Dumb Bugs vs. Bright Noncooperative 
Players: A Comparison 

Thomas Quint^, Martin Shubik^, and Dicky Yan^ 

^ Sandia National Laboratories, Albuquerque, NM 87185 
^ Cowles Foundation, Yale University, New Haven, CT 06520 
^ ATT Bell Laboratories, Holmdel, NJ 07733 



Abstract. Consider a repeated bimatrix game. We define “bugs” as players 
whose “strategy” is to react myopically to whatever the opponent did on the 
previous iteration. We believe that in some contexts this is a more realistic 
model of behavior than the standard “supremely rational” noncooperative 
game player. We compare how bugs fare over a suitable “universe of games,” 
as compared with standard “Nash,” “maximin,” and other types of players. 



1. Background 

In this paper we consider bimatrix games in which the two players are denoted 
as I and II. Both players have a finite number of (pure) strategies. Let Pk{i^j) 
be the payoff to player k if Player I uses pure strategy i and Player II uses 
j {k = 1,2). A pair (i*, j*) form a pure strategy Nash equilibrium (PSNE) 
if majci Pi(i, j*) is achieved when i = i* and also maxj P 2 {i* ^ j) is attained 
when j = j*. 

Suppose Player I has m pure strategies, and Player II has n. A mixed 
strategy for Player I is a probability vector (pi, Pm) in which pi rep- 
resents the probability that he plays pure strategy i {i = 1, rn). Sim- 
ilarly, a generic mixed strategy for Player II is given by an n- vector q, 
A mixed strategy Nash equilibrium (MSNE) is a pair (p*, q*) for which 
p* e aTgmaXp'^^JPiqjPl{i,j), q* 6 argmax, j), and either 

p* or q* is nonintegral. A Nash Equilibrium (NE) refers to a PSNE or MSNE. 

Another way to play a game is to use maximin strategies. Mixed strategy 
vector p* is a maximin strategy for Player I if it maximizes 
min^ Yli j PiQjPiii^j) over all p. Similarly, a maximin strategy for II is any 
maximizer of mmpY^-jPiqjP 2 {i,j) over all q. A player’s maximin payoff is 
his expected payoff when both players play a maximin strategy.^ 



^ Note that under this definition, a player’s maximin payoff may not be uniquely 
defined (if one or both players have multiple maximin strategies). However, 
using the randomly generated cell entries described in Section 4, we find that 
almost always it will be, and this will serve our purposes in that section. 




186 



/(4,1) (0,0) (2,2) \ 

(0,0) (9,9) (0,0) 

V (1,3) (0,0) (3,1)7 

Figure 1: A Bimatrix Game 



Referring to the game in Figure 1, we may also illustrate the idea of op- 
timal response. Suppose that the players were to select moves (strategies) in 
sequence. Given some move for one player, the other can calculate his optimal 
response. For example, if Player II assumes that Player I has chosen his first 
move. Player IPs optimal response is to choose her third move. If Player I 
believes that Player II has chosen her second move then Player Ps optimal 
response is to choose his second move. The optimal response calls for the sim- 
ple minded “greedy” rule of selecting the largest item in a row or column.^ 
Any PSNE in a bimatrix game has the “optimal response property,” i.e., at 
the equilibrium the action of each is the optimal response to the action of 
the other. 

Rather than view optimal response as a computational device, we may 
regard it as a description of the behavior of a primitive decisionmaker (or 
“bug” ) who learns nothing, but is able to perceive the individual worth of all 
outcomes in a row or column. Suppose initial strategies for the two players 
are given. The bug’s first move will be to play an optimal response against 
the other player’s initial strategy. In general, its move in the nth repetition 
of the game will be an optimal response against the opponent’s move in the 
n — 1st repetition. Hence the bug player is reacting myopically to the last 
move of his opponent, without taking into account what that opponent might 
he doing currently. 

We may wish to distinguish between sequential and parallel (simultane- 
ous) motion for our bugs. The spirit of the Nash equilibrium calls for alternate 
one-player adjustments (see Brown 1951); this gives rise to the notion of se- 
quential hugs. When sequential bugs play a game, (after the initial strategies 
are determined) the choice to move is determined randomly, and then passed 
back and forth until possibly both players choose not to move. If this hap- 
pens, they have arrived at a PSNE, and no further motion occurs. If this never 
happens, then the bugs will end up cycling (among four or more outcomes). 

Another possibility is for the players to be simultaneous hugs, i.e., they 
move blindly and simultaneously. This has the feature that the cell to which 



^ There axe several alternatives to the assumption that a player looks down a 
entire row or column of outcomes and picks out the best. For instance, we 
might restrict the individual to only choosing among immediate neighbors. [This 
implies that there is some form of metric which defines neighbors. In particular, 
this would mean that permuting rows or columns would change the game.] 
Alternatively we might regard any choice that leads to an improvement as 
equiprobable, etc. 




187 



they move may be different from what either expect (but neither is assumed 
to learn). 

To clarify matters, consider the 3 by 3 example of Figure 1, and suppose 
the “initial strategies” are Player Ps first strategy and Player II’s second. If 
sequential bugs are playing this game, and Player I is determined to move 
first, then he changes to his second strategy, the PSNE payoff (9,9) is reached, 
and neither player has incentive to move thereafter. If Player II is determined 
to move first, she changes to her third strategy, obtaining payoff (2,2). Now 
Player I has incentive to change to his third strategy, obtaining (3,1), and 
then Player II changes to her first strategy, etc. Continuing this process, we 
see that ultimately the bugs will cycle through the four “corner” payoffs. We 
call this cycle a sequential bug cycle. 

On the other hand, if simultaneous bugs are playing the game, then Player 
I’s first move (his second strategy) and Player IPs first move (her third strat- 
egy) are played simultaneously, resulting in the (0,0) payoff on the right of 
the second row. From this point. Player I switches to his third strategy, and 
Player II to her second, yielding the (0,0) payoff in the bottom row. Again 
the result is a cycle, called a simultaneous hug cycle, this time among the 
four (0,0) payoffs in the bimatrix. 

In this paper we consider several types of player. The first type is the stan- 
dard homo ludens, or “Nash player,” and is wedded to the need for consistent 
expectations. He is capable of computing equilibrium strategies, knows his 
and his competitor’s payoff functions, and always plays “his part” of a Nash 
equilibrium. The second type is the somewhat primitive and unintelligent 
bug described above. The third type is intelligent but highly pessimistic. It 
believes that the optimal policy is to play maximin on the basis that it does 
not trust the other player to act rationally or nonmaliciously. The fourth type 
is a random player, or “Nature,” which selects among its moves with equal 
probability. Finally, we also consider malicious “anti-players,” each of whom 
try to minimize the payoff to the other. 



2. Landscapes and Almost All Games 

The 2x2 bimatrix game has been studied in some detail. There is a small, 
but useful literature on counting and characterizing matrix/bimatrix games. 
References include Rapoport, Guyer and Gordon (1972), O’Neill (1988) and 
Barany, Lee and Shubik (1992). The attraction of the 2x2 bimatrix game is 
basically that its major variants can be enumerated and studied exhaustively. 
Unfortunately the simplicity can also be misleading for the study of many 
questions we might wish to ask about general bimatrix games. In particular, 
because each row or column contains only two entries, the shape of the payoff 
surface is limited in complexity. 

For the rest of this paper, we limit our analysis to the class of 2 x 2 games, 
but with some observations and qualifications for larger bimatrices. 




188 



If we regard the payoffs as being given in von Neumann Morgenstern 
utilities, then there are infinitely many games of any size. But these games 
may be classified into a finite set of classes by the ordinal aspects of the 
payoffs.^ 

We wish to obtain a reasonable measure for all bimatrix games of a given 
size and then study different forms of behavior over the various classes. For an 
nxn bimatrix we generate its payoffs by drawing 2n^ times from a rectangular 
distribution on the interval [0,1]. Topologically speaking, this gives us “almost 
all” nxn games.^ 

Limiting our concern at this point to the 2x2 bimatrix game, we draw 
from 8 i.i.d. uniform random variables to obtain the games and place them 
in the 78 ordinal categories, ignoring ties.^ 

If we consider large nxn bimatrices, we may observe that if we regard 
the n^ payoffs to each player as describing a payoff surface, this surface may 
be arbitrarily rough. In biological applications and in military operations 
research involving duels and search problems, one may wish to consider a 
game between competitors played on an actual two dimensional surface. It 
is recisonably straightforward to show the relationship between matrix games 
and games played over a map.® 

In many applications of game theory there is a special structure to the 
game and its payoffs. Sometimes (such as in the functioning of markets) moves 
may have specific physical meaning, and payoffs vary smoothly with small 



^ We know that of the 576 ordinal 2x2 games, there are 78 classes of games, 
each strategically different. Of these 78 classes, 66 contain 8 members, and the 
other 12 contain only 4 members (due to an extra symmetry). Also, 58 of the 
classes axe of games having a single NE that is pure, 9 having a single NE that 
is mixed, and 11 having three NEs, two pure and one mixed. 

Finally, we note that there are several other ways one might classify all 
games, for example by their optimal response diagrams (i.e., their labelling 
functions), or by the structure of their payoff sets. 

“All” games would be generated from the open set (— oo,co), while we have 
generated the games from the closed set [0, 1]. 

^ If we permitted ties, there would be over 700 strategically different games to 
consider. But the games with ties will form a set of measure zero. In applications, 
it is dangerous to ignore ties, especially if they appear to be a fact of life and 
there is some feature (such as crudeness of perception) which is generating them. 
But if we are considering behavior over all possible games, the ignoring of ties 
can be justified as a close approximation. 

® In particular, in some applications one may want the two dimensional surface 
to be the surface of a sphere. In wargaming, if a flat surface is used, the best 
regular paving is hexagonal. Limiting ourselves to bimatrix games, in order to 
avoid boundary problems it may be useful to regard the matrix as a torus where 
the top row is identified as a neighbor of the bottom row, and the left column 
is identified as a neighbor of the right column. A game with two players and k 
units of space where the payoff to each is a function of the land occupied by the 
player and whether it is occupied alone or by both players gives rise to a k x k 
bimatrix game. 




189 



changes in strategy. For example, the amount of profit a firm may make is 
usually a smooth function of its sales. 

Thus, although it may be desirable to consider all games in some general 
enquiry, applications tend to be constrained to special subclasses. 



3. Two Useful Benchmarks 

Prior to considering the dynamics of the various players, we establish lower 
and upper bounds on the expected payoffs of players in the special case of the 
2x2 bimatrix game where each of the eight payoffs is an iid random variable, 
uniformly distributed on the interval [0, 1]. We first consider two agents who 
play randomly, and then two who play completely cooperatively. 

Random Players: Recall that a “random” player is one who plays each 
of his pure strategies with equal probability. Hence, if both players play ran- 
domly every outcome is equiprobable. Since the expected value to each in a 
randomly selected cell is .5, the payoff from random play by all is (.5, .5). 



Joint Maximizers: To evaluate the yield from complete cooperation, we 
assume the players move so as to maximize the sum of their payoffs.^ Consider 
a 2 by 2 matrix where the entry in cell z is a random variable Xi, {i — 
1, 2, 3, 4), defined as the sum of two i.i.d. uniform random variables on [0,1]. 
The payoff for either player (assuming they cooperate so as to achieve the 
maximum Xi and then split the payoff) is one half the random variable F, 
where Y = max(Xi, X2, X3, X4). The expected value of Y is calculated as 
follows: 



1. It is easy to see that Pr{Xi < x) is equal to 0 if x < 0, ^ if x G [0, 1], 

1 — if X G [1, 2], and 1 if x > 2. 

2. Hence we have 



Pr{Y < x) = ^ 



0 , 

V 2 / — ’ 

1 , 



if X < 0 , 
if X G [0, 1] , 

if xe[l, 2 ], 
if X > 2 . 



3. Since F > 0, its expectation is given by f^(l — Pr(Y < x))dx (Hoel- 

Port-Stone 1971, p. 193). This is equal to J^^l- j^x^)dx-l-/^^(l-(-|x^-f 
2x — l)"^)dx, which is approximately 1.424. 



^ Hence, we axe assuming transferable utility^ i.e., that the players can transfer 
payoff from one to the other, and do so in a manner so as to equalize their 
payoffs. If we wish to consider the “nontransferable utility” case, we have a 
more complicated problem because there is not necessarily a unique optimal 
payoff in a game; instead there is in general a Pareto-optimal surface from 
which any point may be chosen. 




190 



4. Each player expects to gain half of this, or .712. 



Our results from the “random case” and the “joint maximizer” case imply 
that nonmalicious players should expect to receive a payoff of somewhere 
between .5 and .712. However, we should note that it is possible for the players 
to lower their expected payoffs below .5. In fact, it is feasible (but rather 
unlikely) that they could collaborate to achieve their minimal expected joint 
payoff. A calculation symmetric to the one presented above gives minimum 
expected payoffs of (.288, .288). 



4, A Comparison of Players 

In this section we are concerned with comparing the expected performance 
of different players, playing a universe of games against a player of the same 
type. In all instances, we assume players are either not concerned with the 
type of “opponent” they face, or else they assume that opponent is like them- 
selves. This assumption, in essence, indicates that the players do not learn 
from history. 

In each instance it is important to be specific about the nature of the 
optimal response: does Player I or Player II move first, with information 
concerning the behavior of the other, or do they move simultaneously? When 
players of the same type are playing each other one can argue that if reaction 
is sequential, order does not matter as it will average out over all games. 

4.1 Problems in Comparison 

The noncooperative equilibrium solution offers no dynamics, whereas the 
bugs’ optimal response procedure does. It is necessary to take this into ac- 
count when trying to compare the solutions. One way to do this is to average 
over any cycle into which bugs may settle. If they eventually “fall into” a 
PSNE, they are credited with its value. 

In nondegenerate cases, an n x n bimatrix game can have as many as n 
PSNEs, and we conjecture at most 2^ — 1 NEs in total (see Quint-Shubik, 
1994a). Hence, as the theory of Nash equilibrium provides for existence, but 
no selection,® we have several choices as to how to assign payoffs to “Nash 
players.” If PSNEs exist, we can rule out all MSNEs and treat all PSNEs as 
equiprobable, or we can treat all equilibria as equiprobable. If (as some do) 
we have some reservations about MSNEs, we could give them a lesser weight 
than PSNEs. For our comparisons we treat all PSNEs as equiprobable and all 



® Unless one wishes to follow the tracing procedure given by Harsanyi and Seiten 
(1988). We do not do so here, but it might be of interest to evaluate the expected 
payoff from the equilibrium selected by tracing. 




191 



MSNEs as equiprobable, but the probability of a PSNE is not necessarily the 
same as the probability of a MSNE. We then consider the weighted average 
of the associated payoffs. 

In the next section, we propose a dynamic process which produces the 
weighted average of NE payoffs outlined above. 

4.2 Two Players of One Type vs. Two Players of Another Type 

Consider a bimatrix game, in which player I is the row player and player II 
the column player, and where each player has two strategies. Hence the game 
has the following form: 

f {a,b) {c,d) \ 

V (e,/) {g,h) ) 

The game is to be played repeatedly. We consider five types of player: 

1. A Nash Player — this player always plays a (possibly mixed) strategy 
which is “his part” of a NE. If there are three NEs (= 2 pure and 1 
mixed) in the game, then this player plays each pure with probability ^ 
and the mix with probability 1 — a {a G [0, 1]).^ If in this case he plays a 
particular NE and his opponent matches his expectations and plays the 
“other half’ of it, the NE player responds to this by playing according 
to that NE forever. Otherwise, he re-randomizes his NE strategies on the 
next iteration. 

2. A Sequential Bug Player — as described in Section 1. 

3. A Simultaneous Bug Player — as described in Section 1. 

We assume that the “initial strategies” (see Section 1) place the bugs 
into one of the four cells, each with probability In addition, for the 
sequential bugs, we assume either player “goes first” with probability 

4. A Maximin Player — this player always plays his maximin strategy. 

5. A Random Player — this player always plays the mixed strategy 

V2’ 2/* 

Example 4.2.1: Consider the bimatrix game 

M6,2) (1,1) 

(5,3) (2,6) 

and let us assume the parameter a is equal to |. 

Now, if two random players were to play this game, all of the outcomes 
would occur with probability and so the expected payoffs would be (|, 3). 
In addition, we note that the maximin strategy for I is to play his second pure 

® Hence, ot = ^ represents the case where all three NE strategies are played with 
equal probability; a = 1 represents the case where the PSNE strategies axe each 
played with probability - and the MSNE strategies are disregarded. 




194 



for Player I is again For games in ATi, one can see that the only labelling 
cycle (see Section 6 of Quint-Shubik-Yan, 1995a) will contain precisely one 
row and one column, and the row and column will be the ones defining the 
PSNE. Hence, we see that the bugs must end up at the PSNE. 

The analysis of the previous section then gives expected value | for this 
Ccise. 

Finally, for games in N 2 , we see that with probability .25 the bugs end up 
in one NE, with probability .25 they end up in the other, and with probability 
.5 they end up cycling between the two other cells. Hence, if we assume 
(WLOG) that a > e, g > c, b > d, and h> f, the expectation in this case is 
^[.25a-f .25^ + .5(^)|a > e,g > c]= E[j{a -h c e g)\a > e, g > c\ = \. 
Putting this all together, we have E{h^) = + = | = -625. 

4.2.5 Nash Players. Let us again use the same technique. For the analysis 

of games in No, let us assume WLOG that a > e, d > b, g > c, and f > h. The 
unique MSNE in this case is for Player I to play p = and 

for Player II to play q = a-4?f-c )- Given a, 6, c, d, e, /, g, and h, 

the expected payoff for Player I is then pqa-{-p{l—q)c+ (l-p)qe-{-{l—p){l—q)g, 
which turns out to be To find E{h}) over all games in No, we 

evaluate E[ ^^^^~~~ \a > e, g > c\, which turns out to be 

For games in N\, it is clear that the payoffs for the players are merely 
those of the PSNE. Since either type of bug attains the same result, we can 
use the results from the previous sections to state that the expected payoff 
in this case is |. 

For games in N 2 , assume WLOG that a > e, b > d, g > c, and h > f. We 
note that the probability is f that either of the two PSNEs will occur, and 1—P 
for the MSNE, where (3 = 234 ^+^ '^^ Hence, Player I receives payoff a with 
probability g with probability and (sstme analysis as in the A^o 

case) with probability 1-/3. In expectation this gives him f § + f I 
which is equal to | + f • 

Putting this all together, we get E(h^) = + + + which 

is equal to | Hence the Nash players’ expected payoffs range between 

those for the simultaneous bugs (| when = 0) and those for the sequential 
bugs (II when ^ = 1). 

4.2.6 Msiximin Players. To analyze the case of maximin players, we will 
use the same general procedure as above, except, instead of “partitioning”^^ 

This is a rather complicated quadruple integral, which we evaluated both by 
longhand and by using Mathematica. 

In Example 4.2.1, where a = |, /3 was equal to |. 

We did not formally partition 17, because there is a set of measure zero consisting 
of games with an infinite number of NEs. The same comment applies to the 
maximin analysis, where we disregard cases where players have more than one 
maximin strategy. 




193 



M-^o) = IJ'iNi) = f, andiJ,{N 2 ) = 

Lemma 4.2: Let fl and fi be defined as above. Let MM\ he the subset of 
Q in which Player I has a saddlepoint at a, c, e, or g (i.e., e < a < c, 
g <c< a, a<e<g, or c<g<e respectively). Similarly , let MM| be 
the subset of Q in which Player II has a saddlepoint at b, d, f , or h (i.e., 
d<b<fyh<d<h, h<f<b, or f<h<d respectively).^^ Then 

li{MMl) = i^(MM^s) = I- 

4.2.1 Joint Maximizers. If two “joint maximizers” axe playing the game, 
the calculation presented in Section 3 says that the expected payoff for each 
player is approximately .712. 

4.2.2 Random Players. If two “random players” axe playing the game, 
the calculation presented in Section 3 says that the expected payoff for each 
player is exactly .5. 

4.2.3 Sequential Bugs. To find the expected payoff to a pair of sequential 
bugs playing the game, let us consider the (expected) payoffs for Player 
I. [By symmetry, the expected payoff for II will be the same.] We consider in 
turn games in iVo, iVi, and N 2 respectively. 

Given a game is in Nq, WLOG we may assume a > e, d > b, g > c, and 
d > h. In this case the bugs will cycle around the four outcomes, giving an 
average payoff of to Player I. Hence the expected payoff, given the 

game is an element of Nq, is > e, g > c]. By symmetry, this is 

.5. 

Given a game is in Ni, Corollary 6.4 of Quint-Shubik-Yan (1995a) implies 
that the sequential bugs attain the PSNE payoffs. WLOG assume (a, b) is 
the PSNE payoff; hence a> e and 5 > d. In addition, either c> g 01 f > h. 
Either way, the expected payoff to I is E[a\a > e], which turns out to be |. 

Finally, consider the case where the game is in N 2 - WLOG, a > e, b > d, 
g > c, and h > f. Since we assume a) the bugs “start” in any of the four cells 
with equal probability, and b) either player moves first with probability .5, 
symmetry dictates that the bugs axe equally likely to end up in either PSNE. 
Hence the expected payoff for Player I is E[^^\a > e,g > c] = ^. 

Putting all of this together gives that E{h^) = game in 

^k)fJ'{Nk) = || + || + H= ls = 

4.2.4 Simultaneous Bugs. For simultaneous bugs, we use a similar proce- 
dure. 

For games in Nq, it is easy to see that a pair of simultaneous bugs will 
act exactly as do a pair of sequential bugs, and hence the expected payoff 

The significance here is that if a player has a saddlepoint, his maximin strategy 
will be to play the pure strategy corresponding to that saddlepoint. 




192 



strategy, while that for II is to play his first, so that the payoff for maximin 
players is (5, 3). 

Next, let us do the analysis for the case of two sequential bugs playing 
the game. In this case, all is dependent on the initial conditions. If the game 
“starts” in the upper left cell or the bottom right cell, neither bug will ever 
move, so the payoffs are in fact those in the cells. If the game begins in 
the upper right cell with bug player I to move, he will play his second pure 
strategy, giving payoff (2,6), and then neither bug has incentive to move. 
Similarly, if the game were to start in the lower left cell, I would play his 
first pure strategy, and no further moves would be made. A similar analysis 
would hold if player II were to make the first move. All in all, there is a 50% 
chance that payoffs (6, 2) would be obtained forever, and a 50% chance for 
(2,6). Expected payoff: (4,4). 

The simultaneous bug analysis is the same in the cases where the game 
begins in the upper left or bottom right cell. However, if the game were to 
begin in either of the other two cells, both players would then move, and 
the result would be endless cycling between the outcomes (1,1) and (5,3). 
Hence, the expected payoffs for the simultaneous bugs are |(6, 2) -f- 1(2, 6) -f 
which is equal to (|, 3). 



Finally, for the NE player-NE player analysis, we note that there are two 
PSNEs in the game, paying off (6, 2) and (2, 6). In addition, there is a MSNE 
which pays off (|, |). Since a = |, this means that when two NE players 
play this game, the chances are ^ that they both will play their first pure 
strategy, ^ that they will both play their second pure strategy, and j that 
they will both play their MSNE strategy. [The other | of the time, they will 
be playing NE strategies that don’t “match.”] This 1:1:4 ratio means that in 
the long run, they will attain either PSNE payoff with probability i , and the 
MSNE payoff with probability |, giving expected longrun payoff (^, ^). 

Comparing the payoffs for Player I, we see that the maximin game is best 
for him, followed by the sequential bugs game, the NE-NE game, and finally 
the simultaneous bugs and random players game are tied. For II, we see that 
the sequential bugs game is best, the NE-NE game is worst, and the other 
three games are all tied. 



To compare how the various types of players fare over a suitable universe 
of games, let us assume that the eight payoffs a, ..., h are each i.i.d. U[0,1] 
random variables. Define i? as the set of all joint outcomes (a, ..., h). Let fjL 
be the probability measure defined on the set of subsets of i7, defined in the 
natural way with /i(i?) = 1. 

Our aim is to calculate the expected payoff for each type of player over 
the above universe (i7). To this end, we will need the following simple lemmas: 



Lemma 4.1: Let i? and /x be defined as above. Let No, Ni, N 2 be the subsets 
of Q consisting of those games with 0, 1, and 2 PSNEs respectively. Then 




195 



Q into the three sets iVo, and N 2 , we will break it up into the following 
four sets: 

1. Games where both players have a saddlepoint, and Player II’s saddlepoint 
is in the same column as Player Ps. Using Lemma 4.2, the measure of 
this set is 

2. Games where both players have a saddlepoint, but Player II’s saddlepoint 
is not in the same column as Player Ps (/x = |). 

3. Games where Player I has a saddlepoint but Player II does not (/x = |). 

4. Games where Player I has no saddlepoint (/tx = |). 

The Analysis. 

1. Suppose in this case WLOG that Player Ps saddlepoint is a (so e < a < c) 
and so Player IPs is in the first column. Then I’s maximin payoff is a, 
and his expected payoff in this case is £^[a|e < a < c] = 

2. Suppose in this case WLOG that Players Ps saddlepoint is a (so e < 
a < c) and so Player IPs is in the second column. Then Ps maximin 
payoff is c, and his expected payoff in this ca^e is E[c\e < a < c] — 

^fo /e /a 

3. In this case we again assume I’s saddlepoint is a, and that h > b > d > f. 

Then I plays p = (1, 0), and II plays q = h-d+i-f )- Hence I’s 

expected payoff is Since a, c, e, and g are by 

assumption independent of 6, d, /, and h, we get that E{h^) in this case 
is equal to jB[a|e < a < c]E[ ^J^^^_j, \h > b > d > f] + E[c\e <a< 

>b>d> f]. This is equal to || + || = |. 

4. For this case, WLOG assume a > g > c > e. Then I’s maximin strategy 

is P = Suppose II’s maximin strategy is (x, 1-x). 

Then I’s expected payoff is +^ ^- (1 - x)c+ jz^=^xe + 

which is equal to Hence, I’s expected payoff 

in this case is > g > c > e], which is equal to 

Finally putting this all together, we find that Player I’s expected payoff 
for the entire game is || + || + || + || = ^ 5 or approximately .583. 



4.3 “Anti-Players” 

Although we might consider random behavior as providing a lower bound on 
payoffs, we could consider malicious, “looking-glass” versions of our players. 
For instance, we might consider a (sequential or simultaneous) “anti-bug,” 
which always moves to a row (or column) which minimizes his opponent’s pay- 
off (instead of maximizing his own). Or, we might consider an “anti-maximin 



Again we calculated this integral using both longhand and Mathematica. 




196 



player,” whose modus operandi is to minimize his opponent’s maximum pay- 
ofT. Finally, we can define a “anti-Nash player” similarly. Given the generation 
of payoffs described in Section 4.2, the following Theorem is obvious: 

Theorem 4.3: The payoffs to two Anti-Bug players (resp.j two anti-maximin 
players, two anti-Nash players) evaluated over all environments are one mi- 
nus the payoffs to two Bug players (resp., two maximin players, two Nash 
players). 

4.4 A Summary of Expected Payoffs for Different Types of 
Players 

Putting together our results from subsections 4.2 and 4.3, we may state a 
“rank ordering” on all types. The ordering^^’^® is as follows: 

1. Cooperative joint maximization (.712) 

2. Sequential bugs (.646) 

3. Noncooperative players (between .625 and .646) 

4. Simultaneous bugs (.625) 

5. Maximin players (.583) 

6. Random players (.5) 

7. Anti-maximin players (.417) 

8. Simultaneous Anti-bugs (.375) 

9. Anti-Nash players (between .354 and .375) 

10. Sequential Anti-bugs (.354) 

11. Cooperative joint minimization (.288). 



5. A Final Word 

In a separate paper, we explore the types of movement that can occur by 
the two types of bugs in a bimatrix game. Specifically we ask, “What is the 
longest possible cycle on which the bugs can embark?”, “How many cycles 
of what length can there be?”, and “How long can the bugs move before 
entering a cycle or PSNE?” See Quint-Shubik-Yan (1995a) or Quint-Shubik- 
Yan (1995b). 



In Quint-Shubik (1994b), the authors ranked the player types according to the 
criterion: “type x” ranks higher than “type y” if the measure of the subset of Q 
in which the “type x” player I does better than the “type y” Player I is larger 
than vice versa. Not only did this produce a complete ordering of the player 
types, but the ordering is exactly the same as we obtain here. 

In section 4.2 our concern was with the performance of players facing an oppo- 
nent of the same type. We have not covered the “mixed cases,” where a player 
of one type confronts a player of a different type. 




197 



References 



1. Baxanyi, I., J. Lee and M. Shubik (1992) “Classification of Two-person Ordinal 
Matrix Games. International journal of Game Theory 21, pp. 267—290. 

2. Brown, G. W. (1951) “Iterative Solutions of Games by Fictitious Play,” in Ac- 
tivity Analysis of Production and Allocation, T.C. Koopmans. New York; John 
Wiley, pp. 374-376. 

3. Harsanyi, J. and R. Selten (1988), A General Theory of Equilibrium Selection 
in Games. Cambridge, MA: MIT Press. 

4. Hoel, P., Port, S., and C. Stone (1971), Introduction to Probability Theory, 
Boston, MA: Houghton Mifflin Company. 

5. O’Neill, B. (1988) “Notes on Game Counting” Yale University, mimeo, April 
18. 

6. Quint, T. and M. Shubik (1994a) “On the Number of Nash Equilibria in a 
Bimatrix Game,” CFDP #1089, Yale University. 

7. Quint, T. and M. Shubik (1994b) “A Comparison of Player Types in a 2 x 2 
Bimatrix Game,” Yale University, mimeo. 

8. Quint, T., M. Shubik, and D. Yan (1995a), “Dumb Bugs and Bright Noncooper- 
ative Players: Games, Context, and Behavior,” CFDP #1094, Yale University. 

9. Quint, T., M. Shubik, and D. Yan (1995b), “On Optimal Response and Cycling,” 
Yale University, mimeo. 

10. Rapoport, A., M. J. Guyer and D. G. Gordon (1976) The 2x2 Gcmie. Ann 
Arbor, MI: University of Michigan Press. 




Communication Effort in Teams and in Games 

Eric J. Friedman^ and Thomas Marschak^ 

^ Fuqua School of Business, Duke University 

^ Walter A. Haas School of Business, University of California, Berkeley 



Abstract. Quick and cheap communication networks may strengthen the 
case for the dispersal of organizations from one central site to many local ones. 
The case may be weakened if the dispersed persons behaved like members of 
a team before the dispersal but become self- interested players of a game after 
dispersal. To obtain some insight into these matters, the paper explores one 
very simple class of models. Each of N persons observes a signal that partly 
identifies the “week’s” (finite) game and then sends a chosen action (strategy) 
to a Center. Following the receipt of all N chosen actions, the Center puts 
them into force. Each person’s net payoff for the weak is the gross payoff 
determined for the week’s game minus a communication cost. Person z’s cost 
depends on the probabilities with which i sends alternative messages to the 
Center. We use the “information content” measure of classic Information 
Theory, multiplied by 9, the time required to transmit one bit. We study the 
“rule game” in which i chooses a rule that assigns a probability mixture over 
actions to each of z’s signals; in that game z’s payoff is z’s average weekly 
gross payoff minus i’s communication cost. We find that (i) if an equilibrium 
iV-tuple of rules exists, then it has to be an N- tuple in which true mixtures 
are not used; (ii) the existence of an equilibrium A'-tuple is not guaranteed 
and the usual appeal to mixing over the possible AT-tuples does not help, 
since such mixing has no meaning in our setting; (iii) even under the strong 
assumption that the gross payoff is identical for every person, there may be 
equilibria in which the dispersed persons communicate “too little” (from the 
team point of view) as well as equilibria in which they communicate “too 
much”; (iv) for very small 6 and very large 0, equilibria exist in the identical- 
gross-payoff case. A great deal remains to be learned. 



!• Introduction 

It is widely claimed that the revolution in high-speed communication net- 
works will lead organizations to disperse. An organization which previously 
operated at a single site, where its members were sufficiently controlled to 
behave essentially as a team, will now find it advantageous to disperse its 
members among local sites and to let them communicate via the network. If 
local sites are cheap compared to a large central site, then abandoning the 
central site and dispersing to local sites may be good for the organization 
— even if the tasks the organization wishes each member to perform remain 
unchanged. 




199 



But whether dispersal is desirable depends on the way the members use 
the communication network. If the dispersed members continue to behave 
like a team, then they will impose the “correct” demands on the network. 
Deciding what to communicate, and then sending it, uses up members’ time. 
The more time the members spend observing their private environments and 
deciding what to communicate, the better the actions which the organization 
takes when they are finished. But during the time they spend generating new 
actions, obsolete actions — or no actions — are in force. A balance has to be 
struck between the improved payoff due to better actions and the low payoff 
that is earned while the better actions are being found. 

Even if the delay due to communication is not costly, the act of commu- 
nication (deciding what to send and then sending it) still requires effort, and 
that effort has to be paid for. A balance then has to be struck between costly 
communication effort and the improved payoff due to the better actions that 
more effort produces. 

If the dispersed players continue to behave like a team, then they will find 
the balance that is correct from the team point of view, whether it is delay 
or effort that has to be weighed against payoff. The difficulty is that once 
members disperse and leave the controls of the central site behind them, they 
may start to pursue their self-interest instead of sharing a common goal. They 
may start to behave, in other words, like players in a game and not members of 
a team. Will they then choose the “correct” balance between communication 
effort and payoff improvement — or will they make some other choice? If the 
equilibrium choice is the wrong one (from the organization’s viewpoint) then 
the benefits of dispersal may be an illusion: while the network is cheap and 
the abandonment of the central site indeed saves money, the organization may 
have to undertake serious control expenses if it wants the dispersed members 
to make good choices. Consequently the case for dispersal may be a weak one. 
When is there an equilibrium in which the dispersed members communicate 
too little? When is there an equilibrium in which they communicate too 
much? If delay is costly, or if effort depends on how fast messages can be 
sent, then when are the equilibria sensitive to the speed of communication? 

One might study a variety of models to obtain some insight into these 
questions. In this paper we explore, in Section 2, just one class of models. 
Each person communicates a choice of action to a Center, and the resulting 
communication effort is simply subtracted from the sender’s gross payoff. 
Special as they are, the models present a number of challenges. Section 3 
describes, but does not study, a class of models in which the cost incurred 
by communication effort depends on the delay that communication causes. 
Concluding remarks are made in Section 4. 




200 



2. A Class of Models in which Communication Effort is 
Costly 

Consider an iV-person organization. Every week each person chooses an ac- 
tion. The N choices determine, for each person, a payoff that will be earned 
that week. To start with, the organization is not dispersed and behaves like a 
team. It wants the sum of the N payoffs to be large, given the communication 
effort that it makes, and it seeks a good balance between the sum of the N 
payoffs and the organization’s total communication effort. After dispersing, 
the organization becomes a collection of selfish individuals. Each wants his 
individual payoff to be large and seeks a good balance between his individual 
payoff and his individual communication effort. In either case, we begin our 
description of the typical week by specifying a game g belonging to a finite 
set G of possible games, which we shall call the gross-payoff games. 

In the game g each person i has a finite set^ of possible actions. Let 
A* denote A^ x ■ x A^ . If the action A'-tuple (a^, . . . ,a^) G A* is chosen 
by the N persons, then, in the game g, Person i,i = 1, . . . ,iV, receives the 
(gross) payoff . . . , a^) G IR. 

We shall suppose that as soon as each person has decided on his action for 
the week, he communicates that choice to a Center. That happens whether 
or not the organization is dispersed. The Center is not a player but rather a 
coordinator. He does not know what the week’s game is. His function is to 
insure that the chosen actions are carried out. 

We assume that each week’s game is drawn from the same population 
of possible games; this week’s drawing is independent of previous drawings. 
At the start of each week. Person i instantly observes a signal that tells 
i something about the new game g. Each person’s signal is independent of 
every other person’s signal. For each person i there is a finite set of 
possible signals. For generality we shall let Person i respond to the week’s 
new signal by choosing a probability mixture over his |A*| possible actions. 

Specifically, Person i uses a rulep^ : W* where pl^*l denotes the 

probability simplex in |A*|-space, i.e., the set {(pi, • • • ,P|a*|) I Pa > 0,o = 

1, . . . , |A|; Pa = !}• Observing the signal G and using the rule 
p*. Person i chooses the action a G A* with probability Pa(w^)- Person i then 
tells the Center what his chosen action is. 

Given the rule AT-tuple (p^, . . . ,p^), Person i obtains the average weekly 
gross payoff 

u,»€vr* 



N 



•P*(a^...,a^), 



geo 



< 1=1 



^ In what follows, persons will always be identified by superscripts . 




201 



where is the joint probability that the game is ^ G G and i’s signal is 

€ W^. All the joint probabilities are common knowledge. 

Now suppose that we measure i’s average weekly communication effort by 
some real- valued function defined on the possible rules (We defer for a 

moment a discussion of the particular function we shall be using). Then z’s 
average weekly net payoff for the rule A-tuple . . . ,p^) is 

The game in which Person z’s strategy space is the set of rules and the payoffs 
are given by y ^ , . . . , will be called the rule game. It will be useful to define 

N 

i=l 

If the organization is a team, then it uses an iV-tuple (p^, . . . ,p^) that max- 
imizes V. Our central questions will be these: When is there a Nash equi- 
librium of the rule game in which there is “too little” communication — in 
which the total communication is less than it is in the team-optimal 

solution? When is there a Nash equilibrium in which there is “too much” 
communication? The answers will depend on the gross-payoff games and on 
the communication cost measures 

Consider the special case in which tt^ can be written as 7r^(J^(p^), 

. . . , J^(p^)). Here we may view the rule game as a communication- effort 
game., wherein each Person i chooses a communication effort p* and collects 
the net payoff , • • • , — P*- We are then in familiar territory. As first 

noted by Holmstrom (Holstrom 1983), there is no Nash equilibrium in which 
the chosen efforts are team-optimal. Informally speaking, in a Nash equilib- 
rium of the effort game. Person i equates the marginal change in (as 2 varies 
his effort) to one, whereas Person i “should” equate the marginal change in 

TT* to one. Moreover, if, for every {i,j), is increasing in , then in a 
Nash equilibrium Person i undervalues the marginal contribution of his com- 
munication effort to the gross team payoff 53 ^^o little communication 

effort occurs. 

It is possible, however (as Holmstrom noted), to shift the incentives to- 
wards team-optimal effort by modifying the effort game. If person i receives 
7 T^ — when the entire effort vector chosen by the N persons is team-optimal, 
but receives —rf otherwise, then a team-optimal vector is a Nash equilibrium. 
But that version of the effort game would seem to require a drastic reshaping 
of our organization. After the organization disperses, we imagine individual 
payoff to be determined following each person’s action choice in exactly the 
same way as it was before dispersal. In the modified game, by contrast, some 
sort of policing agency would have to determine whether the chosen effort 
iV-tuple is in fact the “correct” one, and would then have to deny everyone 
any part of the gross payoff if the iV-tuple is wrong. 




202 



For the communication-efFort measure we shall use, it will not be possible 
to write tt* in the form tt*. We are, in fact, in quite new territory. 

We take our measure of individual communication effort from classic In- 
formation Theory. We view Person i, at the start of a week, as a source of 
I A* I possible messages to the Center. Given that i uses the rule ), our 
measure is 

6 • ( the information content of p*), 
where 6 >0 and the information content of p* is 

( probability that i sends a*)-log 2 ( probability that i sends a*). 

a*GA* 



P reaches its maximum when all the probabilities are equal and its minimum 
when one of them is one while the others are zero. The probability that Person 
i sends the action to the Center is 

IT Z! 

geGw'eW' 

We shall refer to the game in which each person i’s payoff is 7 r^(p^, . . . ,p^) — 
0P{p^) as the rule game for 6. 

The term “information content”, as just defined, is widely used in Infor- 
mation Theory. To obtain one interpretation of P{p^) and of 9P(p'^), first 
note that some of i’s action messages will be sent more frequently than oth- 
ers. Imagine i to code each of the \A'^\ possible messages as a binary string, 
in such a way that the Center can decode each string without ambiguity; 
the Center can distinguish any string from its predecessor. Suppose that the 
transmission of one bit requires time 6 and that all other elements of the act 
of communication are instantaneous. Then, using an appropriate coding, the 
average weekly length of the binary string that i sends is between P{p^) and 
/^(p*) + 1. So the average weekly time that Person i, or someone in i’s employ, 
has to spend in transmitting the week’s binary string is between 6P{p^) and 
9P{p^)-\-0? We suppose that the transmitter has to be paid for his time, and 
that the payment comes out of i’s “pocket” and has to be subtracted from 
7T* to obtain z’s net payoff. The case 0 = 0 (instantaneous transmission) will 
be an important benchmark. In the rule game for 0 = 0, Vs payoff is just 

7r*(p^ . . . 

^ See, for example, Abramson (1963), Chapter 4. To obtain another interpretation 
of /*(p*), imagine that each week the organization draws not once but K times 
from the population of possible games and their accompanying signals. The K 
games then take place simultaneously during the week. Person i sees one signal 
for each of the K games and chooses an action for each game in response to 
that game’s signal. Since all K games and their accompanying signals are drawn 
independently from the same population of games and signals, Person i will use 
a common rule p* for each of the K games. Then as K increases, the average 
length per game of the weekly message i sends to the Center approaches Pip")- 




203 



Note that P is continuous and strictly concave in Those will be the 
two key properties of P that we shall use. The results we obtain and the 
questions we raise will be the same for any communication-effort measure 
that depends on i’s message probabilities and has those two properties. 

We would argue that probability-sensitive communication-effort measures 
have to be serious candidates in modelling organizations. One could certainly 
consider a measure that does not depend on message probabilities — for ex- 
ample, the total number of possible messages that i can send. Such a measure 
ignores the saving in average communication effort that coding permits. Once 
we choose to recognize that saving — and once we note that equal message 
probabilities are the most demanding case, while having just one possible 
message is the least demanding — we are driven to measures that depend on 
the probabilities. Among possible measures of that sort the classic P seems 
the strongest candidate.^ 

We now establish an important fact about the rule game for 0 > 0. First 
define 

_ pA* X . . . X 
|w*| times 



We have 

Lemma 1 For any 6 > 0; (i) The rule game has a finite number (possibly 
zero) of Nash equilibria; i/ (p^, . . . ,p^)is a Nash equilibrium of the rule game, 
then for every i and every G W^, p*(u;*) is not a true mixture (all com- 
ponents of p^{w^) are zero except one), (ii) There exists a team-optimal rule 
N-tuple (a maximizer ofV = number of optimal rule N -tuples 

is finite; if (p^, . . . ,p^) is team-optimal, then for every i and every G W^, 
p^{w^) is not a true mixture. 

Proof: For both parts of the Lemma, we note that a strategy for Person i 
in the rule game is a function from the finite set to the compact convex 
set Pl^l*, and is therefore a point in P% which is also a compact convex set 
and has a finite number of extreme points. Thus a rule iV-tuple is a point in 
the set X • • • X P^; that set is again compact and ha^ a finite number of 
extreme points. 



^ When i scans his current signal, chooses an action in response, encodes it as a 
binary string, and then transmits it, he is, of course, engaging in time-consuming 
efforts other than transmission itself. But one can argue that P says something 
about those efforts as well. Suppose, for example, that i uses just two possible 
messages in response to his signal. Compare the case where the first message 
is used far more frequently than the second with the case where they are used 
equally often. In the first case, it may be easier for i to scan the current signal, 
since the signals leading to the second message are rare and stand out promi- 
nently. The difference in the signal-scanning effort between the two cases is then 
properly reflected in P. Modellers of information-processing efforts axe very far 
indeed from a consensus about a winning approach. We argue that the classic 
measure fits our setting quite well. 




204 



(i) Consider Person 1. The rule maximizes . . ,p^) on F^. But 

V^{p^,p^,. , . ,p^) is strictly convex in p^, since n^{p^,p^, . . , ,p^) is lin- 
ear in and I^{p^) is strictly concave in p^. That means^ that p^ is 
an extreme point of F^, so that for any e W^, p^{w^) is not a true 
mixture. Similarly for Persons 2,...,N. Thus a Nash equilibrium is an 
extreme point of x • • • x F^ . Since that set has a finite number of 
extreme points, there is a finite number (possibly zero) of Nash equilibria. 

(ii) The function V = is defined on x • • • x F^ and is strictly convex. 

Hence some extreme point of x • • • x P^, say maximizes 

Y2i ^ team-optimal rule A^-tuple. Moreover every team-optimal 

rule N'-tuple is an extreme point, so there is a finite number of team- 
optimal iV-tuples. Now consider the game in which each person i chooses 
a p* and i collects the payoff ^V{p^,. . . ,p^). A team-optimal A^-tuple 
of the rule game must be an equilibrium of the new game. Using the 
argument in Part (i), we conclude that for any i and any G W\p‘^(w^) 
is not a true mixture. ■ 

Now define 

E = {(p^ , . . . ,p^) I p* is an extreme point of F\ i = 1, . . . , N}. 

The set E is also the set of extreme points of P^ x • • • x P^. It is finite. The 
Lemma establishes, for any 0 > 0, that 

— If there is a rule iV-tuple that is a Nash equilibrium of the rule game for 
0, then it belongs to E. 

— Some member of E is team-optimal in the rule game for 6 and every team 
optimum belongs to E. 

Without imposing further conditions, we cannot guarantee that a rule 
A^-tuple which is a Nash equilibrium exists. Theorems on the existence of 
equilibria when strategy spaces are continua do guarantee the existence of 
mixtures of rule N -tuples with the equilibrium property.^ But such a mix- 
ture, if it is not degenerate, has no meaning in our setting. If it is not de- 
generate, such a mixture will require some Person i to mix over two or more 
rules — for example, to use p* with probability | and p*' with probabil- 
ity |. But the resulting expected value of does not properly express z’s 
net payoff after allowing for communication effort. In the expected value 
of the term |P(p^) + |P(p*‘) appears, and that term does not equal 

— probability of sending a^) • (log probability of sending a*)]. 

2.1 Three Examples 

We shall consider three two-person examples. In the first two examples, each 
person has just two possible actions. 

^ See, for example, Luenberger (1973) 

® See, for example, Pudenberg and Tirole (1992), Theorem 3.1. 




205 



In the first example there are four possible gross-payoff games, labelled 
TL, BR, BL, and TR. “T” and “B” stand for Top and Bottom; “L” and 
“R” stand for Left and Right. Each game occurs with probability j. Person 
I’s signal is the first term of the identifying label, and Person 2’s signal is the 
second term. Each of the four gross-payoff games has the strong property that 
the two persons receive the same payoff. We shall call such games identical- 
payoff games. The four identical-payoff games are shown below. Person I’s 
actions are Top, Bottom and Person 2’s actions are Left, Right. If each person 
chooses the action identified by his signal, then each receives 4. If only one 
does so, then each receives 4 ~ 4e. If neither does so, then each receives zero. 
The common payoff received by each person is shown in each box. 



game TL 



game TR 



Left Right 



Top 


4 


4-4e 


Bottom 


4-4e 


0 




game BL 
Left Right 


Top 


vu 

1 


0 


Bottom 


4 


4-4e 



Left Right 



Top 


4-4e 


4 


Bottom 


0 


4-4e 




game BR 
Left Right 


Top 


0 


4-4e 


Bottom 


4-4e 


4 



We assume 0 < e < From the team’s point of view each person ought 
to take the action identified by his signal. 

In view of Lemma 1 we can confine attention to “pure” rule pairs, in 
which each player chooses one action in response to each signal. In the rule 
game each person has four possible pure rules. Two of them are constant- 
action rules, with information content zero. In the other two, a different 
action is sent to the Center for each signal and the information content is 
— (| • log \ \ ' log |) = 1- III game for 0 there are two identical 

rows and two identical columns. Nevertheless, we write out the whole payoff 
matrix. 




206 





Left always 


Right always 


Left if R, 
Right if L 


Left if L, 
Right if R 


Top always 


^ -6 ^ ~e 


3 3 _ 

2 ^’2 ^ 


1 e 1 e 

2 2 > 2 2 

e 


2-e, 2-6-9 


Bottom 

always 


^ -e ^-6 


^-6 ^ -e 
2 ^>2 ^ 


1 e 1 e 

2 2 ’ 2 2 

e 


2-e, 2-6-9 


Top if B, 
Bottom if T 


1 € n 1 

2 2 2 
e 

2 


1 € n 1 

2 2 2 
c 

9 


-e,-e 


2- 6-9,2 - 
6-9 


Bottom if B, 
Top if T 


2 -e-e, 2 -e 


2-e-e,2-e 


3 3 n 3 

2 2^ ‘^»2 


2-6,2-e 



The first two rows and the first two columns are identical. If we delete 
one of the duplicate rows and one of the duplicate columns and if we now 
suppose that 

then we find that the middle column and the middle row are dominated. We 
are left with: 





Constant action 


Left if L, Right if R 


Constant action 


2 ^'2 ^ 


2-e,2-e-e 


Bottom if B, Top if T 


2-e-0,2-e 


2-6,2-e 



This is a Battle of the Sexes with two pure equilibria: {Bottom if B and 
Top if Tj constant action) and {constant action, Left if L and Right if R). 
Each equilibrium has a total information content of = 1 . On the other 

hand, ^ < 6 < e implies 9 < ^ + e and 6 < 2e. That implies in turn that the 
team-optimal rule pair is {Bottom if B and Top if T, Left if L and Right if 
R), with a total information content of two. The two persons, as rule-game 
players in an equilibrium, communicate less than they do as a team. 

In the second example, there are just two possible gross-payoff games, 
labelled 0 and 1. Each occurs with probability Person 1 observes only 
an uninformative signal and does not know the identity of the week’s game. 
Person 2 knows the identity of the week’s game perfectly. 



game 1 game 0 

Left Right Left Right 



Top 


20, 10 


-8,2 


Top 


40,2 


-2,8 


Bottom 


16,-8 


0,0 


Bottom 


-20,6 


0,0 



The rule game is as follows: 




207 





Left always 


Right always 


Left 

if 1, Right if 
0 


Right if 1, 
Left if zero 


Top always 


30,6 


-5,5 


9, 9-6» 


16, 2-9 


Bottom always 


-2,-1 


0,0 


8,-4-6» 


-10, Z-9 



We suppose that 

0 < 6> < 3. 

Then there are no dominated rows or columns, the only (pure) equilibrium 
is {Top always. Left if 1 and Right if 0), with a total information content 
of one. But the team-optimal rule pair is {Top always, Left always), with 
information content zero. This time, the two persons, as game players in 
a rule-game equilibrium, communicate more than they do as members of a 
team. 

If we now change the payoffs in the lower left box of game 1 to (82, -8), 
then in the rule game the payoffs in the first box of the bottom row become 
(31,-1) and the payoffs in the third box of the bottom row become (41, -4-0). 
If 0 equals 2 (for example), there is then no pure equilibrium in the rule 
game. There are equilibria which are pairs of mixtures of rules, but — as we 
argued above — a person’s average payoff in such a rule mixture does not 
incorporate a reasonable measure of that person’s communication effort. 

If we return to the identical-payoff case, can we again have the game 
players communicating more than the team members? It is perhaps surprising 
that even with the strong identical-payoff property this can indeed happen 
again. That is shown in our third two-person example, which, however, has 
substantially more actions, gross-payoff games, signals, and possible rules 
than the preceding two examples. 

There are now sixteen possible gross-payoff games, each identified by one 
of the sixteen possible four-digit binary strings. Person I’s signal is the first 
pair of digits in the string and Person 2’s signal is the second pair of digits. 
Each person observes the four possible signals 00, 01, 10, 11 with probabil- 
ities .3, .3, .2, .2. Each person has seven possible actions which we shall label 

^00 7 ^01 5 ^10 j ^11, Co, Cl, d. 

Define i’s signal action to be the “6” whose subscript equals i's signal. 
Define i’s appropriate c to be that “c” whose subscript equals the first digit 
in i’s signal. Then the typical gross-payoff game may be described as follows: 





signal action 


appropriate c 


d 


other actions 


signal action 


0 


0 


1 


0 


appropriate c 


0 


1 


0 


0 


d 


1 


0 


0 


0 


other actions 


0 


0 


0 


0 




208 



The number in each box of the table is the payoff received by Person 1 
and by Person 2. There are many possible rule pairs. But only three pairs 
guarantee — for 0 = 0 — that in every gross-payoff game each person col- 
lects 1. In two of those pairs one person always takes the signal action while 
the other always takes d. In the third pair each person always chooses the 
appropriate c. If Person i always takes (and sends) the signal action, then P 
equals 

-2(.3(/o^.3) 4- .2{log.2)), 

which equals 1.97 to two decimal places. If Person i always takes (and sends) 
the appropriate c, then P equals 

-{S{log.6) -f A{logA)), 

which equals .97 to two decimal places. 

To analyze the rule game it suffices to study the following three-by-three 
submatrix of the rule-game payoff matrix. The submatrix deals with the rules 
that belong to the three pairs just described. 





always take signal 
action 


always take appro- 
priate c 


always take d 


always take signal 
action 


-1.976,-1.970 


-1.970, -.970 


1 - 1.976, 1 


always take appro- 
priate c 


-.976,-1.976 


1-. 970,1 -.970 


-.970,0 


always take d 


1,1 - 1.976 


0, -.970 


0,0 



Now consider the relation between the three-by-three submatrix and the 
matrix of the full rule game. First notice that for any 0 > 0, the rule pair 
{take appropriate c-action, take appropriate c-action) is a team optimum in 
the game defined by the three-by-three submatrix. For that rule pair, the 
team earns a total average gross payoff of 2 and a net payoff of 2 — 1.94 0. 
Second, note that in any box of the full matrix that is outside the submatrix, 
average gross payoff is less than two. So for 0 = 0, and also for sufficiently 
small positive 0, that rule pair remains team-optimal in the full rule game. 
For sufficiently small 0, moreover, no other rule pair of the full game is team- 
optimal. 

Finally, note that in any box that lies in the extension of a row of the 
submatrix to the full matrix. Person I’s communication cost remains what 
it was in that row of the submatrix, while I’s average gross payoff is no 
higher than it was in that row of the submatrix. Similarly for Person 2 and 
the extension of any column of the submatrix. It follows that for any 6, any 
equilibrium of the three-by-three game remains an equilibrium of the full rule 
game. 

If 9 is less than 2 ^ (sufficiently less to overcome rounding inaccuracies in 
the logarithms), then in the three-by-three game the rule pair whose payoffs 




209 



are in the lower left box, and the pair whose payoffs are in the upper right box, 
are equilibria, li 6 < -^ then a third equilibrium (for any positive 0) is the 
team-optimal pair already noted, whose payoffs are in the middle box. None 
of the three equilibria is Pareto-dominated by any of the others. Moreover, 
for sufficiently small 0, no equilibrium rule pair of the full rule game that 
one might find outside the submatrix can Pareto-dominate any of the three 
equilibria of the three-by-three game. 

Among the three equilibria of the rule game it is the team-optimal pair 
that has the lowest communication effort (for positive 0); that total is 1.94 9. 
The other two equilibria have a total effort of 1.97 9, 

We conclude: for sufficiently small positive 9 there are undominated equi- 
Ibria of the rule game in which total communication effort is more than in 
the team optimum. 

2.2 A General Proposition about Team-Optimal Total 
Information Content. 

We shall establish that the team has a well-behaved “demand curve” for total 
information content. As 9 — the “cost” of transmission — rises, the infor- 
mation content of the team’s messages cannot increase. In fact its “demand 
curve” — the graph of / — is a descending sequence of flat lines. 

We shall need some definitions. The rule A^-tuple (p^, . . . ,p^) will often 
be denoted p. We let 7r(p),(7(p) stand, respectively, for the average total 
gross payoff ^he total information content For 

any ^ > 0 we define: 

L{9) = {p \ p is team-optimal in the rule game for 9} 

L*{9) = {p I p is an admissable equilibrium in the rule game for 9}. 

“Admissable” means Nash and not Pareto-dominated. 

We next define two finite sets whose elements are levels of total informa- 
tion content: 

m = {r e iR I r = C(p);p G L{e)} 

7*(0) = {rGJR|r = C(p);pGL*(0)}. 

(In a more cumbersome notation, we can write C{L{9)) and C(L*{9)) for 
these two sets). We let I{6)^I_{9) denote, respectively, the largest and the 
smallest element of I{9)\ and we let I*{9),I_*{9) denote the largest and small- 
est element of I* (9). 

Theorem 1 For every 9 > 0, the set I{9) is a singleton at all but a finite 
number of non-negative values of 9. Where it is not a singletonj it is finite. 
The functions I_ and I are non-increasing. 

Proof: Prom Lemma 1, we know that the set {p j p G L(6); 0 > 0} is a subset 
of the finite extreme-point set E. For any fixed rule iV-tuple p in E, the total 




210 



team payoff in the rule game for ^ is a function of 0, namely 7r(p) — 0C{p). 
Define 

T{6) = maXp^E{T^{p) - 0C{p)). 

Note that for any 0 > 0, all rule iV-tuples p in L{6) have the same value of T, 
while all other rules have a lower value. The function T is the upper envelope 
of a finite collection of straight lines with nonpositive slopes. The function T 
must therefore be convex; the slopes of the straight lines defining T must get 
smaller in absolute value as we move in the direction of increasing 9. 

For all positive 0 in a neighborhood of zero, T = 7r(p*) — 9C{p*), which 
can also be written 7r(p*) — /(O)0, where p* is a member of L(0) for which 
total information content C{p) is minimal. 

At some 0, the line 7r(p*) — Z(O)0 intersects a new line of the envelope, 
namely 7r(p**) — C{p**)9^ which can also be written 7r{p**) — where p** 

is a member of L{6) for which the total information content is minimal. At 0, 
both p* and p’^* are team-optimal rule AT-tuples. The set /(0), consisting of 
total-information-content levels for team-optimal rule A'-tuples at 9 is finite. 
Its smallest member is C(p**) = I_{9) and its largest member is C(p*) = Z(^). 

The pattern is the same for the remaining corners of the envelope T. For 
any 9 that is at a corner, the set I {9) is finite and its smallest member I_{9) is 
the absolute value of the slope of the new straight line of the envelope. The 
new straight line is smaller, with regard to the absolute value of its slope, 
than the preceding straight line. 

The number of corners — at which the set /( ) is not a singleton — is 
finite since each successive corner is defined by two members of the finite set 
E and one of those members is unique to the corner. 

All parts of the Theorem are therefore established. ■ 

Figure 1 and Figure 2 illustrate the pattern for a very simple case, in 
which there are just three rule A^-tuples, called p*,p“^*, and p; their total 
information content is, respectively, 1,|, and zero. The envelope has just two 
corners. The function / consists of three flat pieces, for the values of 9 that 
are not at a corner of the envelope, plus {|, 1} for the corner value 9 = 2, 
and {0, for the corner value 9 = 4. 

2.3 The Case of Identical Payoffs in the Gross-Payoff Game 

The third of our three examples showed that even if we impose the strong 
identical-payoff condition, there may be equilibria with “too much” commu- 
nication as well as equilibria with “too little” . What general statements about 
equilibria are implied if we add the identical-payoff condition? The following 
Theorem shows that if we add the condition, then for very small 9 and for 
very large 9 the anomalies that appear in the examples disappear. For such 
9, existence of equilibrium is guaranteed. Moreover for very small 9, the rule 
iV-tuples used in a rule game, and those used in a team optimum, are both 




212 



subsets of the rules used when 6 = 0, For very large 6, the set of equilibrium 

iV-tuples is the same as the set of team optima. 

Theorem 2 If all of the gross-payoff games have identical pay offs, then: 

(1) For 6 sufficiently small: (a) an admissahle equilibrium of the rule game 
exists; (b) L^{6) C L*(0) = L(0); and (c) L{6) C L(0). 

(2) For 9 sufficiently large, an admissable equilibrium of the rule game exists; 
L{6) = L*{9); and r{6) = L{0) = F(6) = I{6) = 0. 

(3) Suppose the rule game has an equilibrium for every 9 >0. Then for any 
0>0, I{9)=0=^P{0)=0. 

Proof: 

(l)(a) We consider the finite rule game, in which every Person i chooses from 
the finite set The sets L(0) and L*(0) are identical; they consist of 
the A^-tuples in which the common average gross payoff, received by 
every person, is maximal. We now describe a procedure which leads 
to a rule iV-tuple that is an admissable equilibrium if 9 is sufficiently 
small. Define an A^-tuple to be valid if it belongs to 1/(0). We first note 
that if 9 is sufficiently small, then given any valid N- tuple, any person 
i will only be interested in deviating from that iV-tuple if the resulting 
deviation preserves validity (the new A^-tuple remains in 1/(0)) while 
lowering Vs information effort P. 

We start our procedure with an arbitrary member of T(0), say 
(p^, . . . ,p^). Now we ask Person 1 to replace by p* ^ where 
p^ preserves validity and has a lowest information cost P among all 
the rules in F^ that preserve validity. Next we ask Person 2 to con- 
sider the new N'-tuple (p* ,p^,p^, . . . ,p^) and to replace p^, by p* , 
which preserves validity and has the lowest value of P among all the 
rules in F^ that preserve validity. Continue in this manner until it is 
Person TV’s turn. After Person TV’s turn we have obtained an TV-tuple 

1 N 

p* = (p* , . . . , p* ) . We then go through the same TV steps for p* . If 
no person i replaces p*' with a different rule when it is his turn, then 
p* is an admissable equilibrium. If someone does make such a replace- 
ment, then complete the second sequence of TV steps and start a new 
sequence with the TV-tuple so obtained. The procedure cannot cycle, 
since at least one person’s information content decreases in each com- 
pleted sequence. Moreover there must be a terminal sequence, since 
every T* is finite. 

(l)(b) Suppose that p is an equilibrium in the rule game for 9 but that 
(contrary to assertion (l)(b)), p does not belong to 1/(0). Let p denote 
the rule TV-tuple found by the procedure described in the proof of 
(l)(a). The TV- tuple p belongs to L(0) and yields the TV-tuple of net 
payoffs {H - 6F(ff^),. .. ,H - 91^ {p^)), where H = maxp^E 7r(p) 
is the average gross payoff received by every person for any rule TV- 
tuple belonging to 1/(0). On the other hand, for p, the TV-tuple of net 




213 



payoffs is (Q — 01^ where Q = Since 

p does not belong to 1/(0), we have Q < H. That means that for 0 
small enough, p is Pareto-dominated by p while at the same time p is, 
as shown in the proof of (l)(a), an equilibrium. So while p may be an 
equilibrium, it is not admissable. 

(1) (c) The proof is straightforward. 

(2) If 6 is sufficiently large, then for every Person every rule p^ G 
with P{p^) > 0, is dominated by some no-communication rule (with 
P( ) zz: 0), under which i chooses a constant action, regardless of 
signal. Discarding all dominated strategies, we are left with the gross- 
payoff game. Since every action A^-tuple in that game hats the same 
payoff for every person, an equilibrium exists. Proofs of the remaining 
statements in (2) are straightforward. 

(3) We are given that for some 0, some p* £ E is a, no-communication 
rule A^-tuple which is also team-optimal, i.e., C7(p*) = 0 and 

7t{p*) > 7t{p) — 0C{p), all p e E. 

The identical-payoff property of the gross-payoff games implies that 
= ■ • • = TT^ip*) = T. 

But p* is also an equilibrium. For suppose not. Then for some person 
— say Person 1 — there is a better reply to {p* , . . . ,p* ) than p* . 
Let p^ denote that better reply, i.e., 

v^(p\p'^ . . . ,p*") = z^~ > v^(p*) = T, 

where = tt^ (p^ , p *^ , . . . , p*^ ) . Hence 

(t) > T. 

The identical-payoff property of the gross-payoff game implies that for 
every person 2 > 1, P*(p^,p* , . . . ,p* ) — — P(p^). But, in view 

of (t), that means that (p^,p* , . . . ,p* ) achieves a higher total team 
payoff than p* does, which is a contradiction. Thus zero is the lowest 
total information content among the equilibria of the rule game. ■ 

If we want to obtain similar characterizations of team information content 
and game-equilibrium information content for values of 6 that are not very 
large or very small, then, it appears, we have to add further conditions on 
the gross-payoff games. The identical-payoff condition, strong as it is, does 
not suffice. 



3- Models with Delay 

In the model thus far. Person i is the only person to suffer directly when i 
increases his own communication effort, from which all may benefit. Suppose 




214 



instead that Vs increased effort imposes a cost on everyone because of the 
delay it causes. Suppose that the week’s new actions cannot be taken until 
everyone has been heard from. 

Then in the simplest variation of the preceding model, the Center receives 
the N messages in sequence. It takes the average time 9 until all 

N mesages have been received. We then define Vs net payoff in the rule game 
for 9, when the rule iV-tuple p is used, to be 

1 ^ 

F‘(p)=7r‘(p)--0^r(p‘). 

A penalty equal to the delay is subtracted from the week’s gross payoff, 
and everyone is assessed an equal share of that penalty. (One might imagine 
the organization to have a client who will pay the organization in a given 
week, where the game is g and the actions are (a^, . . . , a^), the net amount 
, • • • , CL^) minus a penalty equal to the time until the week’s actions 
are actually “delivered” to the client). 

If, on the other hand, the N messages are received simultaneously^ then 
the time 0maxi(/*(p^)) elapses until the new actions are taken, and we have 

V\p) = 7t‘(p) - ^(9mp(/’(p*)). 

In a more ambitious variation, it remains the case that new actions cannot 
be taken until everyone has been heard from, but the penalty due to delay 
is now the payoff that is foregone until the new actions have been found 
and taken. Take the week to be the unit in which time is measured. For 
Q > 0, interpret Q . . . , a^) as the gross payoff that accumulates 

during an interval of length Q in which the game is g and the actions are 
(a^, . . . ,a^). Make the rather drastic simplifying assumption that until the 
week’s new actions have been taken, zero payoff accumulates. Suppose (to 
simplify again) that for all possible rule A'-tuples (p^, . . . ,p^), 0maxi P(p^) < 
1 and < 1. (The week’s new actions are always taken before 

the week is up, whether transmissions are simultaneous or sequential). Then 
if message transmissions are simultaneous. Vs net payoff in the rule game 
becomes 

y* = 7T‘(pi, . . . ,p^) • (1 - 0maxr(p*)). 

I 

If messages are received in sequence, then we have 

N 

t=l 




215 



All the questions we have uncovered in our previous efFort-is- costly model 
arise again in all of the four variants.® In the simultaneous- transmission mod- 
els, we have now lost the strict convexity of V\ and have therefore also lost 
the useful fact that the number of pure equilibria is finite. The equilibrium- 
existence question remains in all four models. One can again not use rule 
mixtures to insure existence of meaningful equilibria. In all four models there 
appears to be even more reason to believe that when pure equilibria exist 
they may include both equilibria in which there is too much communication 
and equilibria in which there is too little, even in the identical-payoff case. 
The behavior of the functions I and /* is far from clear a priori even for the 
identical-payoff case. 

Both the models of Section 2 and the delay models just sketched could 
be altered in one respect. Instead of viewing the Center as a passive robot 
who simply turns the action choices of the other persons into actions that are 
physically in force, we could let the Center be a robot who works harder. We 
could let each person’s message to the Center be an arbitrary message about 
the sender’s current signal instead of restricting it to be an action choice. 
The Center translates the messages received into the actions that are to be 
taken, using a rule that is best from the organization’s point of view.^ 



4. Concluding Remarks 

The popular view that cheap and quick communication strengthens the case 
for the dispersal of organizations turns out to rest on rather shaky ground. 
If dispersal means that team members become game players, then they may 

® In a still more ambitious model, one would drop the assumption that the suc- 
cessive weeks’ games are serially independent and would suppose that they are 
generated by a stochastic process — a Markov chain, for example. Then is 
accumulated payoff during the part of the week in which the new actions are 
being found and transmitted is not zero, as in the version above. Rather it equals 
the payoff specified by this week’s game for last week’s actions multiplied by 
0maxt P(p^) in the simultaneous-transmission case and by P{p') iii the 

sequential-transmission case. V* becomes i’s average total weekly payoff — the 
old-actions total payoff plus the new-actions total payoff — over all realizations 
of the process. 

^ That sort of model would be closer in spirit to the Theory of Teams (Marschak 
and Radner, 1971), which compares alternative “information structures” with 
respect to average gross team payoff, though not with respect to communication 
cost. A structure specifies who knows what about the team’s current environ- 
ment (the current game, in our setting). Suppose the Center knows the rule 
used by every person i to select a message given I’s signal. Then when the N 
persons send messages that axe proposed actions, the Center acquires informa- 
tion about the week’s environment (game). When they send arbitrary messages 
instead, the Center acquires different information. Models with arbitrary mes- 
sages would generally be harder to study than ours. The equilibrium-existence 
question, in particular, would not be easier. 




216 



communicate too little (from the organization’s viewpoint) and they may 
communicate too much. We appear to be very far indeed from the familiar 
setting of the effort models first explored by Holmstrom. We have done little 
more here than to open an agenda. 

The most striking finding is that if communication effort is going to be 
measured in a way that depends on message probabilities, then the customary 
role of mixtures in guaranteeing the existence of equilibrium no longer helps. 
Even the Harsanyi interpetation of an equilibrium of mixtures (Harsanyi, 
1973) fails. In the two-person case, that interpretation asks 1 to think of 2’s 
mixture as portraying 2’s alternative types, not observable to 1. For each of 
2’s types, 2 uses a different rule. But 1 cannot meaningfully view the payoff 
he earns when he chooses a rule as the average he earns over 2’s possible 
types — since that average again fails to incorporate a meaningful measure 
of I’s communication effort for his chosen rule. 

We appear to be in a new area where one central issue is “too few” equi- 
libria instead of the more familiar “too many” . It was the work of Reinhard 
Selten that went a very long way in meeting the challenge of “too many”. 
Possibly it will take someone with his vision and ingenuity to clear up the 
present puzzle, using concepts and approaches not yet imagined. 



References 



Abramson, N.: Information Theory and Coding^ McGraw-Hill, 1963. 

Fudenberg, D. and Tirole, J.: Game Theory^ MIT Press, 1992. 

Harsanyi, J.C.: “Games with Randomly Distributed Payoffs: A New Rationale for 
Mixed-Strategy Equilibrium points,” International Journal of Game Theory 2 
(1973), 1-23. 

Holmstrom, B.: “Moral Hazard in Teams,” Bell Journal of Economics^ 13 (1982), 
324-40. 

Luenberger, D.G.: Introduction to Linear and Nonlinear Programming^ Addison 
Wesley, 1973. 

Maxschak, J. and Radner, R. Economic Theory of Teams, Yale University Press, 
1971. 




Endogenous Agendas in Conunittees 



Eyal Winter’ 

The Economics Department and The Center for Rationality and Interactive Decision Theory, 
The Hebrew University of Jerusalem, Jerusalem 91905, ISRAEL. 



1. Introduction 

When a committee faces the task of making a collective decision on several issues 
procedural matters seem to play an enormous role. The process of decision making 
typically involves a preliminary phase of negotiation, which may take the form of 
either an informal behind-the scene bargaining or a round-table discussion prior to 
voting. In this paper we will formulate the process of such decision making as a 
non-cooperative bargaining game, and analyze the effect of the agenda on the 
efficiency and stability of the bargaining outcomes. In Winter (1993) it was shown 
that if the imderlying preferences of the committee members are such that there 
exists a consensus over the importance of issues, then, in order to guarantee 
efficiency, issues should be negotiated in the order of their importance. Our main 
purpose here is to explore two models, in addition to those in Winter (1993), in 
which no agenda is exogenously determined prior to the bargaining. In one model 
the agenda itself is negotiated by the committee members before dealing with issues 
of substance; in the other, the agenda is determined endogenously, i.e., each 
proposer can at any time propose an alternative on any unsettled issue. We will 
show that these models can only partially replace the "important issues first" rule 
for maintaining efficiency and stability. 

Our approach here is one of multilateral bargaining theory. Reinhard Selten and 
John Harsanyi were the first to introduce extensive games into models of multi- 
person negotiations and coalition formation. Harsanyi (1974) and Selten (1981) can 
also be regarded as pioneering contributors to the literature on the non-cooperative 
foundations of cooperative game theory. Selten's (1981) paper introduces a 
bargaining model that involves sequential proposals of a coalition and a payoff 
distribution for its members^. An alternative model, introduced by Selten (1992), 
is the demand commitment bargaining, where players publicly announce then- 
requests from cooperation. Both models are based on an underlying cooperative 



I would like to thank the German-Israeli Foundation for financial support. 

^ Our bargaining models bellow are very much in the spirit of Selten's proposal model. 
However the underlying problems are different since we are interested in committees and not 
in characteristic function games. 




218 



game in coalitional form. Selten's interest in the relationship between non- 
cooperative equilibrium behavior and cooperative solution concepts was motivated 
by his awareness that understanding this relationship is essential for understanding 
the discrepancies between game theoretic predictions and actual behavior in 
experiments. The bargaining models considered here are designed to fit the context 
of committees, and are based on sequential proposals as well. Each issue is 
negotiated through a series of sessions and the outcome on an issue is determined 
once it is supported by a majority. We commence in Section 2, with a formal 
description of committees. In Section 3 introduces the benchmark bargaining 
model, which is based on simultaneous proposals on all issues. In Section 4 we 
review the main result in Winter (1993) regarding issue-by-issue bargaining; 
roughly, this result asserts that the efficiency and stability of the bargaining 
outcome can be guaranteed in agendas where important issues are discussed first. 
Section 5 presents a model in which the agenda is negotiated by the members of the 
committee prior to discussing issues of substance. We show that such a procedure 
fails to guarantee the efficiency of the bargaining outcome. Section 6 concludes 
with a model of endogenous agenda; here, any member can choose an issue to 
propose as long as this issue is not yet settled. We show that such a model can 
guarantee the efficiency and stability of the bargaining outcome, but only for 
committees that deliberate no more than two issues and when each issue involves 
yes-no voting. 



2. Committees 

A committee must reach a decision on k issues {l,2,...,k}. An issue is a finite set 
of alternatives Aj. The final process of the bargaining should induce a Platform a 
= (ai,a 2 ,...,ak), which is an element of the product set A = HAj. Thus, on each 
issue one alternative has to be chosen. The committee consists of a set N of n- 
members. Each member i has an ordinal strict preference relation on A, which we 
denote by Members' preferences on A are not necessarily separable -there 
may be interaction between the issues. The distribution of power within the 
committee is represented by a collection of winning coalitions W. Each coalition in 
W has the power to impose any decision it favors. For example, if the decision is 
taken by means of simple majority voting then W consists of all coalitions with at 
least n/2 members. But the structure of winning coalitions may also allow for 
decision making that requires unanimity, in which case W consists only of the 
grand coalition. The standard property which we assume on W is that there are no 
two disjoint winning coalitions. To summarize, a committee is a represented by a 
4-tuple (N,A,(^)ieN,W). 

The notion of stability that interests us here is based on the concept of 
core. Given a committee C = (N,A,()^ )ieN,W), a platform a is said to be stable if 
it cannot be blocked by any winning coalition, i.e., there exists no S in W and a 




219 



platform b such that all members in S prefer b to a (b>^ i a for all ieN). Note that 
the notion of stability automatically guarantees Pareto efficiency. 

We describe the process of collective decision making to determine a 
platform as a non-cooperative bargaining game. Our aim is to consider several 
bargaining models and determine whether equilibrium behavior in these models 
necessarily leads to an efficient/stable bargaining outcome. We start with a model 
in which all issues are negotiated simultaneously. 



3. The Simultaneous Bargaining Model 

When bargaining commences a chair i in N is determined exogenously. Bargaining 
then takes place in sessions. At the beginning of each session the chair submits the 
first proposal. A proposal is a pair (S,a) where S is a winning coalition containing 
i, and a is a platform in A (i.e., a proposal that specifies an outcome on each 
issue). Once (S,a) is proposed, each member in S either accepts the proposal or 
rejects it. Responses are carried out sequentially according to some exogenously 
specified order^. If all the members accept (S,a), the bargaining terminates with the 
outcome a. If some member j in S rejects the proposal he has to propose an 
alternative proposal, say (T,b), which has to be tabled again. This process 
continues until each committee member has had the opportunity to propose at least 
once"^. When all members have proposed at least once, the chair closes the session 
and a new session opens with an initial proposal by the chair. The whole process 
continues imtil a platform is chosen. 

The procedure described above gives rise to an extensive-form game with 
perfect information and an infinite horizon. The solution concept we use for 
analyzing this game as well as the subsequent ones is subgame perfect equilibrium. 
As we have said before we would like to focus on the occurrence of inefficiencies 
due to the multiplicity of issues. To abstract away all other sources of inefficiencies 
we would confine ourselves to stationary equilibria. In stationary equilibria players’ 
actions can depend only partly dependent on the history of the game. Specifically, 
we will be interested in equilibria satisfying the following properties: 

1. In responding to a proposal, the player may condition his action only on the 
proposed platform, i.e., each player has some cut-off point in his ordering, above 
which he accepts any proposal. A platform bellow this cut-off point will always be 
rejected. 



^ The results of the paper hold for every choice of such exogenous order. 

^ Note that each session can potentially be infinite. We however assume that reaching an 
agreement is the basic common interest (i.e., no agreement is inferior to any other outcome) 
so that indefinite disagreement cannot be sustainable in equilibrium. 




220 



2. In proposing, a player can condition his action on events that occurred within the 
current session, but not on events in previous sessions, i.e., 

Conditions 1 and 2 should not be taken as behavioral assumptions 
regarding players' behavior. They are imposed in order avoid the well-known 
phenomenon of the "Folk Theorem" in multilateral bargaining, by which non- 
stationary equilibria sustain almost every possible outcome (see, Chatterjee et al 
(1993), and van Damme et al (1990), where this phenomenon occur even in 2- 
person bargaining). 

We now show that when all issues are negotiated simultaneously, strategic 
behavior leads to an efficient platform. Furthermore: 



Proposition 1: For every committee C = (N,A,(>^ )ieN,W) a platform a is stable if 
and only if a is sustainable by a stationary subgame perfect equilibrium of the 
simultaneous bargaining game. 

The proof of Proposition 1 is a direct consequence of Winter (1993), 
Proposition 5.1. The simultaneous consideration of all issues in effect reduces the 
multi-issue problem to a single issue. On a single issue, inefficiency or instability 
cannot be sustained because a blocking coalition would necessarily negotiate a 
dominating outcome, which (with a backward induction argument) will be 
accepted. This argument is made formal in Winter (1993). In fact this result (as 
well as the subsequent results) is in some respect robust to the bargaining 
procedure. In particular, the Proposition holds also in a model without sessions. 
Namely, a model in which a rejection always triggers a new proposal by the 
rejector. However, such a model will require a stronger notion of stationarity for 
the results. 

In spite of the advantage, in term of efficiency, in negotiating several 
issues simultaneously, committees are often reluctant to use this type of bargaining 
procedure. Typical arguments against such a regime include the fact that decision 
making is much more complex when all issues are considered simultaneously, and 
that it allows for more "log rolling", i.e., trade of interest (see Riker, 1969). 
Thus, most committees prefer to negotiate one issue at a time. We now examine 
the strategic behavior in this alternative procedure, focusing on the possibility of 
guaranteeing efficiency and stability as a result of equilibrium behavior. 



4. Issue-by-Issue Bargaining 

In this alternative model issues are negotiated in sequence. At the beginning of the 
bargaining an exogenous A genda , i.e., an order of issues, is fixed. The bargaining 
starts by discussing the first issue on the agenda using the same procedure as in 
Section 3, i.e., a proposal on Ai consists of a pair (S,aj) where aj is in Ai and S is 




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in W. If a proposal is rejected, the bargaining continues session after session, 
according to the same rules^ as in Section 3, on Aj alone. If (S,ai) is accepted, then 
ai is determined to be the outcome on the first issue and the bargaining proceeds to 
the second issue in the same manner. Bargaining terminates after all issues are 
discussed and a platform a = (ai,a 2 ,...,ak) is determined. Note that when 
discussing a certain issue the players can keep track of the decisions made on 
previous issues, and are allowed to make their decisions dependent on this 
information. 

Does the issue-by-issue game guarantee efficiency as does simultaneous 
bargaining? Unfortunately not. In Winter (1993) it is shown that the fact that a 
decision on one issue can be made contingent on previous decisions on other issues 
may yield serious inefficiencies as a result of stationary^ subgame perfect 
equilibrium play. However the situation is different if individual preferences 
represent a consensus over the importance of issues. Many committee situations are 
characterized by the fact that all parties share the same view regarding the 
importance of issues even if they disagree on the preferred outcome on each issue. 
In times of war, for example, parliaments would often agree that issues of national 
security are more important than economic issues. We now turn to such 
committees to extend the Winter (1993) results on the efficiency of bargaining 
outcomes in such committees. But first, some notations which will enable us to 
define the notion of "order of importance". 

Let A be a set of platforms and a = (aj,...,ak) in A. Suppose that bj e Aj 
is an alternative on issue j. We denote by a|bj the platform obtained from a by 
replacing aj with bj. Furthermore, we write a | bj,bm = (a | bj) | b^. 

Let i be a committee member with a preference relation j on A. We say 
that issue j is more important for i than issue m (with respect to>- j) if the following 
holds: 

For each platform a, if a | bj,bm>^ i a I c^ for some bjG Aj and c^ in 
Am, then a|bj,b'm>^i a|c’m for every b'm,c’m in A^. That is, the ranking of any 
two platforms that differ only on issues j and m should be independent of the 
choice of issue m. 



^ Specifically; If a player in S rejects (S,ai) then he submits a proposal himself on Aj. The 
session continues until every one had a chance to propose at least once in the current session. 
Then a new session starts again on Aj. 

^ Stationarity in this context means applying properties 1 and 2 of Section 3 on each issue 
separately, i.e., when discussing the jth issue, if at some stage aj was rejected by player i, 
then i will reject aj also in subsequent responses to a^. Recall however, that players are 
allowed to make their action on one issue dependent on the decisions taken on former issues. 




222 



We say that the preference profile (>^i)ieN represents consensus over the 
importance of issues if the order of importance induced by the individual 
preferences is complete, transitive, and identical for all players^. 

In Winter (1993) it is shown that even with imder consensus over the 
importance of issues, stationary subgame perfect equilibria may yield inefficient 
outcomes. However, if issues are discussed in the order of their importance, then 
no inefficiency or instability can result from an equilibrium play. 

Proposition 2 (Winter 1993): Let C = (N,A,()^ )ieN,W) be a committee in which 
(^)ieN represents a consensus over the importance of issues. Consider the issue- 
by-issue bargaining game with an agenda that is consistent with the order of 
importance of issues, i.e., the most important issue is discussed first, the next most 
important second, and so on. Then, a platform a is sustainable by a subgame 
perfect equilibrium outcome if and only if it is stable. 

The intuition behind Proposition 2 and example 1 is quite clear. Suppose 
the bargaining starts with a less important issue. Since the committee members can 
make their decisions on one issue dependent on previous issues, they will be 
"overly concerned" with the effect this decision on the more important issue. This 
may prevent them from making the "right" decision on less important issues, using 
them as signals or threats. If, however, the bargaining is carried out by order of 
importance of issues, then when discussing a certain issue it is known that 
subsequently only less important issues will be considered, and everybody can act 
as if this issue is the only issue on the agenda. 

We have seen that by discussing issues by order of importance one can 
guarantee the stability and the efficiency of the equilibrium outcome. We now 
examine two additional mechanisms that allow for more flexibility in the choice of 
the agenda and check whether they can endogenize the positive effect on efficiency 
seen in Proposition 2. 



5. Bargaining Over the Agenda 

The bargaining model discussed in Section 4 is based on the exogenous 
determination of the agenda. Obviously, since the agenda has a dramatic effect on 
the bargaining outcome, different members may prefer different agendas so that the 
agenda itself becomes an issue. We now consider a model in which committee 
members start by bargaining over the agenda before discussing the issues of 
substance. Let G be the set of all k! agendas; at the beginning of the bargaining G 



^ The requirement that individual preferences induce an order of importance is weaker 
than lexicographic preferences. In lexicographic preferences if two platforms a and b 
coincide on the first j issues, then their ranking depend on aj+, and bj+i only. In our 
preferences this is not necessarily the case. 




223 



is considered as an issue and the agenda is negotiated according to the same 
procedure as in Section 4, i.e., the chair starts with a proposal (S,g), where S is a 
winning coalition and g is an agenda. If the proposal is accepted by all the 
members in S, then g is fixed as the agenda. If some i in S rejects the proposal, he 
comes up with an alternative proposal. This continues in the same manner until 
each player has had a chance to propose an agenda at least once, after which a new 
session is opened by the chair with a new proposed agenda. Once the agenda is 
determined, bargaining continues according to the chosen agenda by means of the 
issue-by-issue bargaining game discussed in Section 4. 

Does this process of negotiation over the agenda necessarily imply that the 
more important issues will be discussed first? Is it at feast the case that this 
mechanism guarantees a stable platform as a result of equilibrium behavior? 
Unfortunately, 

Proposition 3: In the negotiated-agenda bargaining game every stable platform is 
sustainable by some SSPE, but there may be non-stable bargaining outcomes that 
are sustainable by an SSPE. 

Proof: We start with the second assertion. Consider the following committee with 
two issues, two alternatives on each issue, and 4-person simple majority voting 
(three votes win): 

Ai = {ai,a 2 }, A 2 = {bi,b 2 }. The preferences are: 

1 : (ai ,bi) ^ (ai ,b2)* ^ (a2,b2>* ^ (a2,bj) 

2: (a2,b2) (a2,bj) (ai,b2) >-- (ai,bi) 

3: (U2,bi) (U2,b2) ^ (ni,b2) ^ (ai,bi) 

4: (ai,b2)>- (ai,bi))^ (a2,b2))- (a2,bi). 



Note that the outcomes (aj,b 2 ) and (a 2 ,b 2 ) are in the core; the other two 
are non-core outcomes. Note also that all players agree that the first issue is more 
important than the second. There are two possible agendas, one in which the most 
important issue is discussed first (denoted by [1]) and the other in which the second 
issue is discussed first (denoted by [2]). Consider the subgame starting after agenda 
[1] was chosen (the most important issue is discussed first). Since (a 2 ,b 2 ) is a core 
platform of C, we know by proposition 2 that there exists some SSPE on this 
subgame that yields the platform (a 2 ,b 2 ). Let Sj denote this strategy combination, 
and let S 2 denote the SSPE that yields (ai,bi) on the subgame starting after the 
agenda [2] was chosen (as in example 1). We now specify the SSPE in the 
negotiated-agenda game that yields the non-stable platform (ai,bi). 

In the first phase of the bargaining: 

1: Proposes ([2], {1,3, 4}) and accepts only the agenda [2]. 

2: Proposes ([2], {1,2,4}) and accepts every agenda. 




224 



3: Proposes ([2], {1,3,4}) and accepts every agenda. 

4: Proposes ([2], (1,3,4)) and accepts only the agenda [2]. 

After the agenda is chosen: 

If the chosen agenda is [1], then all play Sj. 

If the chosen agenda is [2], then all play S 2 . 

To verify that this is indeed an SSPE note first that this strategy 
combination specifies an SSPE on each subgame that occurs after the agenda was 
determined. It is, therefore, enough to check that the behavior prior to the choice 
of agenda, as specified above, is part of an SSPE. This follows from the fact that 
given Si and S 2 players 1 and 4 prefer agenda [2], while players 2 and 3 prefer 
agenda [1]. 

The first assertion follows from the fact that for any agenda and for any 
stable platform a is supported by some SSPE [see the proof of Proposition 5.1 in 
Winter (1993)]. This means that any such platform is also supportable by an SSPE 
in the negotiated-agenda game: simply take any strategy combination that yields 
some SSPE supporting platform a on every subgame starting after the agenda was 
chosen. 

We have seen that allowing players to bargain over the agenda is not 
sufficient to induce stable outcomes as a result of equilibrium behavior. We now 
move to an alternative model, in which the agenda is not fixed at any stage of the 
bargaining but is determined endogenously in the course of the negotiation. We will 
show that for two-issue committees with yes-no voting such model yields positive 
results. 



6. Open- Agenda Bargaining 

In the open-agenda bargaining game proposers choose the issues on which they 
wish to propose a decision. 

Bargaining starts with a proposal by the chair. A proposal 0»^j»S) now 
consists of an issue j, which has not yet been settled, an alternative to this issue, 
and a coalition S in W. If everybody accepts the proposal the jth issue is settled 
with the outcome aj, and the chair submits a new proposal (i,aj,T), If the 
proposal is rejected by some player m, then m has to submit a new proposal. At 
every stage the proposed issue must be one that has not yet been settled. Bargaining 
terminates when all issues are settled and a platform is obtained. 

We now show that when the committee engages in "simple" decision 
making, i.e., when there are only two issues, each with two alternatives, then the 
open-agenda model yields the same effect as the "important issue first" model. 
Note that the domain of platforms with two alternatives for each issue is an 
important domain because it involves all issues on which yes-no decisions made. 




225 



Proposition 4 : Consider a committee C = (N,A,(^ )ieN»W), where A consists of 
two issues, each with two alternatives, and there is a consensus over the 
importance of issues. In the open-agenda bargaining game, platform a is 
supportable by an SSPE if and only if it is stable. 

Proof: We again start by showing that all SSPE outcomes are stable. Without loss 
of generality assume that the first issue is more important and that (ai,b2> is an 
equilibrium platform. Since the first issue is more important, we can reduce the 
preferences on platforms to preferences on this issue (because they are independent 
of the choice made on the other issue). This means that there is no S in W such that 
all players in S prefer a2 to ai. Otherwise consider the coalition T, which is 
proposed before ai is accepted, and consider i in TnS. i can guarantee a2 by 
proposing (S,a2), which must be accepted by the members of S (because of the 
stationarity). Now suppose, by way of contradiction that (ai,b2) is not in the core. 
Then, given the arguments above, there must be some SeW such that all members 
of S prefer (ai,bi) to (ai,b2). Now let T be the coalition proposed at the beginning 
of the game and take again i in SnT. Consider a proposal of (S,ai) by i. If this 
proposal is accepted, bj will obviously be chosen on the second issue. Otherwise, 
one of the responders to b2 in S should reject it and propose bj, which must be 
accepted. Since all players expect bi to be chosen on A2, all members in S must 
accept ai on Aj. Thus by proposing ai before A2 is discussed player i guarantees 
the platform (ai,bi) which he prefers to the equilibrium outcome. This contradicts 
the fact that (ai,b2) is an equilibrium platform. 

To show the converse, assume without loss of generality that (ai,bi) is 
stable. Consider the following SSPE: Each player proposes (N,ai) as long as Aj is 
not yet settled. Each player rejects (S,aj) if and only if he or some other player who 
has not yet responded to this proposal prefers ai to aj. Once Aj is settled, the 
remaining subgame is identical to single-issue bargaining. Thus we can specify 
here an SSPE in which bj is played after aj was chosen and bj is played after a2 
was chosen, where bj is such that there exists no SeW with 
(ai^bjmodx)^ k (a2,bj) for k in S [see, for example. Winter ( 1993 ) Theorem 5 . 1 ]. 

It can be shown that open-agenda cannot guarantee stability when the 
committee has to discuss more than two issues. 

We use the simplest example to demonstrate this statement, which is 2 - 
person, 3 -issue, 2 alternatives each. 

Example 1 : 

N = { 1 , 2 }, W = (N), A = A1XA2XA3, Aj = {^ 1 ,^ 2 }* ^2 ~ A3 = 

Preferences: 

1 ; (a, ,b, ,C[) >- (a, ,b, , 02 ) >- (a, .bz.Cj) 

(a2,b2,c,)>- (a2,b2,C2);^ (a2,bi,C2)>^ (a 2 .b 1 .C 1 ) 

2: (a2,bi ,C2) >- (a2,bi ,Ci) (a2,b2,c,) (a 2 .b 2 .C 2 ) 

(ai .b2.C2) >- (a, .b2.Ci) (ai .bi .Cl) (ai ,bi .C2) 




226 



Note that with respect to this preference profile both members agree that 
Ai is the most important issue, A2 is the second most important, and A3 is the least 
important. Note also that (a2,bi,Ci) is inefficient since both prefer (a2,bi,C2). 
Nevertheless, this platform is an SSPE outcome in the open-agenda game. We will 
not specify the complete strategy combination here, but briefly describe the way 
one can construct such an equilibrium. Along the equilibrium path, the players start 
with proposing and accepting first Cj then bj and then a2. Now, if the second issue 
is settled before Cj, the players play aj on the first issue. Similarly, if the first issue 
is settled with a2 before the third, the players play a2 on the second issue afterward. 

A similar example can be constructed for the case where only two issues 
are discussed, buf there are more than two alternatives on each issue. 



7. Conclusions 

The fact that the agenda of multi-issue negotiations has a direct influence on the 
equilibrium outcome begs the question how one should construct the agenda. 
Proposition 2 suggests that given consensus over the importance of issues, 
important issues should be discussed first in order to guarantee stability and 
efficiency. Propositions 3 and 4 explore the possibility of attaining efficiency 
through endogenizing the agenda. We have seen that if the agenda is treated as an 
issue negotiable by members of the committee, then a "wrong” agenda can still be 
determined in equilibrium, i.e., some equilibria may yield inefficient outcomes. 
However, if no agenda is fixed and each player can choose the issue on which he 
wants to propose, then if the underlying problem is simple enough (i.e., two issues, 
yes-no voting) efficiency and stability can be guaranteed. With more complex 
platform spaces there would be much more room for contingent behavior. This has 
a negative effect on the efficiency of the equilibrium outcomes. Our results suggest 
that under such circumstances, guaranteeing an efficient outcome requires that the 
agenda be imposed exogenously. 



8. References 

Chatterjee K., B. Dutta, D. Ray, K. Sengupta, (1993) "A Noncooperative Theory of 
Coalitional Bargaining", The Review of Economic Studies, 60, 463-477. 
van Damme E., R. Selten and E. Winter, (1990) "Alternating Bargaining with Smallest 
Money Unit" Games and Economic Behavior 2, 188-201. 

Harsanyi, J. C. (1974): "An Equilibrium-Point Interpretation of Stable Sets and a Proposed 
Alternative Definition," Management Science, 20, 1422-1495. 

Riker, W.H. (1969) "Democracy in the United States"' The Macmillan Company, NY. 




227 



Selten, R. (1981) "A Non-Cooperative Model of Characteristic Function Bargaining 

in Game Theory and Mathematical Economics in Honor of Oscar Morgenstern, Eds. V. 
BohmandH.H. Nachtkamp, 131-151. 

Selten, R. (1992), "A Demand Commitment Model of Coalition Bargaining,” Rational 
Interaction, Essays in Honor of John C. Harsanyi Editor: Reinhard Selten Springer- 
Verlag, Berlin. 

Winter, E. (1993) "Bargaining in Committees" Discussion paper #22, The Center for 
Rationality and Interactive Decision Theory, The Hebrew University of Jerusalem. 




The Organization of Social Cooperation: A 
Noncooperative Approach 



Akira Okada 

Institute of Economic Research, Kyoto University, Sakyo, Kyoto 606-01, Japan 



Abstract. A noncooperative game model of organizations for social cooperation 
is presented based on Selten’s (1973) model of cartel formation. The possibility of 
organization with enforcement costs is studied in an n-person prisoners’ dilenrima 
situation. 

Key words, collective action, cooperation, organization, prisoner’s dilemma 



1. Introduction 



Selten (1973) presents a noncooperative game model of cartel formation among 
Cournot oligopolistic firms. In the model, the cartel imposes quotas on its 
members, if such an agreement is reached. Selten shows that five is the critical 
number of firms for voluntary formation of a cartel. If the number of firms is less 
than five, then a cartel is formed. Otherwise the probability of the cartel is very 
small. The purpose of this essay is to show that Selten’s pioneering model can be 
applied to the general problem of organizations for social cooperation. 

In recent literature on game theoretic analysis of cooperation, two main 
approaches have dominated: repeated game approach and evolutionary game 
approach. The central result, termed the Folk Theorem, in the repeated game shows 
that cooperation can be attained among self-interested individuals if a game 
situation is repeated without any limit, and if players do not heavily discount 
future payoffs. The mechanism of attaining cooperation is mutual punishment 
among individuals, for defection from cooperation. See Aumann and Shapley 
(1976) and Taylor (1987) for the repeated game approach. 

In evolutionary game theory, it is assumed that a game is played within a 
population of players from generation to generation. The distribution of strategies 
over the population is emphasized. Tte higher the reproductive success one 
strategy attains, the more likely that strategy is inherited by the next generation. 
One of the main interests is what kind of strategy can survive in such a selection 
process, similar to the Darwinian process of natural selection. In his well-known 




229 



computer tournaments of an iterated prisoners’ dilemma game, Axelrod (1984) 
shows us that the TIT FOR TAT strategy (cooperate first, and imitate what the 
opponent did) is very successful in a certain environment. For surveys on 
evolutionary game theory, see Maynard Smith (1982) and Hammerstein and Selten 
(1994). 

Although the two approaches have contributed much to our und^standing of 
social cooperation, some aspects seem not to be fully understood. Both approaches 
assume players to be “passive” to the social environment they face in the sense 
that they regard the rules of a game as given. In our view, however, one of the 
important activities of the players as “social beings” is to change and reconstruct 
the rules of their game, if necessary. The purpose of our research beginning in 
Okada and Kliemt (1991) is to develop a game theoretic framework for considering 
the problem of social cooperation from this aspect of institutional arrangements. 
In particular, we consider the situation where individuals in a society voluntarily 
form an organization for collective action of cooperation. 

We define an organization as a social relationship created by its members for 
their common benefit. A primary function of the organization is enforcement of its 
objectives by punishing the opportunistic behavior of members. We explicitly 
formulate an agency to perform enforcement, and introduce costs associated with 
establishing it. In our model, the possibility of organization dqpends upon such 
enforcement costs in the organization and free-riding behavior of nonmembers. 

This essay is organized as follows. Section 2 describes a society as an n-person 
prisoners’ dilemma and presents a noncooperative game model of social 
organizations. Some implications of the model are given about the formation of 
organizations. Section 3 extends the model to the case of heterogeneous players. 
The last section provides two applications of the model: environmental pollution 
and emergence of the state, and critically discusses our model with regard to future 
research. 



2. A Game Model of Social Organizations 



Our society is described as an n-person prisoners’ dilemma. Let N = { l,...,/i) be 
the set of players. Every player has two actions: C (cooperation) and D (defection). 
Player /’s payoff is given by 



fl(ai, h) ai - C,D and A = 0, ...» w-l, 

where a/ is player I’s action and h is the number of other players who select C. In 
this section, we assume that all players have identical payoff functions. Player 
index / is omitted in notations when no ambiguity arises. This assumption of 
homogeneous players will be relaxed in Section 3. 




230 



The payoff function of every player has the following properties: 



Assumption 2.1. h) >f(C, h) for all A = 0, .... n-1, {2)f(C, n-l) > 

f(D, 0), (3)/fC, h) and f(D, h) are increasing in h. 



The implications of the assumption are as follows. Property (1) means that action 
D dominates action C for every player, i.e., each is better off by taking D than C 
regardless of what all the other players select. Thus the action combination 
(D,.../)) is a unique Nash equilibrium point of the game. The equilibrium payoff 
f(D, 0) is called the noncooperative payoff in the prisoners’ dilemma. On the otha- 
hand, property (2) means that if all players jointly select C, they are all better off 
than the Nash equilibrium point. Cooperation is beneficial in the society. Finally, 
property (3) means that the more other players cooperate, the better every player’s 
life is, regardless of that player’s action. In other words, cooperation by one player 
gives positive externality to other players’ welfare. An example of the payoff 
function is illustrated in Figure 2.1. It is easily seen from Figure 2.1 that there 
exists a unique integer Sq (2<Sq< n) such that 



/(C, So-2) < f(D, 0) < f(C, 5o-l). (2. 1) 

The integer Sq shows the minimum number of cooperators that guarantees each of 
them a payoff higher than or equal to the noncooperative payoff. 

Society as the prisoners’ dilemma is in an anarchic state of nature in which there 
exists no social order to enforce collective actions upon self-interested players. 
Therefore, the natural outcome of that society is the Nash equilibrium point where 
no players cooperate. Scholars in the theory of Social Contract have presumed 
that players in society enter, with their free will, into a contract to organize a 
social institution for enforcing cooperation. We investigate the possibility of 
voluntary formation of such a social institution, by using game theoretical 
models. In this paper, we call a social institution created by players for cooperation 
an organization. 

An organization in society is formulated by a quadruple = (S, p, i*. C). A 
subset S of N is the set of all members composing the organization. We often 
refea- to the cardinality of the set 5 as the size of the organization. The secaid 
element p, a nonnegative real number, is a punishment imposed on any member in 
case of defection. When punished, the payoff of a member is reduced by the 
amount p. The third element i* is the special agent, called the erf or cement agent, 
that does enforcement work for the organization. The final element C is a function 
assigning to each size s of the organization the costs C(j) of enforcing cooperation 
on its members. In general, there are many kinds of enforcement cost. One of the 




231 



major enforcement costs is the payment, denoted by w, to the enforcement agent 
by the members of the organization/ Others include costs of monitoring members’ 
actions and of punishing deviators. Enforcement costs minus the payment to the 
enforcement agent is denoted by M{s). Then, the enforcement cost CC^) can be 
represented by C{s) = iv + M(s). The existence of enforcement costs plays an 
important role in our study of the formation of the organization. 

The constitutional rules of an organization are regarded as rules for defining the 
four components above. For instance, there are at least two rules, free and 
compulsory^ for defining its members. Undo- the free-participation rule, evoy 
player in the society is at a liberty to decide whether to participate in the 
organization. In contrast, under the compulsory rule, all players must participate in 
the organization. The punishment level may be determined by a certain mechanism 
of collective decision (voting and negotiations, etc.) within the organization. Th^ 
are two types of enforcement agent of the organization: external and internal. In the 
case of the external agent, an agent outside the society is employed for enforcement 
work in the organization.^ The payment to the external agent may be determined in 
a labor market outside the society. For example, if the labor market is 
competitive, the payment to the enforcement agent will be detemined at the 
opportunity cost of enforcement labor. In the case of the internal agent, one 
member (or members) of the organization is selected as the enforcement agent by 
political, social or cultural mechanisms. Rules for defining the payment to the 
internal agent tend to be more complicated than for the external agent. This is due 
to the emCTgence of a distributional conflict between the enforcement agent 
(enforcer) and other members (enforcees) of the organization. In general, as with 
the punishment, the payment to the internal enforcement agent may be determined 
by a collective decision mechanism in the organization. Exploitation by either side 
(enforcar or enforcees) is an extreme case. Finally, the constitutional rules of the 
organization should include a rule for distributing the enforcement cost among 
members. 

The research agenda in the formation of organizations is divided into two levels: 

(1) the constitutional choice problem: how is a constitutional rule of the 
organization selected?, 

(2) the organization formation problem: under a given constitutional rule, is the 
organization formed? If it is, what components does it have? 

Since an answer to the first problem naturally dq)ends upon that to the second 
problem, it is reasonable to consider first the organization formation problem. In 
the rest of the section, we consider the constitutional rule of free participation and 



^ We assume that players have transferable utility. This assumption can be relaxed in a 
more elaborate model. 

^ An example of the external agent is where a country employs foreign workers for 
various kinds of public service. 



232 



the external enforcement agent as a benchmark of our analysis, and assume that the 
enforcement cost is exogenously given. Our analysis focuses on what kind of 
organization can be created undo* this constitutional rule. We will discuss the 
constitutional choice problem in the last section. 

Our basic model of organization formation is described below as a multistage 
game. 

(1) The participation decision stage 

All players / (= l,...,n) in the society decide indqjendently and simultaneously 
whether to participate in the organization. Let 5 be the set of all participants who 
become the members of the organization. 

(2) The bargaining stage for punishment and cost allocation 

Enforcement cost C(^) of the external agent is exogenously given. The 
punishment p and the allocation of enforcement cost C(5) aie determined through 
negotiations among all members. If no agreement is reached, the organization is 
not formed. 

(3) The action decision stage 

Every player in the society selects an action, C or £>, with or without the 
organization (p = (S, p, /*, C), dq)ending upon the bargaining outcome in stage 
(2). The punishment is imposed only on defecting members of the organization. 

In each stage, players know perfectly the results of previous stages when they 
make decisions. Our primary solution concept in analyzing the game is that of a 
subgame perfect equilibrium point. 

To simplify analysis, we assume that punishment p has only two possible 
values: 0 and p*? The punishment level p* is sufficiently high to the extent that 
cooperation gives a higher payoff to every member of the organization than 
defection. We assume that the punishment level is determined by unanimity 
among the members. We also assume that the total enforcement cost C(5) is 
equally allocated among all members. By these assumptions, the bargaining stage 
of the model is reduced to the following game. Every member in S selects 
independently either 0 or p* as the punishment level. The payoff of each member 
is given by 



F(C, (2.2) 

if all members select p*, wAf{D, 0), otherwise. To derive these payoffs, we use 
the fact that in the final stage of action decision every nonmember of the 
organization always selects defection and that every member selects cooperation if 



^ See Okada (1993) for a detailed analysis which allows a continuum range of 
punishment levels. 



233 



and only if the punishment level is p*. We call the payoff F(C, s-l) in (2.2) the 
cooperative payoff of the organization. 

We assume that the condition of the prisoners’ dilemma (Assumption 2.1) holds 
true if one rqplaces /(C, h) with F(C, h) as the payoff function in case of 
cooperation. 



Assumption 2,2. The cooperative payoff F(C, .y-1) of the organization is 
increasing in size s, and F(C, n-l) >f(P, 0). 



Figure 2.1 illustrates the properties of the payoff functions /(C, A), f(P, h) and 
the enforcement cost function C{s) satisfying Assumptions 2.1 and 2.2. Similarly 
to (2.1), we can see that there exists a unique integer (2 < ^* < n) such that 

F(C, - 2) < /(D, 0) < F(C, - 1)/ (2.3) 

The integer s* shows the size of the organization such that (i) cooperation in the 
organization is more beneficial to each member than in the noncooperative 
equilibrium point in the prisoners’ dilemma, and (ii) this property never holds if 
one member deviates from the organization. We call the integer s* the minimum 
size of the organization. 

We first analyze the Nash equilibrium point of the reduced bargaining stage with 
the payoffs given in (2.2). Since the effective punishment level p* is detOTnined 
by unanimity, there are many “trivial” Nash equilibrium points leading to 
disagreement (or zero punishment level) regardless of the organization size. For 
example, all situations where two groups each consisting of more than one 
member make different choices, are Nash equilibrium points leading to 
disagreement We want to exclude these trivial equilibrium points peculiar to the 
unanimous voting rule. A simple way to do this is to employ a strict equilibrium 
point (a Nash equilibrium point in which every player is worse off in case of 
unilateral deviation from the equilibrium) as the (local) solution in this stage.^ In 
what follows, we mean a strict equilibrium point whenever we refer to an 
equilibrium point in pure strategy. We are mainly concOTied with when agreement 
on effective punishment can be supported in equilibrium. The following theorem 
is easily proved. 



^ It may be possible that equality F(C, 5*-l) 0) holds in (2.3). In this case, all 

members of the organization with size s* are indifferent about establishing the 
organization or not, but their decision matters much to nonmembers. In order to avoid 
the problem of which indifferent action is selected, we simply assume as a regularity 
condition that the strict inequality holds in (2.3). 

^ In general, we can employ the “trembling -hand” perfect equilibrium point of Selten 
(1975). 




234 



payoff 




Figure 2. 1. An n-person prisoners' dilemma 



Theorem 2.1. Agreement on effective punishment p* is reached in a unique 
equilibrium point of the reduced bargaining stage if and only if the organization 
size is greater than or equal to the minimum size s*. 

The theorem implies that the success of an organization dqiends upon a simple 
criterion, size, in our framework. The minimum size s* of the organization is the 
critical size for its success. If the size of the organization is smaller than s*, then 
it fails to achieve cooperation among members. 

In view of Theorem 2.1, the participation stage is reduced to the following 
game. Every player i (= !,...,«) has two actions: participate (1) or not (0). For an 
action combination b = (bj,...,b„), let s(b) be the number of all participants. Then, 
every player i obtains the payoff Efb) below. 




235 



Case (i) 0 < s(b) < s*-l: Effy) 0), 

Case (ii) s* < s(b) : 

E ( ^(C.s(b)-l) if bi = l 
I ms(b)) if bi^O. 

The payoff structure of the participation stage is summarized from the point of 
view of each player as follows. There are three possible outcomes: (i) an 
organization is formed without the player’s participation, (ii) an organization is 
formed with participation, (iii) no organization is formed. The following theorem 
characterizes pure equilibrium points of the participation stage. 



Theorem 2.2. An action combination b in the participation stage 

is a (strict) equilibrium point if and only if exactly s* players participate in the 
organization. 



Proof. The if part is proved from inequalities /(D, 0) < F(C, ^*-1) and F(C, 

< f(P^ s*). Any action combination with s (> s*) participants is not a Nash 
equilibrium point because every member is better off under unilateral deviation 
from the organization. If s < s*, then the action combination is not a strict 
equilibrium point.^ Q.E.D. 

The theorem first implies that if s* = /i, the full organization can be attained in 
an equilibrium point of the participation stage. In this case, moreover, we can see 
that from the point of view of every player the action of participation weakly 
dominates that of nonparticipation. Therefore, there is no Nash equilibrium point 
in mixed strategy. We can derive from these results the conclusion that all players 
in the society participate in the organization for cooperation in this case. Second, 
the theorem implies that in general cases only a minimal organization with size s* 
can be supported in a pure equilibrium point due to every player’s incentive to free- 
ride the organization. Then, a further question arises: “Who participates in the 
minimal organization?” Various kinds of formal and informal institutions such as 
moral attitudes, seniority-rule, leadership and sequential moving may be needed to 
answer the question. In an institution-free society, a symmetric equilibrium point 
seems to be more appropriate as our solution than an asymmetric one.^ A trivial 
symmetric (non-strict) equilibrium point is the situation that no players 
participate. The next theorem shows the other symmetric equilibrium point 



® If 5 = .y*— 1, it is not a Nash equilibrium point, and if y < y*-l, it is a Nash equilibrium 
point, but not a strict one. See Okada (1993). 

^ Roughly speaking, a symmetric equilibrium point is a Nash equilibrium point in 
which identical players employ identical strategies. 



236 



Theorem 2.3. There exists a unique symmetric equilibrium point in mixed 
strategy in the participation stage. Participation probability t of every player is the 
solution of the equation 



( I f(P,k)-F(C,k) 



(2.4) 



Proof. For a mixed strategy combination q = let Efq) be the expected 

payoff of player i. A symmetric mixed strategy combination with participation 
probability t (> 0) is a Nash equilibrium point if and only if Ef^qll) = Ef^q/0) for 
every /, where q!l and qlO mean the strategy combinations obtained from q if only 
player I’s strategy qi is rq)laced with the pure strategy of participation and of 
nonparticipation, respectively. We can see that this condition is equivalently 
reduced to (2.4). The left-hand side of (2.4) is a continuous and increasing function 
of t over [0, 1), and moreover it takes zero if r = 0, and diverges to infinity as t 
goes to 1. Therefore, (2.4) has a unique solution. Q.E.D. 

The theorem implies that participation probability of every player depends upon 
three factors: population n, the minimum size s* of the organization and the ratio 
of payoff diffidences, {/(Z>, k) - F(C, k)]l[F{C, s*-l) - f{D, 0)} for each k = 
The last factor, called the incentive ratio, means the ratio of playd* 
incentive (payoff gains) to deviate from the organization of size k + \ ovct 
incentive to join the minimal organization. 

Since the pioneering work of Olson (1965), the effect of group size on the 
likelihood of group success has been one of the central questions in the theory of 
collective action. The next theorem gives one implication of our model to the 
group size problem in terms of player participation probability. For the proof, see 
Okada(1993). 



Theorem 2.4. Assume that the minimum size s* of the organization and the 
incentive ratios are constant with respect to population n. Then, participation 
probability t(n) decreases and converges to zero as n goes to infinity. 



When a society is so small that the population is equal to the minimum size 5* 
of the organization, all players join the organization for cooperation. If the 
population grows, the probability of every player participating in the organization 
decreases, becoming close to zero in a very large society. In this sense, social 
cooperation is more likely in a small society than in a large society. 




237 



3. Heterogeneous Players 



In the previous section, all players are assumed to be identical: they have the 
same payoff functions. We here consider how heterogeneity among players affects 
the formation of an organization. 

One simple way to introduce heterogeneity among players into our model is to 
assume that the minimum size s* of cooperation defined in (2.3) is not necessarily 
common to all players. Taking this idea, we assume that for every player / (= 1 , 
..., n) there exists a unique integer ^ • (2 < < n) such that 



F,.(C, ^ - 2) < 0) < F,.(C, 5,-1) (3.1) 

where the functions and correspond to player z’s payoff functions / and F, 
respectively, in the previous section. 

The game model of organization formation in the previous section can be easily 
extended to the heterogeneous case. Keeping the same assumptions about the 
enforcement costs and their allocation rule, our analysis of the last two stages, the 
bargaining stage for punishment and cost allocation and the action decision stage, 
can be applied with minor modifications. Therefore, it suffices to analyze only the 
participation decision stage. 

The participation decision stage of the organization game with heterogeneous 
players is described as follows. Each of n players has two actions: participate in 
the organization (1) or not (0). Player i (= l,...,n) is characterized by the minimum 
size S{ (2 < Sj < n) of cooperation defined by (3.1). For each subset S of N and 
each t = 2,...,n, let g^O) be the number of members of 5 whose minimum sizes of 
cooperation are t. The function gg(t) represents the distribution of members’ 
characteristics in organization S in terms of their minimum sizes of cooperation, 
and we call it the characteristic distribution of organization 5. 

Unanimous agreement on effective punishment is reached in the bargaining stage 
for an organization S if and only if its size satisfies Si < ISI for every member /. 
Such an organization is called succes^ul. From the fact above, we can daive the 
payoff of every player in the participation decision stage as follows. For an action 
combination b = (fey,...,6„), let S{b) denote the set of all participants. Then, the 
payoff E^b) of player i is given by: 

(i) if S(b) is successful, 

E (*)=( if ^ = 1 

\ fiiD,\S(b)\) i/6.=0, 



(ii) otherwise, £,(6) =f^D, 0). 




238 



Theorem 3.1. An action combination b = (bj,...,b^) in the participation 
decision stage with heterogeneous players is a (strict) equilibrium point if and only 
if the set S(b) of participants satires one of two properties below. 

(1) S(b) is a minimal successful organization, 

(2) S(b) satiates g^^l) + ... + g^dSD = jSj and g^lS/) >2. 



Proof. (1) Let b = (6y,. be a (strict) equilibrium point. Suppose, on the 
contrary, that property (1) does not hold. Consider first the case that S{b) is not 
successful. Then, every player Vs payoff is ffp, 0). If one participant deviates 
from 5(6), the payoff is either /;</>, 0) orffD, IS(6)I-1), dqjending upon whether 
the remaining organization is successful. Since ffD, \S(b)\-l) > ffD, 0), this 
contradicts that the action combination 6 is a strict equilibrium point. ConsidCT 
next the case that S(b) is a successful organization but not a minimal one. Then, 
there exists some member / of S(b) such that the organization remains successful 
despite individual deviation. In this case, player / obtains the payoff ffD, \S(b)\-l) 
higher than the payoff F,<C, \S(b)\-l). A contradiction. It can be easily proved that 
if property (1) holds, the action combination b is a strict equilibrium point. 

(2) We prove that two properties in the theorem are equivalent First, we can see 
without much difficulty that an organization S of is successful if and only if 
g^(2) + ... + gsOS\) = 151. Let 5 be a successful organization. Suppose that ^5(151) 
> 2. If any one member deviates from 5, the resulting organization is no longer 
successful. Thus, 5 is a minimal successful organization. Suppose that ^5(151) = 1. 
Then, if player i with si = 151 deviates from 5, the resulting organization is still 
successful. Finally, suppose that g^OSl) = 0. Then, the minimum size Si of 
cooperation for every member of 5 is less than or equal to 151 - 1. This implies 
that if any member deviates from 5, the resulting organization is still successful. 
Q.E.D. 

Theorem 3.1 shows that a (strict) equilibrium point of the participation decision 
stage necessarily leads to a minimal successful organization. The theorem also 
characterizes a minimal successful organization by using the characteristic 
distribution function of organizations. An organization is successful if and only if 
its size can meet the minimum size of cooperation for every member. The 
minimality of a successful organization is given by the very simple condition of 
g5(l5l) > 2. That is, the organization must have more than one member to whom 
its size is at the minimum level for beneficial cooperation. Figure 3.1 illustrates 
an example of the characteristic distribution of players and a minimal successful 
organization. In the figure, each block corresponds to one player and the shaded 
area shows a set of players forming a minimal successful organization. 




239 




Figure 3.1. The characteristic distribution of players 



The following corollary about the possibility of full cooperation is easily 
obtained from Theorem 3.1. 



Corollary. The largest organization N can be formed in an equilibrium point of 
the participation decision stage if and only ifgf/IN/) >2. 



The corollary says that if there are at least two players whose minimum sizes of 
cooperation are n, it is possible in equilibrium that all players cooperate in the 
society with heterogeneity. In contrast, such full cooperation can be attained in the 
society with homogeneous players only when all players’ minimum sizes of 
cooperation are n. These results seem to suggest that full cooperation may be more 
likely in a heterogeneous society than in a homogeneous society. 



4. Discussion 



We provide two applications of our model of organizations for social 
cooperation. 

(1) Environmental Pollution 

Environmental pollution is a typical and serious example of social dilemma in 
which cooperative actions by players are needed in order to escape tragedy. In 




240 



Okada (1993), we considered voluntary organization for environmental regulation 
by applying our model to the “Lake” game (Shapley and Shubik 1969), in which 
factories decide whether to treat wastes before discharging them into a lake. A 
simulation of the model shows that probability of organization converges to some 
positive value (around 0.69) when the number of factories becomes large, despite 
the asymptotic behavior of each factory’s participation probability converging to 
zero (Theorem 2.4). It may be possible that (partial) cooperation can be attained 
even in a large society. 

(2) The emergence of the state 

The state is a complex political organization. Although there have been divCTse 
points of view on it in the literature, one of the traditional explanations for the 
state is the idea of the Social Contract People in a society make a contract to 
create the state in order to avoid “Wane of every one against every one” (Hobbes, 
1651) in the state of nature. The primary function of the state is to ddend its 
constituents from invasion by foreigners and injuries inflicted on one another, so 
that they may nourish themselves and live contentedly. To achieve such a common 
benefit, the state has enforcement power to control the constituents’ behavior. If 
one takes this point of view to the state based on the Social Contract, a theoretical 
issue arises: how is formation of the state consistent with the free-rider problem? 

In Okada and Sakakibara (1991), we investigated the emergence of the state as a 
tax enforcement institution in a simple dynamic model of a production economy. 
In the economy, individuals of nonoverlapping generations produce private goods 
by utilizing social overhead capital as public goods. Due to the free-rider problem, 
there is no accumulation of capital stock without the state. We consid^ed a 
democratic state of which constitutional rule is characterized by free participation 
and an internal enforcement agent. The enforcement agent is elected from among 
participants. All participants decide by unanimous voting the payment to the 
enforcement agent before the election. 

It is shown that there exists the critical level of social productivity, measured in 
terms of the capital stock gifted by the previous generation. If the society is so 
“rich” that social productivity is beyond the critical level, the state is nevCT 
established. If not, the formation of the state is stochastically detOTnined (see also 
Theorem 2.3). Under the assumption that population and marginal productivity of 
the public good are constant over generations, we can show that social productivity 
stochastically converges to (and beyond) the critical level over a sufficiently large 
number of generations. In this sense, the end of the state necessarily comes in the 
long run. The main reason for this result is that the payment to the enforcement 
agent increases as the capital stock accumulates. If this enforcement cost exceeds 
the critical level, no enforcees find it beneficial to be burdened with such cost. 

To conclude this essay, we present some critical discussion of our analysis. 
First, our model assumes that any nonmember of an organization can perfectly 
free-ride cooperative actions by its members. Of course, the possibility of free- 
riding dq)ends upon specific social contexts in which our game situation is 




241 



embedded. For example, in the case of the state, it may be practically difficult for 
nonmembers of the state to free-ride a common benefit provided by the state. Once 
established, the state tends to be a “Leviathan” possessing enough physical pow^ 
to prevent free-riding. On the other hand, in the environmental pollution problem, 
many empirical studies show that it is quite difficult for voluntary associations for 
regulation to prevent nonmembers’ free-riding. We think that our model allowing 
free-riding by nonmembers can serve as a benchmark of our analysis of 
organizations. 

Second, our model does not treat the enforcement agent as a real player. We 
assume that the enforcement agent does its work sincerely. A serious problem in 
our framework of institutional arrangements is: “Who will take care of the care- 
takers?” (Arrow 1974, p.72).* To investigate this problem of controlling the 
enforcement agent, we will have to elaborate our model of organizations with a 
richer internal structure. The division of authority is an important issue. 

Third, our model considers the formation of only one organization in a society. 
It will be interesting to extend our model so that we can considCT the formation of 
multiple organizations, and to analyze conflict and cooperation among them. 

Finally, the most criticized aspect of our analysis is probably its static nature. 
The central questions in the theory of organizations are: how does the institutional 
structure of the organization evolve in the process of social development; and how 
do organizations affect the development of a society? In a recent work (Okada, 
Sakakibara and Suga 1995), we extend our analysis of the emergence of the state in 
this direction. We consider the centralized political system (monarchy) of the state 
as the alternative system to democracy and investigate how the political system of 
the state evolves in the process of economic development It is shown that 
monarchy may be selected as the result of a social contract by individuals when 
social productivity is low, and democracy is selected when social productivity 
becomes sufficiently high and close to the critical level. The political system tends 
to evolve from monarchy to democracy in the long run of economic development 
An interesting theoretical issue in our future research is to advance our analysis of 
organizations toward a game theory of social development. 



References 



Arrow, Kenneth J. (1974): The Limits of Organization. New York: W.W. Norton & 
Company. 

Aumann, Robert J. and Lloyd S. Shapley (1976): “Long-Term Competition-A Game- 
Theoretic Analysis,” preprint. Also, in N. Megiddo (1994), Essays in Game Theory- 
In Honor of Michael Maschler, 1-15. Berlin: Springer-Verlag. 



8 



Kliemt (1990) discusses some issues arising when the enforcer is a strategic player. 




242 



Axelrod, Robert (1984): The Evolution of Cooperation. New York: Basic Books, Inc. 

Hammerstein, Peter and Reinhard Selten (1994): “Game Theory and Evolutionary 
Biology,” in R.J. Aumann and S. Hart, Handbook of Game Theory Vol. //, 929-993. 
Amsterdam, Elsevier Science B.V. 

Hobbes, Thomas (1651) 1991: Leviathan, ed. by R. Tuck. Cambridge: Cambridge 
University Press. 

Kliemt, Hartmut (1990): “The Costs of Organizing Social Cooperation,” in M. Hechter, 
et al.. Social Institutions. New York: de Gruyter. 

Ma3mard Smith, John (1982): Evolution and the Theory of Games. Cambridge: 
Cambridge University Press. 

Okada, Akira (1993): “The Possibility of Cooperation in an n-Person Prisoners’ 
Dilemma with Institutional Arrangements,” Public Choice 77, 629-656. 

Okada, Akira and Hartmut Kliemt (1991): “Anarchy and Agreement - A Game Theoretic 
Analysis of Some Aspects of Contractarianism,” in R. Selten, Game Equilibrium 
Models II - Methods, Morals, and Markets, 164-187. Berlin: Springer- Verlag. 

Okada, Akira and Kenichi Sakakibara (1991): “The Emergence of the State: A Game 
Theoretic Approach to Theory of Social Contract,” The Economic Studies Quarterly 
42, 315-333. 

Okada, Akira, Kenichi Sakakibara and Koichi Suga (1995): ‘The Dynamic 
Transformation of Political Systems through Social Contract: A Game Theoretic 
App)roach,” DP No. 426, Institute of Economic Research, Kyoto University. 
Forthcoming in Social Choice and Welfare. 

Olson, Mancur (1965): The Logic of Collective Action. Cambridge: Harvard University 
Press. 

Selten, Reinhard (1973): “A Simple Model of Imperfect Competition, where 4 Are Few 
and 6 Are Many,” International Journal of Game Theory 2, 141-201. 

Selten, Reinhard (1975): “Reexamination of the Perfectness Concept for Equilibrium 
Points in Extensive Games,” International Journal of Game Theory 4, 25-55. 

Shapley, Lloyd S. and Martin Shubik (1969): “On the Core of an Economic System with 
Externalities,” American Economic Review 59, 678-684. 

Taylor, Michael (1987): The Possibility of Cooperation. Cambridge: Cambridge 
University Press. 




Reinhard Selten Meets the Classics 



Werner Giith^ and Hartmut Kliemt^ 

* Department of Economics, Humboldt-Universitat zu Berlin 
^ Department of Philosophy, Gerhard-Mercator-Universitat GH, Duisburg 



Abstract: Most of the more empiristically minded among the modem classics of 
political philosophy tried to give a rational choice account of the emergence and 
maintenance of social order. Subsequently we shall relate some aspects of 
Reinhard Selten's work to this tradition. His insights can further our 
understanding of fundamental classical queries: Jean Bodin's quest for 
understanding the implications of the sovereign's lack of commitment power, 
Benedict de Spinoza's scmtiny of opportunism in trust relationships, John 
Locke's problem of founding all legal institutions on individually rational 
consensus in anarchy, and Thomas Hobbes' efforts to explain all aspects of social 
order in terms of individually rational choice. Relating simple game models to 
excerpts from classical texts sheds modem game theoretic light on the works of 
the classics and some classical light on game theory. 



Introduction and Overview 

All modem classics of social theory were stmggling in one way or other with the 
question whether, and if so, how the emergence and maintenance of social order 
can be explained in terms of individually rational choice. Recent non-co-operative 
game theory offers new and fruitful tools for addressing this fundamental problem 
of all social theory. At the same time going back to the modem classics and their 
most fundamental problem may put into perspective the game theoretic enterprise 
as well. 

We hope to demonstrate subsequently that some of the basic insights of 
Reinhard Selten and the classics are quite intimately related to each other. We 
shall start by putting Jean Bodin's argument that sovereign power of necessity 
must be absolute into Reinhard Selten's game theoretic perspective (I.). It is then 
illustrated that this perspective also sheds some new light on Benedict de 
Spinoza's theory of contracting (II.), the Lockean idea of social contract (III.) and 
finally the so called “Hobbesian problem of social order“ (IV.). 




244 



I. Reinhard Selten Meets Jean Bodin 

Jean Bodin is generally regarded as a propagandist of absolutism, as a kind of 
ideologue favoring the absolute and unrestricted power of the sovereign. 
Adherents of limited government therefore tend to reject Bodin's arguments for 
normative reasons. But rather than dismissing the unwelcome arguments out of 
hand they deserve to be taken seriously. Modem game theory suggests that 
Bodin's basic insight was a much deeper one than presumed by most political 
theorists. Bodin neither assumed that sovereigns were power seeking individuals - 
- at least not to a greater extent than ordinary individuals ~ nor did he suggest that 
they should act as such. He rather doubted that a rational sovereign could in fact 
restrict his own sovereignty even if he wanted to: 

“If then the soueraigne prince be exempted from the lawes of his 
predecessors, much lesse should he be bound vnto the lawes and 
ordinances he maketh himselfe: for a man may well receiue a law from 
another man, but impossible it is in nature for to giue a law vnto himselfe, 
no more than it is to command a mans selfe in a matter depending on his 
owne will: For as the law saith. Nulla obligatio consistere potest, quae 
voluntate promittentis statum capit, There can be no obligation, which 
taketh state from the meere will of him that promiseth the same: which is a 
necessarie reason to proue euidently that a king or soueraigne cannot be 
subiect to his owne lawes. And as the Pope can never bind his owne hands 
(as the Canonists say;) so neither can a soueraigne prince bind his owne 
hands, albeit that he would.“ (Bodin 1576/1606, 92). 

Since the sovereign prince cannot bind his own hands he and his subjects suffer a 
utility loss. The prince as well as his subjects could be better off if only he could 
bind himself by his own word. But since the sovereign reigns supreme, he is 
beyond bounds (or absolutus). As the highest (or sovereign) normative authority 
in a state he cannot bind himself by (self-enacted) norms. He cannot give 
“binding“ promises and therefore is “bound“ to be non-trustworthy. Subjects 
know this (as does the sovereign himself). Therefore the sovereign's subjects 
know that their wealth and property are unsecure. 

As long as the supreme power in the state cannot restrict its own future 
exploitative actions and interests, the threat of future exploitation cannot be 
controlled -- not even by the sovereign himself Economic history illustrates that 
this may lead to severe problems for all. In particular, economic growth may be 
reduced considerably whenever individuals are expecting that they might fall prey 
to their governments' exploitative acts. Citizens will under-invest in long run 
projects. 'The accumulation of capital and wealth will be sub-optimal. Resources 
will be misallocated. Capital will be used in ways that cannot be taxed with ease 
rather than in the most productive ways. 

On the other hand, if people could trust their prince distortions in the allocation 
of resources would be reduced. Citizens could then rationally incur the risk of 
investing into activities that in principle would make them vulnerable to 




245 



exploitation. If only they could trust that their prince — or, for that matter, their 
parliament ~ will not act opportunistically this might make all better off — 
including prince or parliament (cf Jones 1981 and Rosenberg and Birdzell, 1986 
for historical evidence that restrained governments do better). 

Still, who would trust an unconstrained sovereign? The prince may well 
understand that all including himself could conceivably be better off could his 
subjects rationally trust him. However, since he cannot commit to a future course 
of action merely by his own prior acts of will, he and his subjects are facing a 
dilemma: Being rational they understand that trustful cooperation among rational 
individuals would be better for all individuals concerned and at the same time 
they understand that, being rational, this course of action is closed off. 

In a more formal vein the underlying dilemma can be illustrated by the 
elementary game of trust of figure 1 : 




t> 1 >o>s 



Figure 1 

In this tree it is assumed that all individuals of the general population are identical 
twins who receive the same payoff. The first component of each pay off vector 
refers to the sovereign while the second indicates the preferences of the people. 
Since t>l the sovereign will exploit his people if they choose to work. 
Anticipating the sovereign's opportunistic behavior the people will shirk or more 
generally shy away from “promising“ investments (cf. for an alternative approach 
the „peasant-dictator game“ Van Huyck et al. 1995). 

The proper subgame of figure 1 has exactly one equilibrium. In the 
terminology of Reinhard Selten (1965): The strategy combination “shirk, break 
promise“ is the only subgame perfect equilibrium of the game of trust. A rational 
sovereign will exploit his people if they offer him a chance. Should he declare in 




246 



SO many words that he has no ill intentions this will not change the situation. 
Enacting some law will not work either, since the sovereign being endowed with 
supreme legal authority can renounce the law whenever this suits his purposes. 
Thus, the sovereign cannot alter the expectations that his subjects rationally 
entertain with respect to his behavior in the proper subgame. The strategy 
combination (“work“, “keep promise“) is not an equilibrium if the rational 
sovereign is unbound by his own orders. 

The preceding argument seems quite compelling from the point of view of 
modem non-cooperative game theory. Nevertheless, modem philosophers tried to 
cast some doubts on it. Again their argument can be illustrated by an elementary 
game tree (cf figure 2): 




t> 1 >r>0>s 
Figure 2 

Now the sovereign first decides whether or not he should drink or pour out his 
best wine. This decision seems quite unrelated with his later decisions as a ruler. 
Nevertheless, some modern theorists suggest that the sovereign can overcome his 
lack of commitment power, by pouring out the wine and thus signaling his 
“good“ intentions (cf for instance Reny 1992 and Bicchieri 1989): Everybody — 
including the prince and his subjects — anticipates that a rational prince is going 
to behave opportunistically rational in the final subgame should people decide on 
working rather than shirking. Anticipating his people's anticipation of his breach 
of promise the rational prince will drink his wine. The prince also anticipates that 
his people expect this. But the prince can falsify the latter anticipation by pouring 




247 



out the wine. Doing this he signals that he is willing to behave in non-subgame 
perfect ways. He signals that he is going to violate Jean Bodin' s precepts of 
“princely“ behavior, or so the argument goes. 

Put that bluntly the modern philosophers' argument does not seem very 
appealing, to say the least. But if any further defense of Bodin against his 
disrespectful colleagues is needed one can trustfully turn to Reinhard Selten 
again, and bring into play the concept of a perfect equilibrium (cf 1975). 

Princes may be quite old before they can get on the throne, even the youngest 
after seizing power may once in a while loosen their grip and some might drink 
more than their due. In any case, their hands like any other person's may tend to 
tremble. Thus, though trying to drink his wine, many a prince will once in a while 
pour, rather than drink it. If this is so, the citizens, after observing that the wine 
has been spoilt rather than being drunk, cannot infer that a prince intended to 
behave contrary to his preferences. Maybe he was drunk and after the sobering 
effect of tragically loosing his best wine will be even more eager to tax his 
citizens because he wants to buy new wine. 

More seriously, rational individuals in analyzing a game will always 
discriminate between “plan and play“. From their point of view, introducing some 
“trembles“ such that all moves are “kept in play“ so to say, is motivated by their 
analytical aims rather than by empirical assumptions. To put it slightly otherwise, 
among individuals who ascribe full rationality to each other, the assumption of 
small trembles — in particular uniform trembles — is not justified by potential 
deficiencies of rationality, but quite to the contrary rather expresses that fully 
rational individuals will always analyze and keep in mind the whole game tree 
when planning their play. 

More realistically, the sovereign body of a modem parliamentary democracy is 
constituted anew every election period. Every freshly elected parliament is a new 
agent that is only loosely linked with the old one. Therefore any sovereign body 
faces seemingly insurmountable difficulties in signaling intentions for a 
subsequent one. Citizens who are aware of this could not rationally assume that 
“declarations of intent“ of a former parliament signal any intentions of a later one, 
unless there would exist some commitment mechanism transcending the powers 
of the sovereign parliament. With shifting majorities this would hold true even if 
the people themselves formed the sovereign. As Bodin had it, if the ultimate 
power to make the law is vested with the sovereign and if the sovereign is 
assumed to make choices according to the preferences prevailing in each instance 
of choice, then it can hardly be imagined how a sovereign (person, parliament, or 
people) could choose to become committed. 

This again may be captured well by a game theoretic notion frequently 
employed by Reinhard Selten, namely that of an agent normal form (cf. Harsanyi 
and Selten 1988, 34). In the agent normal form of the game of figure 2 the 
sovereign is split into two players. One player chooses between drinking and 
pouring the wine and one chooses between keeping or breaking his promise. If we 
apply the “present motives“ interpretation of utility, the game tree and, for that 




248 



matter, the agent normal form show the preferences governing decisions of 
decision makers at every instance of choice. Therefore the notion of the agent 
normal form is almost implied by that of a utility index representing all relevant 
motives of the players at all their decision nodes. 

Clearly, if Bodin is taken by his word, the agent normal form captures the 
essential feature of his concept of sovereignty. We might therefore feel that his 
concept of sovereignty takes us to extremes that conceivably may have some 
relationship with princely behavior but certainly not with our more pedestrian day 
to day problems. However, if we strictly stick to the assumptions of forward 
looking rational choice, Bodin's notion of a sovereign prince directly applies to 
anybody who is entitled to make some decision as a “sovereign“ decision maker. 
Making that “democratic“ move we automatically move on to the general 
problems of contracting and promising faced by all individuals in their several 
rational capacities. Benedict de Spinoza addressed these problems in most 
perceptive and in any case “classical^ ways. 



II. Reinhard Selten Meets Benedict de Spinoza 

In his discussion “of the foundations of the state“ in the “theologico-political 
treatise“ Benedict de Spinoza says: 

“Now it is a universal law of human nature that no one ever neglects 
anything which he judges to be good, except with the hope of gaining a 
greater good, or from the fear of a greater evil; nor does anyone endure an 
evil except for the sake of avoiding a greater evil, or gaining a greater 
good. That is, everyone will, of two goods, choose that which he thinks the 
greatest; and of two evils, that which he thinks the least. I say advisedly 
that which he thinks the greatest or the least, for it does not necessarily 
follow that he Judges right. This law is so deeply implanted in the human 
mind that it ought to be counted among the eternal truths and axioms. 

As a necessary consequence of the principle just enunciated, no one can 
honestly forego the right which he has over all things, and in general no 
one will abide by his promises, unless under the fear of a greater evil, or 
the hope of a greater good... Hence though men make promises with all 
the appearances of good faith, and agree that they will keep to their 
engagement, no one can absolutely rely on another man's promise unless 
there is something behind it. Everyone has by nature a right to act 
deceitfully, and to break his compacts, unless he be restrained by the hope 
of some greater good, or the fear of some greater evil.“ (Spinoza, 203-204) 
This is a remarkably clear statement of basic assumptions of a homo oeconomicus 
model of human behavior. It, quite strikingly, alludes to subjectivism and at the 
same time spells out some of the implications of the assumptions of individually 
rational choice behavior for the institutions of promising and contract: Without 
sanctions promises and contracts will be void. 




249 



Traditionally, at this juncture, some external enforcement mechanism based on 
the sovereign power of the state is wheeled in. Since, however, the sovereign — 
reigning absolute ~ suffers from the same deficiencies and problems as his 
rational subjects, such an exogenous explanation will eventually not suffice. An 
account of how promise keeping may be explained endogenously in social 
interaction should be given. 

Under Spinoza's behavioral assumptions a social institution of promising can 
emerge and be maintained only if under most circumstances no individual of the 
relevant reference group (family, tribe, nation, human society in general) can gain 
from breaking promises. In each instance it must pay off in terms of expected 
future rewards or punishments to keep one's promises. Evidently, without an 
external enforcement mechanism, this can be achieved only if individuals are 
engaged in ongoing interactions or repeated games. 

To have something specific in mind, imagine that two individuals 1 and 2 have 
to play the game of trust of figure 1 repeatedly. On each round of play with equal 
probability both adopt the roles of the first and second mover respectively. 
Clearly, if the interaction is repeated indefinitely, it follows from the Folk 
Theorem that on each round of play choosing to work in the role of the first and 
keeping the promise in the role of the second mover can be justified as a subgame 
perfect equilibrium outcome. 

Those who argue that subgame perfectness captures the most fundamental 
aspect of future directedness of individually rational choice are at least to a 
certain extent right. However, they better listen to what else the inventor of the 
concept has been saying. 

First of all Reinhard Selten would in all likelihood reject that the assumption of 
an infinite number of interactions is a reasonable approximation of any real world 
process. Although some experimentalists (e.g. Weg, Rapoport and Felsenthal 
1990) claim to study such situations, they in fact cannot, as long as any round of 
play requires a positive minimum time. 

Secondly, the objection against the assumption of indefinite repetition can be 
strengthened by drawing attention to Reinhard Selten's seminal paper 
“Spieltheoretische Behandlung eines Oligopolmodells mit Nachfragetragheit“ 
(Selten 1965) where he considered a dynamic game with both an infinite and a 
finite horizon. Selten insisted that the solution for the infinite horizon game has to 
be asymptotically convergent, i.e. must be approximated by the solutions of the 
finite horizon games as the number of repetitions goes to infinity. However, as is 
well known, arguments following the logic of the Folk Theorem do not comply 
with this reasonable requirement of approximation. 

'Thirdly, Reinhard Selten could also object to the use -- or abuse — of the Folk 
'Theorem in recent social theory by drawing attention to his concept of subgame 
consistency (cf Selten and Giith 1982, as well as Harsanyi and Selten 1988). In 
an infinitely repeated game of trust all subgames, where the second mover has to 
choose initially between keeping and breaking the promise, are strategically 
equivalent. Subgame consistency requires that therefore their solutions should be 




250 



the same (up to isomorphisms). But, then, the second mover's choice must either 
be always to keep or always to break the promise. Since the previous behavior of 
a second mover does not influence later decisions of a first mover, always to keep 
one's promise is suboptimal in these subgames. Thus Folk 'Theorem arguments 
cannot provide a normatively convincing potential explanation of how the 
institution of promising could emerge and be maintained among rational 
individuals. 

It seems that any institution similar to our real world institutions of promise 
and contract will violate some plausible rationality requirement or other (cf 
Lahno 1994 for a proof that the requirement of subgame perfectness necessarily 
conflicts with our real world practices of promising). But if the premise that all 
rational actions are chosen in view of the subjectively expected causal conse- 
quences of the actions (each taken separately) must be given up in any case, we 
may quite naturally wonder whether a more fundamental change of perspective or 
approach may not be desirable. 

It may be worth recalling first that classical empiristic modeling of social and 
in particular political interaction was followed by evolutionary modes of thought 
(cf. on the to a large extent undeservedly ill-reputed history of social Darwinism 
for instance Hofstadter 1969). Analogously, perhaps, “classicaT‘ game theoretic 
modeling will be followed by a switch towards evolutionary game theoretic 
modeling. 

Again we find Reinhard Selten on the forefront of research (cf Selten 1983 
and Hammerstein and Selten 1984). But rather than considering his views in the 
abstract let us return to the example of the game of trust, though now in an 
evolutionary setting. If in such a setting both individuals can adopt both roles — 
that of a second and that of a first mover — with equal probability, the disposition 
to work rather than to shirk should be driven out by the evolutionary process. As 
long as there is a positive probability that any individual will work, exploitation 
will be superior. If it is superior to exploit others the proclivity to exploit will 
spread. 'This in turn will make it even less rewarding to work. There seems to be 
but one stable state in which this process comes to an end: a state in which 
nobody works. However, if work is avoided completely all strategies prescribing 
shirking in the role of the first and arbitrary behavior in the role of the second 
mover fare equally well against each other; i.e. no such strategy is evolutionarily 
stable. 

Thus the concept of an evolutionarily stable strategy evidently has its own 
limitations. These can be overcome if we take recourse to Reinhard Selten's 
concept of a limit evolutionarily stable strategy (Selten 1988). Applying that 
concept to the elementary trust game of figure 1 we assume that “work“ is chosen 
with some (arbitrarily small) positive minimum probability. Since in a complex 
environment phenotypes will tend to make mistakes, this assumption will be 
empirically sound in general. Under this assumption always exploiting others will 
be evolutionarily stable. 




251 



More generally, considerations like the preceding ones may give rise to a more 
indirect evolutionary approach (cf. on applying this to the evolutionary game of 
trust Giith and Kliemt 1995). Since it is possible in this approach to treat as 
endogenous parameters like preferences or behavioral dispositions it offers 
promising avenues for future research; but rather than speculating about the 
future, let us stick to the past and turn to how the classics in fact did explain the 
emergence of social institutions, namely by a social contract to that end. 



III. Reinhard Selten Meets John Locke 

Modem political theorists like Buchanan, Nozick and Rawls have revived the 
basic ideas of the so-called old contractarians Hobbes, Locke and Kant. In 
particular they have scrutinized to what extent a social order could be justified by 
a conceivable consensus of rational individuals. Much of the discussion has, of 
course, focused on what kind of consensus is conceivable under what kinds of 
circumstances. Less interest was devoted to the problem whether, and if so, under 
which circumstances, a contract is itself conceivable, if contracting must take 
place in a state of nature. But how could individuals conceivably contract with 
each other in a state of nature in which the institution of contracting is lacking? 
Moreover, if this is not conceivable, how could the standard contractarian logic 
that a conceivable contract provides normative criteria for right and wrong in real 
world matters be valid? 

None of the contractarians has yet managed to solve these problems 
convincingly. Still, since all normative political theories seem to be deficient in 
one way or other, and contractarianism is at least a well developed one, it is 
worthwhile to explore how far it goes. In doing so we can turn to John Locke and 
conceive of social contracting basically as a process of club formation. In his 
discussion "of the beginning of political society" John Locke says 

"(m)en being, as has been said, by Nature, all free, equal and independent, 
no one can be put out of this Estate, and subjected to the Political Power of 
another, without his own Consent. The only way whereby any one devests 
himself of his Natural Liberty, and puts on the bonds of Civil Society is by 
agreeing with other Men to joyn and unite into a Community, for their ... 
greater Security against any that are not of it. This any number of Men 
may do, because it injures not the Freedom of the rest; they are left as they 
were in the Liberty of the State of Nature. When any number of Men have 
so consented to make one Community or Government, they are thereby 
presently incorporated, and make one Body Politick ... And thus that, 
which begins and actually constitutes any Political Society, is nothing but 
the consent of any number of Freemen..." (Locke 1690/1963, 374-377). 

One should note that according to this passage it is any number that may consent 
to form a body politic. Those who do not want to join are free to stay out. At the 
same time, they do not have a veto against any number of other individuals going 




252 



ahead in forming such a club. This differs for instance from the Buchanan- 
Tullock notion of unanimous consent granting a veto to each individual of a 
predefined collective (cf. 1962) as well as from Kantian contract notions starting 
from all rational beings as an exogenously determined collective of normatively 
relevant individuals. 

Having a veto implicitly presupposes a collective monopoly over actions to 
which the veto applies. But from the point of view of individualist values the 
assumption of a normatively justified collective monopoly seems a non-starter. 
The monopoly is exactly what should be justified as a result of the contractarian 
argument. Likewise, the Kantian assumption that the collective of normatively 
relevant individuals is exogenously determined, is stronger than necessary within 
a contractarian approach. At least in a way it seems conceivable that moral and 
legal communities form as self-selected groups or clubs. 

A most austere process of forming associations in anarchy is again suggested 
by one of Reinhard Selten's models. In his study of cartel formation (cf. on this 
1973) he suggested how such a club may form — not necessarily an all inclusive 
one. First of all, individuals decide whether they are in principle interested in 
joining the club. Those who are will bargain then about the terms of the club's 
bylaws or “constitution^. Finally the interaction itself takes place. When 
interacting with each other, insiders of the club are bound by the rules of the club, 
while outsiders interact unrestrictedly. 

The game of trust under random matching of partners as previously discussed 
may serve as an illustration here. Assume that the game is played by n (>2) 
individuals in the following way: As before, each individual expects with 
probability 1/2 to adopt the role of a first mover and with probability 1/2 that of a 
second mover. However, now, after formation of the club, members are certain to 
be matched with a club member, whereas non-members will be excluded from 
interaction with members and therefore interact with non-members exclusively. 

The sole purpose of forming the club is controlling individual behavior such 
that “work and keeping promises“ will emerge as the general behavioral pattern 
of interaction between insiders. Control is costly, though. Whenever an individual 
performs an act of control it incurs a positive monitoring cost c. Since every 
member must monitor every other one the monitoring costs incurred by each 
member of a club with m, 2<m<n, individuals are (m-l)c. A non-member 
receives a pay off of 0 while members get 1 on each round of play. Therefore 
individuals will want to join such a club if l-(m-l)c>0 or c< l/(m-l). Thus only 
in case of c< l/(n-l), the all inclusive club will form (cf for some further details 
of applying Reinhard Selten's ideas on club formation to Lockean 
contractarianism, Okada and Kliemt 1991). 

'This simple model sheds some light on social contracting as a process of club 
formation like the one envisioned by Locke and Nozick. Unless c is extremely 
small relative to “1“ the size of clubs with decentralized monitoring is severely 
restricted. Thus the standard contractarian assumption that eventually an all- 
inclusive club will form spontaneously in anarchy, seems quite doubtful. For, the 




253 



formation of larger groups in which individuals can trustfully interact with each 
other in situations of the structure of the game of trust seems to depend on some 
central monitoring and control (cf for a model in which “extrinsic“ is substituted 
by “intrinsic control“, cf Giith and Kliemt 1995). If, however, central monitoring 
and control is necessary from the outset the aforementioned latent circularity of 
contractarian reasoning immediately re-emerges. Since this problem of circularity 
is nothing but a variant of the so-called Hobbesian problem of social order, let us 
finally turn to Hobbes. 



IV. Reinhard Selten and the Hobbesian Problem of Social Order 

Alfred North Whitehead once remarked that after Plato all philosophy became a 
commentary on Plato's work. Though it would be somewhat exaggerated too, it 
would not be grossly misleading either to say that all modem political philosophy 
after Hobbes in one way or other turned into commentary on Hobbes. In any case, 
Hobbesian views are deeply entrenched in economics and game theory. Above 
all, like Spinoza, modem theorists in these fields still tend to accept the 
Hobbesian view that 

“(i)t is also manifest, that all voluntary actions have their beginning from, 
and necessarily depend on the will, and that the will of doing, or omitting 
aught, depends on the opinion of the good and evil of the reward or 
punishment, which a man conceives he shall receive by the act or 
omission; so as the actions of all men are mled by the opinions of each“ 

(De Give 1651/1982, 75) Therefore, “insomuch as when they shall see a 
greater good, or less evil, likely to happen to them by the breach, than 
observation of the laws, they will wittingly violate them.“ (De Give 
1651/1982, 63) “We must therefore provide for our security, not by 
compacts, but by punishments.“ (De Give 1651/1982 72/73) 

However, taking into account what has been said before, how could we, as 
rational Hobbesian men, ever hope to “provide for our security“? As rational 
citizens (cive) we are aware of Bodin's problem. We know that ,unless we can 
answer the perennial question of “who will guard the guardians“ (“quis custodiet 
ipsos custodes?“), there will be no security against an enforcement agency vested 
with the ultimate power of imposing punishments. We also know that unless we 
can solve Spinoza's contract problem we cannot as rational individuals 
deliberately create that supreme power in the first place. So, are we as rational 
men due to fall prey to our own rational capacities? 

We certainly would if we were merely rational beings (the most forceful 
critique of the economic model applied to law still being Herbert Hart's 1961 
analysis of “the concept of law“). But we are also endowed with some extra 
rational faculties which we can exploit in rational ways. In assessing the scope 
and limits of these faculties and their several relationships to the rational choice 
perspective of pure game theory, it is illuminating again to turn to Reinhard 




254 



Selten's work. Very markedly he endorses the view that we should not try to 
patch up pure game theory such that it better conforms with social reality. We 
should rather acknowledge that the assumption of strictly rational forward 
looking individual choice — except perhaps for extreme cases -- is not suitable for 
explanatory purposes (for experimental evidence on this cf Selten 1979). If we 
aim at explaining real world processes we better stay clear of counterfactual 
assumptions of ideally rational behavior. 

Not only cognitive limitations of human decision makers, who can at best be 
boundedly rational, cast into doubt the assumption that pure or strict rational 
choice approaches can provide true explanations; it is also obvious that human 
subjects often are in one way or other intrinsically motivated by their values 
rather than by the consequences of their acts. Rational choice analysis may model 
all human individuals as sovereign decision makers who are normatively unbound 
in their rational decision making. But even Bodin's sovereign prince is human 
after all. As a human being he is not only cognitively limited but also operates 
under some more or less moral restraints. He like his subjects will tend to keep his 
promises if “t-l“ is small and thus the temptation to act “opportunistically 
rational“ is not too strong. As experienced social beings his subordinates will 
anticipate this. Likewise, if they encounter other authorities facing low 
opportunity costs, as for instance judges or bureaucrats, they may expect norm 
guided rather than opportunistic behavior from those and so will they themselves 
be guided primarily by norms and rules of thumb if opportunity costs are low (cf 
on the relationship between costs and utilitarian ethics Selten 1986). 

Working with idealized rational choice models may serve several non- 
explanatory purposes, though. Thinking through their implications may be of 
philosophical interest in its own right. As (boundedly) rational beings we quite 
naturally wonder what a world of fully rational beings would look like and in 
particular whether there can be some theory of rational behavior in such a world 
that could become common knowledge without becoming self-defeating 
(Harsanyi and Selten 1988 being the best answer to such queries yet). Moreover, 
rational choice analyses can serve as a bench mark telling us where the crucial 
problems of social order presumably are located. After all, without rational choice 
analyses, those critics of the approach who point to the unsolvability of the 
Hobbesian problem of social order — and quite rightly so -- would not even know 
what the problem is. Finally, acknowledging that there are certain limits to 
rational choice approaches does in no way preclude using them within their limits 
wherever we can. Reinhard Selten contributed much to shifting out those limits 
and at the same time to our understanding of their precise nature. But he also 
taught us some lessons about the merits of engaging in truly “experiental“ and in 
the last resort experimental science of human behavior. 




255 



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Equilibrium Selection in Linguistic Games: 
Kial Ni (Ne) Parolas Esperanton? 

Werner Giith^ , Martin Strobel^ , and Bengt-Arne Wickstrom^ 

^ Humboldt University Berlin, Institute of Economic Theory 
^ Humboldt University Berlin, Institute of Public Economics 



Resumo. Kun elirpunkto en artikolo de Selten kaj Pool uzante la teorion de 
ekvilibroselektado evoluigitan de Harsanyi kaj Selten, ni analizas la risksuper- 
ajn ekvilibrojn de lingva ludo. En modelon kun du grupoj da homoj kun mal- 
samaj gepatraj etnaj lingvoj ni enkondukas kroman interetnan planlingvon, 
esperanton, kies lernado estas multe pli facila ol tiu de etnolingvo. El la kvar 
striktaj kaj simetriaj ekvilibroj de la ludo - rezignado pri internacia komu- 
nikado, uzo de esperanto au unu de la etnaj lingvoj - du au tri, depende de la 
valoroj de la parametroj de la modelo, povas kunekzisti. Uzante la koncepton 
de risksuperado por selekti inter tiuj kunekzistantaj ekvilibroj, ni montras, 
ke la enkonduko de esperanto por kelkaj realismaj parametrovaloroj estas la 
ununura solvo. Gi ankaii estas efika solvo, sed ciuj efikaj solvoj pere de es- 
peranto ne realigeblas pere de tia risksupera ekvilibro. Fine ni spekulative 
pritraktas la eblon analizi la lingvouzon pere de evolua ludo. 

Abstract. Taking our point of departure in an article by Selten and Pool 
and using the equilibrium selection theory developed by Harsanyi and Sel- 
ten, we analyze the risk dominant equilibria of a language game. Into a model 
with two groups of people with different native ethnic languages, we intro- 
duce a further interethnic planned language, Esperanto, the learning of which 
is much easier than that of an ethnic language. Of the four strict and sym- 
metric equilibria of the game - absence of international communication, use 
of Esperanto or one of the ethnic languages - two or three might coexist, 
depending on the values of the parameters of the model. Using the concept 
of risk dominance in order to select from those coexisting equilibria, we show 
that the introduction of Esperanto for some realistic parameter values is the 
unique solution. It is also an efficient solution, but all efficient solutions with 
Esperanto are not realised by such a risk dominant equilibrium. Finally, we 
explore in a speculative manner the possibility of analyzing the use of lan- 
guage with an evolutionary game. 



1. Introduction 

If strangers with different native languages want to communicate, at least 
one has to invest in learning a new language. Traditionally this would have 
meant that at least one party would learn another ethnic language. 




258 



Since an ethnic language has evolved over a long time span, it is usu- 
ally rather unsystematic due to imported words, phrases and grammatical 
rules. Famous examples - terrifying millions of pupils - are irregular verbs 
and famous tongue breakers (like chaiselongue or portemonnaie which were 
imported from the French into the German language in spite of the phonetic 
differences between the two languages). To cut a long story short: learning 
another ethnic language is tedious and often fails (our use of English in this 
essay may be a good example). 

Although the community of Esperantists also favors a more cosmopoli- 
tan view of the world, one of the essential reasons for developing Esperanto 
has been to allow for communication without the enormous investment of 
learning another ethnic language. Esperanto as a planned language relies on 
considerably fewer rules and allows for nearly no exceptions. Through its 
clear and systematic agglutinate structure, it achieves with simple means a 
great precision and flexibility of expression. This economy of communication 
is unmatched by just about any ethnic language. It is estimated that to learn 
Esperanto one needs less than one third of the efforts required for learning 
English by continental Europeans in spite of the close relationships between 
nearly all European languages. See for instance Prank and Szerdahelyi (1976) 
or Formaggio (1990). 

But, of course, if two strangers want to communicate in Esperanto, both 
have to learn it. Although one can imagine that children are brought up 
learning Esperanto, which indeed is the case in many families where parents 
from different countries have met through the Esperanto movement, we will 
here neglect such a possibility. Esperanto was not developed to be a substitute 
for ethnic languages. 

There are many obvious reasons why Esperanto till now has failed to be- 
come a frequently used means of communication, e.g. the success of American 
English as the international language of science, business, and pop(ular) mu- 
sic. We do not want to discuss such reasons in general, but rather develop 
a simple linguistic game by which some of these reasons can be exempli- 
fied and explored more thoroughly. Since most linguistic games are usually 
special coordination games (every generally coordinated behavior is a strict 
equilibrium) , our main methodology is the theory of equilibrium selection as 
developed by Harsanyi and Selten (1988). 

To the best of our knowledge there exist only two studies trying to explore 
the strategic (dis) advantages of relying on Esperanto as a communication 
device, namely the ones by Selten and Pool (1991 and 1995) who both are 
Esperanto activists. The latter of these studies reviews the related literature 
which we do not want to repeat here. In section 2 we will discuss these con- 
tributions before introducing our own, simpler model which will be analyzed 
in sections 3, 4, 5, and 6 before concluding in section 7. 




259 



2. The Study by Selten and Pool 

The study by Selten and Pool (1995) was originally written in Esperanto. The 
present edition, however, has a German translation and there are extended 
abstracts written in English and other languages. Players in their linguistic 
game all speak an ethnic language. To communicate with foreigners they can 
learn other ethnic languages or a planned language whose learning costs are 
far below those of any ethnic language. 

One can describe the linguistic game of Selten and Pool as one with n 
countries and different ethnic languages I = 1, ..., n plus the planned language 
1 = 0. Denote by £ = {0, 1, ...,n} the set of all languages. An individual i (/) 
whose native ethnic language isZG£\{0} thus has the strategy set 

%) ={LU{Z}:LC£\{/}}. 

Two individuals i and j using Si G 5i(/) and sj G can communicate if 

Si n Sj 7 ^ 0, i.e. if they have one language in common. If we denote by 
the cardinality of the set A', the total communication gain of individual i is 
measured by 

# {j ■ Sj nsij/^0}. 

Selten and Pool (1991) allow that communication benefits depend on the 
home country of those whom one can communicate with. For the sake of 
simplicity we neglect the possibility of weighing communication benefits in 
this way. Individual Z (Z)’s language learning costs are the product of the gen- 
eral learning costs g (5i(/)) of the language set \ {/} , which it wants to 
acquire in addition to its native ethnic language /, and its inverse individual 
language learning ability or, reciprocally, its learning cost . Since inhab- 
itants of the same country can differ in language learning skills, they may 
choose different strategies even in a symmetric equilibrium of the linguistic 
game in normal form whose strategy sets are 5i(/) and whose payoff functions 
for all individuals i{l) are 

#{i : Sj n Sj(j) 0} - Cj(i)5 (sj(/)) . 

For this game model it is shown, in general, that in any equilibrium s* = 
quite naturally the payoff of an inhabitant i{l) of country I G L 

decreases if his learning cost parameter Cf(/) is higher whereas this is neither 
true for its general learning cost nor for its communication benefit 

#{j : S*J n 0}. 

For the special case of n = 2 and symmetric cost conditions for both 
countries (inhabitants of both countries have the same general learning costs 
for the other ethnic language as well as for the planned one and, furthermore, 
individual learning costs Ci(/) are equally distributed in both countries I = 
1,2) an interesting further result confirms the “Esperanto Solution”: In any 
symmetric equilibrium s* requiring each individual i (1) to learn a language. 




260 



i.e. ^ {/} for all i (1) , I = 1 , 2 , everybody learns the planned language 
/ = 0, i.e. = {0,/} for all i (Z) and Z = 1,2. To rule out learning of the 
planned language at all, its cost advantage has to be low enough and the 
general costs p({Z,l}) = p({Z,2}) of learning ethnic languages have to be 
limited. 

Unlike Selten and Pool (1991) we will rely on a simpler game model with 
a finite number of players who, furthermore, will not differ in their language 
learning skills. Whereas the first assumption needs no justification, the sec- 
ond one is highly restrictive. It has been imposed to simplify our analysis 
when selecting among equilibria. This assumption is, however, no restriction 
imposed by our general approach. 

It should be noted that the assumption of individual learning skills which 
apply to all possible languages an individual may want to acquire, as imposed 
by Selten and Pool (1991), is also rather restrictive. Especially, when includ- 
ing a planned language which tries to avoid all the complications of ethnic 
languages and therefore does not require as many skills like pronunciation 
and grammatical background as ethnic languages, individual differences may 
not matter in the same way as when learning an ethnic language. 

In summary, it can be readily accepted that Selten and Pool (1991) an- 
alyze a richer game model. The main excuse for relying on a simpler game 
structure is that we want to investigate our game model more thoroughly, 
by discussing efficiency aspects and, more importantly, by trying to select a 
unique solution for all possible parameter constellations. 



3. Game Model 

As in the main part of the analysis of Selten and Pool (1995) we focus on the 
special case of two countries with ni, respectively ti 2 inhabitants. Denote by 
n^(> 0 ) the number of inhabitants in country i learning language j where 
j = 0 means to learn the planned language and j ^ 0 , j ^ ^ to learn the 
other ethnic language. Communication benefits are measured as in the model 
of Selten and Pool. The cost of an inhabitant of country i to learn language j 
is denoted by for all i = 1,2 and all j = 0 , 1 , 2 and j ^ i with 0 < < c- 

for 2 = 1,2 and j 7 ^ 0 , j ^ i. 

To compare the situation with and without a planned language two game 
versions will be analyzed, namely game model with strategy sets Sk = 
{ 0 , 1 }, where sjk = 1 {sk = 0 ) stands for (not) learning the other ethnic 
language, and the game model with strategy sets Sk = { 0 , 1 }^ where 
Sk = {pk-i^k) means (not) to learn the planned language in case of = 1 
(p^ = 0 ), respectively (not) to learn the other ethnic language in case of 
5k = \{5k= 0 ). 

When analysing the various games in normal form we will impose non- 
degeneracy assumptions ruling out border cases of no practical relevance. 




261 



4- The World Without Esperanto (Game G^) 

In the game for every vector s = {sk)k the payoff of an individual i of 
country 1 is 

Ui(s) = ni - 1 + Si{ri2 - cl) + (1 - Si) ^ sj (4.1) 

3 

where summation over j runs over all inhabitants of country 2. Similarly, the 
payoff of an inhabitant j of country 2 is given by 

Uj{s) = ri 2 - 1 + Sjini - 4) + (1 - Sj) Si (4.2) 

i 

where now summation runs over all inhabitants i of country 1. 

Let us assume that side payments are possible and that all inhabitants of 
the same country use the same strategy (the first assumption requires means, 
e.g. international money, to transfer utility, the second can be justified by a 
convincing symmetry requirement). Given these restrictions one can show 

Lemma 4.1. The efficient strategy vector s = {sk)^ 

fij Sk 0 generally for min{uic\^n2c\} > 2nin2 
(ii) Si = l,Sj = 0 for mzn{n2C2, 2nin2} > nicf 
(in) Si = O^Sj = 1 for minjnicf , 2nin2) > ri2C2 

where i{j) runs over all inhabitants of country 1(2). 

Proof. Since = 1 = Sj is inefficient due to >0, one only has to compare 
the three strategy vectors (i), (ii), and (iii) which imply the following welfare 
levels: 



W{i) — ni(ni — 1) -f n2(ri2 — 1) 


(4.3) 


W(ii) = W (i) 2nin2 — nicl 


(4.4) 


W (iii) = W(i) 2n\n2 — n2c\ 


(4.5) 


Thus the lemma is obvious. 


■ 



What can be expected, however, in strict (symmetric) equilibria is stated 
by 

Lemma 4.2. The strict and symmetric equilibria of are 

(i) Sk = 0 generally for cj > ri 2 and c\ > rii 
(a) Si = 1, Sj = 0 for n 2 > cf 
(iii) Si = 0, Sj = 1 for ni > c^. 

Proof. According to (i) learning costs exceed communication benefits whereas 
according to (ii), respectively (iii), the opposite is true if inhabitants of coun- 
try 1, respectively 2, learn the other language. Since learning costs are posi- 
tive all three equilibria are strict in the sense that one loses when deviating 
unilaterally. ■ 




262 



According to the theory of equilibrium selection, developed by Harsanyi 
and Selten (1988), the solution of the game is the strict and symmetric equilib- 
rium which risk dominates the other^ . Only equilibria {ii) and {in) described 
in Lemma 4.2 can coexist, namely in the case of condition 

ri 2 > c\ and rii > c\. (4.6) 

Applying the Harsanyi and Selten theory of equilibrium selection to all games 
satisfying condition 4.6 yields 

Theorem 4.1. If condition 4-^ holds j the strict equilibrium {ii) risk domi- 
nates the strict equilibrium {Hi) if 

ri 2 cl > riicj (4.7) 

whereas {Hi) risk dominates {ii) in case of 

nicl > n 2 c\. (4.8) 

The formal proof of Theorem 4.1 can be found in Appendix A. One first 
derives the bicentric prior strategies by imagining that all other players know 
already whether the solution is {ii) or {Hi). The vector of best replies is 
already a strict equilibrium, namely {ii) in case of (4.7) and {Hi) in case 
of (4.8). Thus it is not necessary to compute the tracing path according to 
Harsanyi and Selten (1988). 

The solution of Theorem 4.1 is also the efficient solution of G^. It means 
that the inhabitants of that country with lower total learning costs should 
learn the other country’s language. 



5. When Esperanto is Available (Game G^) 

If the planned language ^ = 0 is available, the payoff of an individual i of 
country 1 for every vector s = {pk^^k)k 

Ui{s) = ni - 1 -h 6 j{n 2 - cj) 4- (1 - Sj) Sj 

3 

Uj{s) can be derived analogously by exchanging indices. The planned lan- 
guage is the only efficient generally spoken language if condition 

min{2nin2,niCi,n2C2} > nicj + ri2C2 (5.1) 




^ Although Harsanyi and Selten (1988) gave priority to payoff dominance they 
now both seem to accept that payoff dominance should be neglected, see 
Selten (1995) and Harsanyi (1995). 




263 



holds. 2 n\ri 2 > nic\ +ri 2 C^ guarantees that general learning of Z = 0 provides 
more communication benefits than costs. If nicf, respectively ri 2 C 2 exceed 
the cost n\c\ -f ri 2 C 2 of learning I = 0, this is, furthermore, the cheapest way 
of guaranteeing maximal communication benefits. 

The strict and symmetric equilibria in game are described by 

Lemma 5.1. The not necessarily unique strict and symmetric equilibria 
Sk = (pk.Sk) ofG^ are 

(0) Sk = (0,0) generally for c\ > U 2 and c^ > ni 

(1) Sk = (1,0) generally for U 2 > cj and n\ > C 2 

(a) Si — ( 0 , 1 ), Sj = ( 0 , 0 ) for U 2 > cf 

(in) Si = (0, 0), Sj = (0, 1) for ni > c\ 

where i{j) runs over all inhabitants i(j) of country 1(2). 

Proof. It does not pay to learn another ethnic language if Z = 0 is universally 
spoken. General learning of Z = 0 requires ri 2 > cj and ni > c^. If one 
of the inequalities is wrong, nobody would learn I = 0 in equilibrium since 
cj, C 2 > 0. The remaining part of Lemma 5.1 follows from Lemma 4.2. ■ 

All possible constellations of equilibria are shown in Table 5.1. We still 
assume that cf > c? and c^ > c^. 



country 1\2 


cf > C? > 722 


cf > ri 2 > Cl 


Zl2 > cf > Cl 


cl > C 2 > ni 


(o) 


(o) 


(ii) 


cl > ni > c° 


( 0 ) 


(oUi) 




rzi > C 2 > C 2 


{Hi) 


(i),(m) 





Table 5.1. Possible equilibria in 



6. When is Esperanto (Universally) Risk Dominant? 

Assume that at least two equilibria, stated in Lemma 5.1, exist. The equi- 
librium (i) is called (universally) risk dominant if it in pairs risk dominates 
(o), (ii) respectively (Hi). Since risk dominance is, in general, no transitive 
relation, there may not exist a universally risk dominant strict equilibrium. 
We first neglect such possibilities and concentrate on the condition for the 
“Esperanto Solution” (z) to be (universally) risk dominant. Afterwards it 
will be shown that for the game at hand intransitivity of risk dominance is 
impossible. 

The risk comparison between (z) and (zz) is described by 




264 



Lemma 6-1. Assume that both equilibria (i) and (ii) of game G'^ coexist. 
Then (i) risk dominates {ii) if 

n\c\ + U2C2 < nic\ ( 6 . 1 ) 

whereas {ii) risk dominates (z) if 

niCi -f U2C2 > nic\. (6.2) 

A detailed proof can be found in Appendix B. By exchanging indices one can 
derive from Lemma 6.1 also the results for risk comparison between (i) and 
{Hi): 

Corollary 6.1. Assume that both equilibria {i) and {Hi) of game coexist. 
Then (z) risk dominates {Hi) if 

niCi -f ri2C2 < n2C2 (6.3) 

whereas {Hi) risk dominates (z) if 

me? 4- ri 2 C? > ri 2 C 2 . (6.4) 

Now the possibilities for learning Esperanto (z) to be the unambiguous 
solution are: 

— (o) and (z) are the only strict equilibria, i.e. 

c? > ri 2 > c? and C 2 > ni > c? (6.5) 

— and either (zz) or {Hi) or both, (zz) and (zzz), coexist. 

Since for the latter case the conditions for (z) to be (universally) risk 
dominant are stated by Lemma 6.1 and Corollary 6.1, it only remains to 
explore the risk dominance relationship between (o) and (z) for condition 
(6.5). The result for condition (6.5) is 

Lemma 6.2. Under condition (6.5) the strict equilibrium (z) risk dominates 
(o) if 

TI 1 TI 2 ^ ^1^1 + *^2^2 
whereas (o) risk dominates (z) if 
Uin2 < nic? + ri2C?. 

The detailed proof of Lemma 6.2 can be found in Appendix C. We now can 
state the conditions which, according to game-theoretic reasoning specified 
by the concept of (universal) risk dominance, imply the Esperanto Solution 
(z). 

Theorem 6.1. The Esperanto Solution (z) is the unique solution of the game 
in case of one of the following conditions: 

1. c\> n 2 > c? and cl > ni > c? and nirz 2 > nic? + ri 2 C? 




265 



2. ri2 > c\ > c\ and cl > ni > C 2 and nicf > nicj + 7i2C2 

3. Cl > U2 > Cl and ui > cl > C2 and U2cl > nic? + ri 2 C 2 

4. U2 > Cl > Cl and ni > cl > C2 

and niCi > riicj + ri 2 C 2 and n2cl > riicj 4- n2C^ 

Theorem 6.1 is nothing more than a summary of our previous results. For 
each component of Table 5.1 with (z) being an equilibrium we report the 
conditions of universal risk dominance of {i). 

For the case of two equilibria it is obvious that the respective condition 1, 
2, or 3 is both necessary and sufficient. For the case of three equilibria, namely 
(i), (ii), and {in), we have imposed the rather stringent solution requirement 
that {%) has to risk dominate the other two strict equilibria. According to the 
Harsanyi and Selten theory of equilibrium selection the strict equilibrium (z) 
could also become the solution in case of cyclical risk dominance relationships. 
Such a possibility has been neglected so far since we wanted to explore the 
conditions when the Esperanto Solution (z) is the unique solution in the sense 
of that it pairwise risk dominates all other strict equilibria. 

Since the minimal formation of (Harsanyi and Selten, 1988) spanned 
by the two strict equilibria (zz) and (zzz) is G^, Theorem 4.1 implies that (zz) 
risk dominates (zzz) if rzicf < n2cl and vice versa. From this result it follows 
that risk dominance in is always transitive if three equilibria coexist. If, 
for example, (z) risk dominates (zz) and (zz) risk dominates (zzz) which in turn 
risk dominates (z) this would require 

UiCi + rz2C2 < niCj < n2cl < riic? + ri2C2 

and thus a contradiction. Hence (z) (learning Esperanto) is the solution of 
whenever it risk dominates (zz) and (zzz). 

Corollary 6.2. The conditions of Theorem 6.1 are not only sufficient but 
also necessary for the solution (z) of the game G'^ . 



7 . Discussion 

In most cases the risk dominant equilibrium is the more efficient one. The 
only exception is when the strict equilibria (o), i. e. no learning at all, and 
(z), i. e. learning of Esperanto, coexist. In this case (z) is more efficient than 
(o) . For the parameter region satisfying 

2niri2 > rzicj + n2(^ > nirz2 

the more efficient equilibrium (z) is, however, risk dominated by (o). Thus 
the solution is not always efficient. In all other comparisons of two strict 
equilibria, the more efficient equilibrium risk dominates. 

In order to explore the implications of Theorem 6.1, we make the simpli- 
fying assumptions that 




266 



Cl = C 2 =: c and = cl =: /3c. 
We further define 



n := 7ii 
and write 



TI2 = OiTl. 

That is, /? measures the learning costs of an ethnic language in terms of the 
cost of learning Esperanto and a the size of country two in terms of country 
one. Without loss of generality we can assume that a > 1. 

Under these assumptions, the conditions for the Esperanto Solution in 
Theorem 6.1 reduce to the following: 

j3 > 1 a and n > (7.1) 



and the efficiency condition (5.1) to 

P > 1 -i- a and n > f 1 + — 
2 V a 



(7.2) 



As also explained above, (7.1) implies (7.2). In the introduction we noted that 
the size of /3 is about 3. Hence, in our two-country world the Esperanto Solu- 
tion is the unambiguous solution if the second country is not more than twice 
the size of the first one and if the advantage of communicating is sufficiently 
large in relation to the language learning costs, indeed, greater than what is 
necessary in order to make language learning efficient. Hence, an imposed so- 
lution would under certain conditions be more efficient than the (universally) 
risk dominant equilibrium. We conjecture that in the many-country case the 
condition for the Esperanto solution would be even less strict than condition 
(7.1) since the largest alternative ethnic language would have a smaller frac- 
tion of total population than in the two-language case where it, by necessity, 
has more than half. 

This brings us back to the question in the title. We feel that the answer 
may have to be sought outside equilibrium selection theory, but not neces- 
sarily outside game theory, namely in an evolutionary approach. Language 
learning decisions are made individually, gradually and in a decentralized 
manner under the influence of the prevailing status quo, not the final equi- 
librium. The usage of various languages will hence develop gradually, as an 
individual, acting according to our payoff function in sections 4 and 5, will 
only decide to learn a language if a certain minimum critical mass of speakers 
already exists. The more users a language has, the larger is the incentive for 
an individual to learn it. This is probably the main reason for the rapid in- 
crease in the use of the American idiom in international communication in the 
world of today and the restriction of Esperanto to a relatively small group of 
specially motivated users. The move to the risk dominant equilibrium would 
require a coordinated collective action in the international arena. 




267 



A. Proof of Theorem 4.1 



To determine the bicentric priors for the equilibria (n) and {iii) one has to 
compute the minimal value of z for which 

ni — 1 -f n2 — Cj > ni - 1 4- (1 - z)ri2 



holds. The value ^ ^ determines the a priori-probability pi = by 

which the inhabitants i of country 1 learn country 2’s language. By symmetry 
one derives p 2 = - as the a priori-probability by which inhabitants j of 
country 2 learn the language of country 1. 

Given that an inhabitant i of country 1 expects all other players to use 
their bicentric prior strategy his best reply is Si = 1 if 



ni — 1 4- ri2 — Cl > ni — 1 






The sum corresponds to the expected value of the binomial distribution 



-P)" *=p-n. 



Thus (A.l) is equivalent to 

ni — 1 4- n2 - Cl > ni - H ri2 

ni 

Ti2cl > nic\. (A. 2) 

Condition (4.8) can be derived analogously. ■ 



B. Proof of Lemma 6.1 



Assume that ri 2 > cf and thus ri 2 > cj as well as ni > C 2 are satisfied, i.e. 
both (i) and (it) are strict equilibria of G^. Prom 

Til — 1 4- ZU2 — c? > ni — 1 4- ri2 — Cl 



one can derive the minimal z with 



z = 



ri 2 — cf 4- Cl 
ri2 



and obtains the bicentric prior probability 



Pi = 1 - 



7^2 — Cl 4 - Cl 



Cl Cl 



712 ^2 

for learning Z = 0 in country 1. For country 2 the corresponding condition is 
ri2 — 1 4- ni — C 2 > n2 - 1 4- (1 — z)rii^ 




268 



i.e. the a priori-probability of player j in country 2 for learning Z = 0 is 

1 _ 1 ^ ni-c° 

Til rii 

The condition for 5^ = {pi.^i) = (1, 0) versus Si = (0, 1) to be a best reply to 
the prior strategy combination is 

■ • ri2 

(B.l) 

whereas for Sj = (pj^Sj) = (1,0) versus sj = (0,0) as a best reply one must 
have 



ni — 1 -f- 712 — Cl 


< Til — 1 — ^1 “b 


712 - cj 




Til 


riiCi + 712C2 


< riicl 



ri2 — 1 -f ni — C2 > ri2 -1 



.2 _i_ ^0 



k=0 



0 ^ n 2 - cf + c\ 

Til — C2 > • Til 

Tl2 

riicl > nicJ+n2C2 



,0 \ 



(B.2) 



since people in country 1 either learn / = 0 or / = 2. 



C. Proof of Lemma 6.2 

Prom 

ni — 1 + Z7i2 — Cl >rii — 1 

the prior probability of players i in country 1 to learn / = 0 is 




Similarly, for country 2 



is players j’s prior probability for learning I = 0. The condition for Si = 
{pi, Si) = (1, 0) versus Si = (0, 0) to be a best reply to p = {Pi,Pj) is 




269 



ni — 1 
0 

nicj -f- 712C2 



< 


rii — 1 


-c\ + \ 


< 


0 1 


m - 


-Cl + 


ni 




< 


niri2. 





ri2 

A:=0 
- -712 



^ri2 
i k 



J \ riij \nij 



,0 \ Tl2—k 



(C.l) 



The condition for sj = (pj^Sj) = (1,0) versus sj = (0,0) to be a best reply 
to p is exactly the same. It follows from exchanging indices in (C.l). ■ 



References 



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en la fremdlingvoinstruado in Internacia Pedagogia Revuo 20, pp. 1-9 
Prank, Helmar; Szerdahelyi, Istvan (1976): Zur pddagogischen Bestimmung relativer 
Schwierigkeiten verschiedener Sprachen in Grundlagenstudien aus Kyber- 
netik und Geisteswissenschsft 17, Heft 2, pp. 39-44 
Harsanyi, John C; Selten, Reinhard (1988): A General Theory of Equilibrium 
Selection in Games MIT Press, Cambridge Massachusetts 
Harsanyi, John C (1995): A New Theory of Equilibrium Selection for Games with 
Complete Information in Games and Economic Behavior 8, pp. 91-122 
Selten, Reinhard (1995): An Axiomatic Theory of a Risk Dominance Measure for 
Bipolar Games with Linear Incentives in Games and Economic Behavior 
8, pp. 213-263 

Selten, Reinhard; Pool, Jonathan (1991): The Distribution of Foreign Language 
Skills as a Game Equilibrium in Reinhard Selten (ed.) Game Equilibrium 
Models IV, Springer- Verlag, pp. 64-84 

Selten, Reinhard; Pool, Jonathan (1995): Enkonduko en la Teorion de Lingvaj 
Ludoj — Cu mi lernu Esperanton (in Esperanto and German with extended 
abstracts in English, French, Italian, and Polish) Akademia Libroservo, Berlin, 
Paderborn 




Are Stable Demands Vectors in the Core of 
Two-Sided Markets? Some Graph-Theoretical 
Considerations 



Benny Moldovanu ^ 

* Department of Economics, University of Mannheim, Seminargebaude A5, 
68131 Mannheim, Germany 



Abstract. Stable demand vectors for games in coalitional form satisfy three 
requirements: 1) Players* demands are such that there are no leftovers in any 
coalition. 2) Each player’s demand is such that he can find at least one 
coalition, including himself, that can satisfy the demands of its members. 3) 
No player i is dependent on player j in the sense that i always needs j to form a 
coalition satisfying its members’ demands, while j does not need i in order to 
do so. Since the stability condition 3 focuses on pairs, we look at NTU games 
of pairs such as those arising from two-sided markets. For each demand 
vector we define a bipartite graph, and we use the Kbnig-Hall Theorem and 
its corollaries to study properties of those graphs. In particular, stable 
demands are shown to be in the core if the number of agents on each side of 
the market is not greater than four. 



1. Introduction 

The main purpose of this paper is to illustrate the use of graph - theoretical 
methods for the study of relations between the set of stable demand vectors 
and the set of core allocations in games arising from two-sided markets. 

The definition of stable demand vectors for general games in coalitional 
form is based on the following intuition: Each player sets a demand for his 
participation in a coalition. Three stability criteria are imposed: 1. 
Maximaliiy requires that the demand are maximal in the sense that no agent 
can raise his demand without excluding himself from any coalition that can 
satisfy the demand of its members. 2. FeasihilHy requires that each agent can 
find a coalition that includes him, and that satisfies the demand of its 
members. 3. Independence requires that no agent is dependent on another one 
in the sense that the first agent always needs the second to ensure feasibility of 
his demand, but the second agent can find a coalition that includes him and 
that satisfies its member’s demands, but excludes the first agent. The 
terminology we use here is due to Selten (1981). 




271 



Demand vectors exhibiting the above mentioned properties have been 
studied, among others, by Albers (1979) , Bennett (1983), Bennett and Zame 
(1988), Moldovanu and Winter (1994), and Reny and Wooders (1993). 
Bennett and Zame proved existence of stable demand vectors for a large class 
of NTU games in coalitional form. Moldovanu and Winter audomatized the 
set of stable demand vectors using consistency axioms. The axiomatization is 
practical identical to the axiomatization of the core of NTU games, but the 
deHnition of reduced games is different, reflecting the fact that demand 
vectors need not be feasible for the grand coalition. We also note that stable 
demand vectors often appear in relation to equilibrium payoffs in 
non-cooperative games of coalition formation and payoff division (see Selten, 
1981, Binmore, 1985, Moldovanu, 1992). 

Since the Independence property focuses on pairs, it is natural to look at the 
implications of the three requirements described above for NTU games where 
the agents are divided in two disjoint sets (say, buyers and sellers), and where 
the effective coalitions are those involving exactly one buyer and one seller. 
Since many real-life markets are in fact bilateral, these types of games have 
drawn a huge amount of attention. In particular, Shapley and Shubik (1971) 
and Kaneko (1982) show that these games are balanced (for the TU and NTU 
case, respectively). Hence, the core is no empty, and, moreover, the core 
coincides with the set of competitive equilibrium allocations. 

Each stable demand vector induces a graph where the vertices are the 
agents, and where a buyer and a seller are connected if their demands are 
compatible. By studying the properties of such graphs we can establish 
whether the given stable demand vector is in the core or not. The main tool is 
the celebrated Konig-Hall Theorem that establishes a necessary and sufficient 
condition for the existence of a perfect matching in a bipartite graph. The 
conditions on the induced graphs implied by the definition of stable demand 
vectors are, in general, rather weak. We first derive several properties of 
induced graphs that are sufHcient for the respective stable demand vector to 
be in the core. We then use these results to show that stable demand are 
necessarily in the core if the number of agents on each side of the market is not 
greater than four. The result does not hold anymore for games with larger sets 
of agents on each side of the market. 

The paper is organized as follows : In section 2 we introduce NTU games in 
coalitional form and stable demand vectors for these games. In section 3 we 
look at the graphs induced by stable demand vectors in NTU games of pairs, 
and study their relations to graphs induced by core allocations. 



2. Games in Coalitional Form 

Let N = {l,2,...n} be a set of players. A coalition S is a non-empty subset of 




272 



N. I S I denotes the cardinality of S. A payoff vector for N is a function 

S 

x: N — > R. We denote by x the restriction of x to members of S. The 

S S 

restriction of R to vectors with non-negative coordinates is denoted by R+. 

The zero vector in R^ is denoted by 0^ . For x , y G R^ we write x > y if x^ > y^ 

c 

for all i € S . Let K be a subset of R^ . Then, intK denotes the interior of K 
relative to R^ and 5K denotes the set K \ intK. 



Definition 2.1: A non— transferable utility {NTU) game in coalitional form is 
a pair (N,V) where V is a function that assigns to each coalition S in N a set 



V(S) C R such that : 

g 

V(S) is a non-empty, closed, and bounded subset of R ^ ; (2.1) 

ViGN, V(i) = 0* ; (2.2) 

If y G R+ , X G V(S) zuid x > y then y G V(S) ; (2.3) 

If X , y e dV(S) and x > y then x = y . (2.4) 



A game with transferable utility (TU) in coalitional form is a pair (N,v) where 
V is a function that assigns to each coalition S in N a real non-negative 
number v(S). 

Condition 2.2 is a normalization. Condition 2.3 ensures that utility is freely 
disposable. A set V(S) satisfying 2.3 is said to be comprehensive. Condition 
2.4 requires that the Pareto-frontier of V(S) coincides with the strong 
Pareto-frontier, and it implies that utility cannot be transferred at a rate of 
zero or infinity. A set V(S) satisfying 2.4 is said to be non— leveled. 

We next define the two concepts that play the major role in our analysis: 

N 

Definition 2.2: Let (N,V) be an NTU game, and let x G R . x can be improved 

upon if there exists a coalition S and a vector y^ G V(S) with y^ > x^ for all i G 
S. The core of (N,V), C(N,V), is the set of all x G VfN) that cannot be 
improved upon. 



N 

Definition 2.3: Let d G R . Denote: 

V k G N, Fk(d) = {S I S C N, k G S, andd^ G V(S)} 

A vector q is a stable demand vector'll it has the following properties: 



(2.5) 




273 



1) d cannot be improved upon 


(2.6) 


2) V i 6 N, 3 S C N such that i 6 S and d® € V(S) 


(2.7) 


3) V i,j € N .with i ^ j, 




it is not the case that Fi(d) C Fj(d) 


(2.8) 



3. Wliere4 AreFeWjButS AreMany 

The fact that the stability condition 2.8 focuses on pairs suggests a study of 
"games of pairs" such as those arising from two-sided markets . In such games 
there are two distinct groups of players, say buyers and sellers , and the only 
relevant coalitional "pies'* are those for singles, and those for coalitions 
consisting of exactly one buyer and one seller. Demands can be interpreted as 
reservation prices at which the agents are willing to buy and sell, respectively. 

We first need some notation : Let B a non-empty set of buyers, and let S be a 
non-empty set of sellers. We assume that B fl S = 0 . We assume here for 
simplicity that | B | = | S | = n . We denote agents by i, j, etc..., unless we 
want to be specific about their type, in which case we use the notation bi , sj, 

etc... To each agent i6N = BUS we associate the set V(i) = 0^ . To each 

N 

pair T= {bi,sj} we associate a set V(T) C R+ , where V(T) satisfies the 

standard conditions for NTU games. In a two-sided market V(T) represents 
the set of utility vectors arising from feasible trades between the members of 

the pmr. If Q is a pair of two buyers, or a pair of two sellers, then V(Q) = 0^ . 
For a coalition S with more than two players let II(S) represent the set of all 
possible partitions of S in in mixed pairs and singletons . We define then V(S): 
V(S) = U{ * V(T)|<S)Gn(S)} (3.1) 

TGt(S) 

For the definition of a TU game of pairs we proceed in a similar manner : To 
each coalition T with no more than two players we associate a non-negative 
real number v(T). If T is a singleton, or if T is a pair of two buyers, or a pair of 
two sellers, then v(T) = 0 . For a coalition S with more than two players we 
define v(S) as follows : 

v(S) = max ( E v(T) | x(S) 6 n(S) ) (3.2) 

T6t(S) 

The core of any game of pairs (TU or NTU) is non-empty (see Shapley and 
Shubik, 1971 , and Kaneko,1982) since these games are balanced. Moreover, 
for two-sided markets, the core coincides with the set of competitive 
equilibrium allocations. 

We now recall several graph-theoretical concepts (see, for example, Berge 




274 



1976). An undirected graph G is a pair (X,E) where X is a set of vertices and 
E is a family of elements of the Cartesian product X x X called edges. Two 
vertices are adjacent if they are connected by at least one edge. Two edges are 
adjacent if they have at least one endpoint in common. The degree of a vertex 
X is the number of edges with x as an endpoint. A graph is simple if it has no 
edges of the form (x,x) and if no more than one edge joins any two vertices. A 
graph is bipartite if its vertices can be partitioned into two sets Xi and X 2 such 
that no two vertices in the same set are adjacent. Given a simple graph 
G = (X,E), a matching is a set Eq C E such that no two edges in Eq are 
adjacent. A vertex x is saturated by a matching Eq if an edge of Eq contains x 
as an endpoint. A perfect matching is a matching that saturates all vertices of 
G. 

N 

For a game of pairs (N,V), and for a vector d G R + we define the graph 

G(N,V,d) as follows; the set of vertices is the set of agents (buyers and 
sellers) ; a buyer bi and a seller sj are connected by an edge if their demands 

are feasible, i.e., if (d^\ d*^) G V(bi, sj). Note that G(N,V,d) is a simple 
bipartite graph. Given a graph G(N,V,d), we define sets of options as follows: 

OPd(bi) = { sk I sk G S and d“^) 6 V(bi, sk) } (3.3) 

OPd(sj) = { bk I bk G B and d®->) G V(bk, sj) } (3.4) 

(The set of options depends of course also on the game (N,V), but we omit this 
dependence whenever confusion cannot arise.) Clearly, the set of options for a 
an agent i is simply the set of vertexes adjacent to vertex i in G(N,V,d) , and 
the degree of the vertex i , degd(i), is given by | OPd(i) | • 

We now use the graph-theoretical methodology to establish some relations 
between the core and the set of stable demand vectors for a game of pairs 
(N,V). To avoid a tedious case-differentiation, we look in the sequel only at 

stable demand vectors d such that d^ > 0 for all i G N. Since V(i) = 0^ , we 
have for such demand vectors that 

Vi G N, degd(i) > 1 in G(N,V,d); (3.5) 

Proposition 3.1: Let (N,V) be an NTU game of pairs, and let d be a stable 
demand vector such that Vi G N, degd(i) > 1 in G(N,V,d). The induced graph 
G(N,V,d) has the following property: For any pair i,j of adjacent vertices 

1. either degd(i) = degd(j) = 1 , or 

2. degd(i) > 2 and degd(j) > 2. (3.6) 

Conversely, let d G R^ such that V b^ G B, V s j G S , (d^\ d^^) ^ intV(bi, sj), 

and such that the graph G(N,V,d) satisfies conditions 3.5 and 3.6. Then d is a 
stable demand vector. 




275 



Proof: Assume that bi and sj arc connected by an edge in G(N,V,d) , and 

that, without loss of generality, degd(bi) > 2. Hence (d^^, d®J) G V(bi, sj), 

and, moreover, there exists sk ^ sj such that (d^\ d^^) G V(bi, sk). Assume 
that degd(sj) = 1. Then = ({bi,sj}) , while D ({bi,8j},{bi,sk}) . 

Hence F C F, , and this yields a contradiction to condition 2.8 in the 
8j ^ bi ’ 

definition of stable demand vectors. 

The proof of the converse part is inmiediate by the definition of stable 
demand vectors. Q.E.D. 

Proposition 3.2: Let (N,V) be an NTU game of pairs, and let d G such 

that V bi G B, V s j G S , (d^\ d*^) intV(bi, sj). Then d G C(N,V) if and only 
if the graph G(.N,V,d) has a perfect matching. 



Proof: A vector d G R^ such that V bi G B, V sj G S , (d^\ d®J) intV(bi, sj) is 

in the core if and only if d is feasible for the grand coalition, i.e., if and only if 
d G V(N). The vector d is feasible for the grand coalition if and only there are 
n distinct pairs (each containing a buyer and a seller) such that the demands 
of the members of each pair (given by d) are feswible for that pair. Clearly, the 
graph G(N,V,d) has a perfect matching if and only if n such distinct pairs 
exist. Q.E.D. 



N 

Proposition 3.3: Let (N,V) be an NTU game of pairs, and let d G R + such that 

V bi G B, V s j G S , (d^\ d^*^ ) ^ intV(bi, sj). Then d G C(N,V) if and only if 

for all sets B’ C B , | U OPd(i) | > | • 

iGB * 

Proof: The result follows immediately from Proposition 3.2 and from the 
Konig— Hall Theorem (see Berge, 1976). Note that the "only ifi* part is trivial. 
The interesting part is the "if" one. Q.E.D. 

Having in mind Proposition 3.3, we observe that the special condition 
imposed on a graph G(N,V,d) whenever d is a stable demand vector 
(condition 3.6) is very weak. Accordingly, we cannot generally expect stable 
demand vectors to be in the core for games of pairs. However, such a result 
can be obtained if the number of agents on each side of the market is not too 
large. Denote by [A] the integral part of a real number A. We first derive some 
preliminary results which yield sufficient conditions for the existence of 
perfect matchings in G(N,V,d). 




276 



Proposition 3.4: Let (N,V) be an NTU game of pairs, and let d be a stable 

demand vector such that d^ > 0 for all i G N, and such that 

Vi 6 N , degd(i) < 2 (3.7). 

Then there exists a perfect matching for G(N,V,d) , and d G C(N,V). 

Proof: By condition 3.5, degd(k) > 1 for each vertex k m G(N,V,d). Let i be a 
vertex such that degd(i) = 1 . Hence i is adjacent to a unique vertex j. By 
condition 3.6, we know that j must be adjacent to a unique vertex, hence j is 
only adjacent to i. Denote by N’ the set of all vertexes k such that degd(k) = 
1. The previous argument shows that we can find a matching Eq that 
saturates N’. We now look at N \ N’ . Using again condition 3.6, we obtain 
that no vertex in N \ N* is adjacent to a vertex in N*. We now look at the 
graph G(N \ N*,V,d) which is the restriction of the graph G(N,V,d) to the set 
N \ N*. By construction, the degree of any vertex in G(N \ N*,V,d) is equal to 
its degree in G(N,V,d). By assumption, for any vertex k in N \ N* we have 
degd(k) = 2. Hence G(N \ N’,V,d) is a graph where all vertexes have the same 
degree. By a corollary of the Kbnig-Hall Theorem, (see Berge, 1976) any 
simple bipartite graph has a matching that saturates all vertices with 
maximum degree. Hence there exists a matching Ei that saturates N \ N’ . 
Since no two elements of Eq and Ei, respectively, have endpoints in common, 
the set Eo U El is a perfect matching for G(N ,V,d). By Proposition 3.2, 
dGC(N,V). Q.E.D. 

N 

Proposition 3.5: Let (N,V) be an NTU game of pairs, and let d G IR + such 

b* s* 

that V bi G B, V sj G S , (d d ^) ^ intV(bi, sj). Moreover, assume that: 

V i e N, degd(i) > - - • (3-8) 

Then there exists a perfect matching for G(N,V,d) , and d G C(N,V). 

Proof: By Propositions 3.2, 3.3, it is enough to show that for all sets B’ C B , 

I U OPd(i) I ^ |B’| . The assertion is clearly satisfied for sets B’ such that 
iGB * 

I B* 1 < ' Assume then that B’ is such that |B’| > 

now claim that | U OPd(i) | = n ^ | B’ | * To see that, assume, on the 

iGB » 

contrary, that there exists a vertex j G S, such that j is not adjacent to any 
vertex i G B*. This means that OPd(j) fl B’ = 0 . We have then 

I OPd(j) U B’ I = I OPd(j) I + I B’ I > 2- [ “^^] ^ = i B I • This yields a 

contradiction to (OPd(j) U B’) C B . Q.E.D. 




277 



We now have: 



Proposition 3.6: Let (N,V) be a game of pairs with four or less agents of each 



type, and let d be a stable demand vector such that d^ > 0 for all i 6 N. Then 
d 6 C(N,V) . 



Proof : By condition 3.5, we know that, for each vertex k in G(N,V,d), 
degd(k) ^ 1 . As in the proof of Proposition 3.4, we can find a matching Eq 
that saturates the set N’of vertices k such that degd(k) = 1. By condition 3.6, 
there is no vertex in N \ N’ that is adjacent to a vertex in N*. We now look at 
the graph G(N \ N*,V,d) which is the restriction of the graph G(N,V,d) to the 
set N \ N*. By construction, the degree of any vertex in G(N \ N*,V,d) is equal 
to its degree in G(N,V,d). For any vertex k in N \ N* we have degd(k) > 2. 

I I 1 . obtain by Proposition 3.5 that 



Since 2 



- [ JdtL' 
“ [ .2 



there exists a matching E\ that saturates all vertices in N \ N’ . As in the proof 
of Proposition 3.4, the set Eq U Ei is a perfect matching for G(N ,V,d). By 
Proposition 3.2, d G C(N,V). Q.E.D. 



The next example shows that the bound on the number of agents of each 
type in Proposition 3.6 is tight. 



Elxample 3.7: Let (N,v) be a TU game of pairs with five agents on each side of 
the market defined as follows: 

v(bi) = v(sj) = 0 , 1 < i,j < 5 (5.7) 

{ i = l,2 and j = 3,4,5 

(5.8) 

i = 3,4,5 and j = l,2 

v(bi,sj) = 1, otherwise (5.9) 

Then d = (l,l,...l) is a stable demand vector, but d(N) = 10 > v(N) = 9 . 
(four pairs can form obtaining 2, one pair obtaining 1). Thus d is not feasible 
and d C(N,v). Q.E.D. 

If a payoff vector (and in particular a stable demand vector) is not in the 
core, then, at the given "prices" some players are "over-demanded" and some 
are "under-demanded". Respective adjustments should be made in the 
demands (over demanded players increase their demands, under-demanded 
players decrease). In Example 3.7 , sellers si,S 2 and buyers bi,b 2 are 
over-demanded , while the other players are under-demanded. Kamecke 
(1988) constructs an algorithm where the above intuition is repeatedly used 
for making adjustments in the agents* payoffs. His algorithm arrives after 
finitely many steps to an allocation that is the most favorable core allocation 




278 



for one side of the market. 

Finally, note that several possible extensions of the stability condition used 
in the definition of stable demand vectors (condition 3.6) become now 
apparent. Here we regarded as unstable situations in which one agent in a 
potential trade has at least one outside option, but his partner has none. 
Similarly, we could define new notions of stability requiring that the numbers 
of outside options available to each trader in a matched pair are, in some 
sense, close to each other (say, equal). Assume, for example, that a demand 
vector cannot be improved upon and that each agent has exactly k options 
(this could be such a “new" definition of stability). Then, by the corollary to 
the Konig-Hall Theorem that we used in the proof of Proposition 3.4, that 
demand vector must be in the core. 

References 

Albers, W., (1974) "Zwei Lbsungskonzepte fiir Kooperative 

Mehrpersonenspiele, die auf Anspruchsniveaus der Spieler Basieren", 
OR-Verfahren 21, 1-13. 

Bennett, E., (1983) "The aspiration approach to predicting coalition 
formation and payoff distribution in side-payments games". International 
Journal of Game Theory 12, 1-28 

Bennett, E., Zame, W.R., (1988) "Bargaining in cooperative games". 
International Journal of Game Theory 17, 279-301. 

Berge, C., (1976) "Graphs and hypergraphs", North-Holland, Amsterdam. 
Binmore, K.G., (1985) "Bargaining and coalitions" in "Game Theoretic 
models of bargaining" ed. Alvin Roth, Cambridge University Press, 
Cambridge. 

Kamecke, U., (1988) "Modeling competition in matching markets with 
non-cooperative game theory", dissertation, University of Bonn. 

Kaneko, M., (1982) "The central assignment game and the assignment 
markets", Journal of Mathematical Economics 10, 205—232 
Moldovanu, B., (1992) "Coalition-proof Nash equilibria and the core of 
three-player games". Games and Economic Behavior 4, 565-581 
Moldovanu, B., Winter, E., (1993) "Consistent demands for coalition 

formation", in "Essays in Game Theory in honor of M. Maschler" ed. N. 
Megiddo, Springer Verlag, Mannheim. 

Reny, P., Wooders, M.H., (1993) "The partnered core of a game without side 
payments", forthcoming in Journal of Economic Theory 
Selten, R., (1981) "A non-cooperative model of characteristic function 
bargaining" in "Essays in Game Theory and Mathematical Economics in 
honor of O. Morgenstern" eds. V. BOhm, H. Nachtkamp, 
Bibliographisches Institut Mannheim. 

Shapley, L.S., Shubik, M., (1971) "The assignment game: The core". 
International Journal of Game Theory 1, 111-130 




The Consistent Solution for 
Non- Atomic Games* 

Guillermo Owen 

Department of Mathematics, Naval Postgraduate School, Monterey, CA 93943 



Abstract. We study here the relation between the consistent solution to 
large games, and economic equilibria. Using a non-atomic approximation, 
we show that homogeneous Pareto-optimal allocations are consistent if and 
only if they correspond to competitive equilibria. 



Solution 

We will consider here a market game with m types of players and n goods. 
Each player of type « (1 < ^ < m) has an initial endowment 

of goods, and a utility function 



Hi I ^ R 



defined on bundles (n-tuples) of goods. We assume that each Ui is monotone 
non-decreasing in each variable, concave, and differentiable (at least at most 
places) . 

A bundle is simply an n-tuple = ( 2 ^ 1 , •••, We will allow negative 
values (i.e. short sales are permissible), though in practice the utility func- 
tions can be used to make short sales (at least beyond some point) very 
disadvantageous. An allocation is an m-tuple 

Z =< > 

of bundles. (This allocation assigns the bundle to each player of type i.) 

A coalition is represented by a vector X = {x \ , ..., Xm), where the X{ are 
non-negative integers: xi is the number of players of type i in coalition X. 
An allocation Z is feasible for coalition X if it satisfies 
( 1 ) 

m m 

i=l t=l 

for each j {1 < j < n). 




280 



For a given coalition X, the usual definition of Pareto- optimality holds: Z 
is Pareto-optimal for X if it is feasible for X, and there is no F, also feasible 
for X, giving at least as much utility to each type as Z does, and strictly 
more utility to at least one type. (Essentially, this means that no mutually 
profitable trade is possible starting from X.) Assuming differentiability, the 
condition for a feasible Z to be Pareto-optimal is the existence of two vectors, 
V - (Pi,-,Pn) and q = (gi, satisfying 

(2) 

for each player type i and good j, where Uij is the partial derivative 

Under the assumptions of monotone utility functions, the vectors p and q 
can be assumed positive (or at least non-negative). Moreover, since it is only 
the products qipj that matter, it is always possible to normalize one of the 
two vectors p and q. 

In equation (2), the components pj are shadow prices. The qi represent 
the rate at which utility can be transferred between players: a player of 
type i can receive a small amount eqi of utility if one of type k gives up the 
corresponding small amount eqk (and this can be done by a physical transfer 
of goods worth e units of money at the going shadow prices). The meaning 
of the shadow prices, together with Pareto-optimality, is the standard one: if 
the goods axe for sale at the going prices, and players are holding the bundles 
X, then no player can gain utility by a costless trade (i.e one which does not 
increase the shadow price- value of his bundle) . 

A Pareto-optimal allocation X is a competitive equilibrium if, apart 
from (1) and (2), it satisfies 

( 3 ) 

n n 

F Pi^J 

j=l j = l 

for each i. 

A consistent solution to an n-person NTU game (X, U), as defined 
in [Maschler/Owen 1992] and characterized in [Hart/Mas Colell 1992], is a 
collection of payoff vectors y{S) = {yi{S))i^s, one for each possible coalition 
S C N, which satisfies the conditions of Pareto-optimality (each y{S) is 
Pareto-optimal in set V{S)) and balanced contributions: for each S C N, 
and each i G 5, 

( 4 ) 

hi{S) iviiS) - ViiS - k)} hk{S) {yk(S) - y,(5 - *)} 
kes kes 

k^i k^i 

where h{S) is the normal vector to the Pareto-optimal surface of V{S) at 
point y{S). 




281 



For our games, we can think of the consistent solution as, instead, a 
function defined over the set of non-negative m- tuples of integers, which 
assigns, to each coalition (m-tuple) X, an allocation Z{X), These allocations 
must all be feasible and Pareto-optimal (for the respective X). Condition (4) 
will now take the rather complicated form that, for each X, and each i such 
that Xi > 1, 

(5) 

Xkhi{X){ui{z\X)) - Ui{z\X - e*=))} = 

keM 

Xkhk{X){uk{z'‘iX)) - Ukiz'^iX - e'))} 

keM 

where, as before, h{X) is the normal vector to the pareto-optimal surface 
of V{X) at point u{Z{X))^ and where is the vector (in R^) whose kth 
component is 1, while all other components are 0. 

Although the X are naturally restricted to being m-tuples of integers, we 
may, for very large games, assume that the X are, for all practical purposes, 
m-tuples of reals. ^ Thus, for large games, we assume that the consistent 
solution is a mapping from the space into the space R^^^ of all alloca- 
tions Z. We can then replace the increments by derivatives, so that (5) now 
takes the form 

( 6 ) 

Xkhi{X)^{z\X)) =Y xMX)^{z>^{X)). 

keM ^ keM " 

Now, the standard chain rule for differentiation, 

dui dui 



when combined with (2), takes the form 

(7) 



dui 

dxk 



Y ^iPi4k 

jeN 



where is the partial derivative, 

We note, next, that the vector h is defined only up to multiplication by 
a scalar. Now the components of q (the vector of rates of motion along the 
Pareto-optimal surface), and those of h, the normal vector along this same 
surface, will be inversely proportional. Thus we can assume that each h* = 
and equation (6) now takes the form 
( 8 ) 

Y Y ^kPjZjk =Y ^kPjZji 

keM jeN keM jeN 




282 



for each i such that X{ > 0. 

The allocation mapping Z is homogeneous^ (of degree 0) if, for every 
coalition X and any constant A; > 0, 

( 9 ) 

Z{kX) = Z{X). 

Assuming differentiability^, this is equivalent to saying that, for every X, 
every i E M and every j E N, 

( 10 ) 

— 9. 

keM 

We have thus defined three interesting properties which an allocation 
mapping (i.e. a mapping which assigns a Pareto-optimal allocation to each 
coalition) can have: it can be a competitive equilibrium mapping (equations 
1-2-3), it can be consistent (1-2-8), and it can be homogeneous (1-2-9). The 
question, now, is as to the logical relationship among these properties. In par- 
ticular, we would like to know whether consistent mappings are equilibrium 
mappings, and conversely. It turns out that, in the presence of homogeneity, 
the two are equivalent: 

Theorem. Let Z be a diflPerentiable, homogeneous allocation mapping. 
Then Z is consistent if and only if it is an equilibrium mapping. 

Proof: Consider the feasibility condition (1). Differentiating both sides 
of this equation with respect to Xk , we obtain 

= Wj 

ieM 



or, interchanging i and k, 
( 11 ) 



IT "^*4 = 4 - 

keM 



Suppose now that Z is consistent, so that (8) holds. By the homogeneity 
condition (10), the left-hand side of equation (8) vanishes. On the other 
hand, the right-hand side of (8) can be written as 




and, using (11), this reduces to 



j£N 




283 



Thus, equation (8) reduces to the statement that this last expression 
vanishes. But this is precisely equation (3), and so Z is an equilibrium 
mapping. 

Conversely, suppose Z is an equilibrium mapping, so that (3) holds. Sub- 
stituting (11) in (3), we have 

Y = 0 - 

j£N k^M 

Now, from (10), we also have 

Y Y Pi^kZ% = 0 

jGAT k£M 

and these two together give us (8). Thus Z is consistent. 

The question of the relationship between consistent and equilibrium allo- 
cation mappings thus reduces to the question of their homogeneity. 

Now, the conditions for equilibrium (1-2-3) are point-wise conditions, i.e. 
they hold (or fail to hold) at a single point regardless of what happens at other 
points. Moreover, it is easy to see that if an allocation Z is in equilibrium for 
coalition X, it is also in equilibrium for any coalition kX {k > 0). Thus, given 
an equilibrium mapping (defined on all of i^!^) it is always possible to obtain 
a homogeneous equilibrium mapping: it suffices to consider its restriction 
to some subdomain such as the unit simplex expand this 

restriction to the entire space by equation (9). Thus, from an equilibrium 
mapping we can always obtain a consistent mapping. 

On the other hand, given a consistent mapping, there is no guarantee 
(at least not at this time) that it will be homogeneous. (Note that [Owen 
1994] shows that in general the consistent solution need not be unique, even 
for finite games.) Thus there may be consistent mappings which are not 
equilibrium mappings. 

Of course, it is also important to remember that these are, essentially, non- 
atomic approximations to large but finite games. The question is whether 
the consistent allocations to the finite game will “eventually “ converge to the 
consistent allocation for the non-atomic game. Our analysis leaves that ques- 
tion open, but seems to say that, if it does converge, then the allocation 
thus obtained will be an equilibrium allocation. 



Footnotes 




284 



of mathematical analysis. Certainly, in physics, the answer seems to be in the 
affirmative: fluid dynamics treats both gases and liquids as continuous media 
rather than as consisting of very large numbers of “minuscule billiard balls “, 
and the results obtained seem to flt well with observations. In economics, 
the idea was first developed in detail by Aumann and Shapley [1974] though 
the idea of a continuum of players (or economic traders) had been introduced 
earlier than that; vide the bibliography in [Aumann/Shapley 1974] for prior 
articles in this vein. Of course the question of the validity of such an approxi- 
mation (large finite games replaced by non-atomic games) is a difficult one. 

^ The importance of (degree 0) homogeneity lies in the fact that, if a 
solution is homogeneous, then the economy can expand with no dislocations. 
In effect, it says that, so long as the economy expands in scale (i.e. the 
relative numbers of each type of trader remain constant), then the returns 
to each class of agent will be unchanged. In the non-homogeneous case, 
expansion of the economy, even in scale, would increase the returns to some 
agents while decreasing the returns to other agents. This would give rise to 
the standard cries of unfairness, etc. which characterize economic activities 
nowadays, and generaly create all sorts of difficulties. 

^ Starting with equation (7), we have introduced the partial derivatives 
as important factors in our analysis. Now, in the consistent solution, the 
utility payoffs Ui{X) are, by equation (6), differentiable functions (almost 
everywhere) of the coalition components Xk- The question is whether we can 
assume that, in that same solution, the bundles are differentiable functions 
of the Xfc. 

Certainly, in the most general case, we canot even expect the to be con- 
tinuous, let alone differentiable, functions of the Xk- Let us assume, however, 
that the utility functions U{ are strictly concave. In this case the values of 
being Pareto- optimal, determine the uniquely, and it will follow (by an 
argument similar to that in [Owen 1995, pp. 72-73]) that are continuous 
as functions of the In such case the differentiability (almost everywhere) 
is a reasonable assumption. 

References 

[1] Aumann, R. J., and L. S. Shapley. Values of Non- Atomic Games. 
Princeton University Press, 1974 

[2] Hart, S, and A. Mas-Colell. “Bargaining and Value. “ Discussion Pa- 
per, Center for Rationality and Decision Theory, Hebrew University of 
Jerusalem, 1994. 

[3] Maschler, M. and G. Owen. “The Consistent Shapley Value for Games 
without Side Payments. “ In Rational Interaction, ed. R. Selten, 
Springer- Verlag, 1992, 5-12. 




285 



[4] Owen, G. “The Non- Consistency and Non-Uniqueness of the Consistent 
Value. “ In Essays in Game Theory, ed. N. Megiddo, Springer- Verlag, 
1994, 155-162. 

[5] Owen, G. Game Theory. Academic Press, 1995. 




Finite Convergence of the Core in a Piecewise 
Linear Market Game 



Joachim Rosenmiiller 

Institute of Mathematical Economics (IMW), University of Bielefeld, 
P.O.Box 10 01 31, 33501 Bielefeld, Germany 



Abstract 

We consider a pure exchange economy or "market” such that every player/ agent has 
piecewise linear utilities. The resulting NTU-market game is a piecewise linear corres- 
pondence. Using a version of "nondegeneracy" for games in characteristic function form, 
we exhibit conditions to ensure the finite convergence of the core towards the Walrasian 
equilibrium. 



SECTION 0 : Introduction and Notation 

Equivalence theorems in General Equilibrium Theory and Game Theory state that for 
increasing or large sets of players/agents some solution concepts like the Walrasian 
equilibrium, the Core, or the Shapley value approximately coincide. Earlier, these re- 
sists where formulated within the framework of replicated markets or games (e.g. 
DEBREU - SCARF j3] ). Beginning with AUMANN’s paper [1] , the measure space of 
agents /players turnea out to be a very fruitful concept, see also HILDENBRAND [4] 
and MAS-COLELL [5] . 

However, to some extent there is a third approach based on combinatorial or number- 
theoretical .considerations related to the concept of a "non-degenerate" characteristic 
function in the sense of Cooperative Game Theory. Models of this type deal with a 
finite framework. There is a fixed number r of types (similar to the replica-model) of 

players and integer vectors of the grid IN^ are interpreted as distributions of players over 

the types. The task is to describe certain areas in and certain subgrids such that , 
within these areas the elements of the subgrids yield distributions of players such that a 
certain equivalence theorem holds true. 

An overview over some applications of nondegeneracy and homogeneity as "surrogates" 
for nonatomicity is presented in [9] . More recent evidence is provided in PELEG- 
ROSENMULLER [6] and ROSENMULLER-SUDHOLTER [10] , where it is shown 
that the nucleolus is as well a suitable object for an equivalence theorem in as much as, 
for sufficiently large sets of players in a homogeneous simple game, it coincides with the 
unique representation of this game. 

The above-mentioned methods appear to work smoothly mainly in models which exhi- 
bit a "side-payment" or "transferable utility" character. Clearly, this is so since some 
version of "optimization" or "linear production" is always involved, at least implicitly. 
The present paper is meant to provide a first approach to the NTU-case. We want to 
study a pure exchange economy and the NTU-game it generates in a framework with 




287 



finitely many types and piecewise linear utility functions. Can we exhibit areas in 
such that the Core and the Walrasian Equilibrium coincide by suitably extending the 
definition of a nondegenerate game to the NTU-case and solving the appropriate combi- 
natorial problem? 

As yet the result is only a partial one: for certain classes of pure exchange economies 
the approach is successful, thus, we may formulate a "finite convergence"-theorem. 
However, since nondegeneracy and piecewise linearity of an exchange economy and the 
corresponding NTU-game viewed as a correspondence on distributions of players over 
the types can be established as consistent concepts, it may be that a departure point for 
more insight into the combinatorial structures of equivalence theorems has been 
reached. 

The paper is organized as follows. SECTION 1 studies the NTU-market-^ame resulting 
from a pure exchange economy viewed as a correspondence from generalized profiles of 
coalitions into utility-space.lt turns out that piecewise linearity of the utility functions 
of types renders this correspondence also to be piecewise linear. The regions of linearity 

are cones in - the space of idealized distributions of players. Hence, in SECTION 2, 
we may study the behavior of a face of the correspondence within a region of linearity. 
Such a face is of course a polyhedron of lower dimension and the behavior of its normal 
as a function of profiles of coalitions is crucial. The extremals of the faces again can be 
identified as piecewise linear function (SECTION 3). Thus , gradually we approach the 
behavior of prices corresponding to faces of the NTU-game-correspondence. Eventually, 
in SECTION 4, we link Equilibrium-prices and Core-payoffs. SECTION 5 provides an 
extended example. 

We now start out to provide the setup for our discussion. The piecewise linear pure 
exchange economy and the NTU-game derived from its data. 

Let us consider a pure exchange economy with finitely many types of agents or players, 
for short such an economy shall be called a market Types are indicated by 

p G R = {l,...,r}. The commodity space is IR^, thus j € J = {l,...,m} refers to a commo- 
dity. Each player of type p commands an initial allocation dP 6 (strictly positive). 

We assume the preferences to be represented by a piecewise linear utility function 
which, for type /?, is denoted by u^ : — » IR . More precisely, let us assume that there 

is a finite set L (not depending on 6 R without loss of generality) and vectors c^^ € 

(/o € R, 1 € L) such that, for any x 6 R^, 

(1) u^(x) = min c^^x + d^^ = min h^\x), (where h^^(x) := c^^x + d^^ for x € R^). 

l€L l€L 

We assume c^^ > 0, d^^ > 0 (p 6 R, 1 6 L) thus, each u^^ is positive, strictly monotone 
and, of course, continuous. 



If we introduce the convex closed polyhedron = {x G R[J^ | h^^(x) < h^^ (x) (1* GL)} 
for some /O G R and 1 G L, then u^^ equals h^^ within H^^ and R^ is covered via 




288 



for each /? € R. There is no loss of generality involved in assuming that 
l€L 

each has nonempty interior. 

Let k = (kj,...,k^) € be a vector of integers. Then the resulting market consists of 
kj players with utility u^ and initial allocation a^,..., k^ players with utility u^ and 
initial allocation a^; that is, is the "k-replica" of (e = (1,..,1) €W^). 

Let s = (sj,...,s^) € Wq := U {0} (some coordinates allowed to be zero) be an integer 
(or 0) vector such that s < k. Then s is the profile of a coalition having s^ players of 
type 1,..., Sj. players of type r. Without offering an interpretation, we consider also 

"generalized profiles", i.e., vectors t6lR^. For any t e (and in particular for 
profiles), the (aggregate) initial endowment is 

(2) a^= E t a^ 

PGR ^ 

and the feasible allocations (assuming equal treatment) are given by 

(3) := {X = I x^ = a‘}. 

Switching to utility space we introduce the notation 

u(X) = (u^(x^),...,u''(x'’)), t * u(X) = (t^ u^(x^),...,tj. u^(x’’)) 
for t G K_j_ , , X G 01.. Finally consider, for t G IR , 

“r“r V “T 

(4) F(t) = {u G I u < t ® u(X) for some X G 01^}, 

then F is a correspondence that maps generalized profiles into subsets of K^, the ele- 
ments of which can be considered as feasible utility vectors to a (generalized) coalition 
with profile t, coordinate p denoting the joint utility for all players of type p. 

The restriction of F to all profiles, i.e., to {s G Wq | s < k} for some k G is denoted by 
k k u 

V = V ’ and called the characteristic function of (the nonside-payment- or NTU- 
game corresponding to) u^. 

SECTION 1: The Market Game as a Piecewise Linear Correspondence 

Since all utilities are piecewise linear, the correspondence F should exhibit a similarly 
simple shape. We expect F(t) to be a convex compact polyhedron for fixed t; thus our 
first aim should be to describe the nature of the extremals. Next, with varying t, it 
turns out that F also behaves "piecewise linear" in the sense that within the interior of 

certain well defined cones in , the extremals are linear functions of t. This expo- 
sition is the aim of the first section. 




289 



For fixed type p the utility function u^ is linear within each of the convex polyhedra 
as depicted in Figure 2., and these polyhedra provide a polyhedral covering of K°^.If 
we consider an (equal treatment) allocation X = > then each x^ is somehow 

located in some , thus refers to the system of polyhedra generated by u^ as ex- 
plained above. 

Next, consider a utility vector u which is an extremepoint of F(t) for some t £ 

Suppose that this vector is generated by some X e via u = t ® u(X). Then we expect 

a tendency of the x^ {p e R) to move towards the boundaries of the roughly 
speaking. That is, next to the inevitable equation E t^x^ = a^, there will be "many" 

equations of the types = 0 and t^h^^(x^) = u^ to be satisfied by X and u respective- 
ly; say for j ^ and \ 

More precisely, let us first formulate the appropriate property for a system of sets of 
indices 3= (Jj^,...,Jj.,L^,...,Lp. 

Definition 1.1. Let 3 = (Jj^,...,J^,L^,...,Lj.) be such that C J (/? e R) and L^ C L 
(p £ R). 3 is said to be an admissible system if the following equations are satisfied. 

(1) E |J |<r(m-l), 

p=i p 

|L |>l(/>eR), S |J |+ E |L I =mr+r-(m-min |J I). 

P p=i P p=i P P&. P 

Now we may establish the connections between extreme points of F(t), admissible 
systems, and generating elements of 01 ^ as follows. 

Lemma 1.2: Let t £ and let u € F(t) be a Pareto efficient ("P.E.") extreme point 

of F(t). Then there exists ^^ 01 ^ and an admissible system 3 = (J^,...,Jj.,Lp...,L^) such 

that (X,u) is the unique solution of the linear system of rm + r equations (in variables 
(X,u) € X R^) given by 

(2) x^ = 0(p6RJeAt h^V)-% = 0,(peR,l€L^), E t x^ = a^ 

Proof: As u is Pareto optimal, it is easy to show that 

(3) 0!j:={XeO!j. t®u(X) = u} 

is a nonempty, compact, convex polyhedron. Let X be an extreme point of 0!^ and pnt 
:= {j € J I x^ = 0} (p €R), l’ := {1 €L I t^ h^\x^) = y (p 6R). 




290 



It is then verified by standard arguments that (X,u) is the unique solution of the corres- 
ponding system (2) (- where has been replaced by Lp. 

Now inspect the coefficient matrix of (2) (note that h^^(x) = c^^x + d^^). Observe that 
the first and last group of equations yield a submatrix of rank 

I J^l 4-...+ 1 J^l + m - min | J | < r m. 

/?€R P 

As the rank of the original matrix is mr + r, it is possible to complete this submatrix in 
order to obtain a square submatrix by choosing (at least r) rows from the middle group 
of (2) in such a way that for each p at least on row corresponding to a variable u^ 

appears. This procedure defines index sets c (p = l,...,r), |L^| >1. Note, that for 
each j there is p such that j i. J^; this justifies the first equation (1). The last equation 

(1) is obtained by counting rows. Note that X is a P.E. allocation which is extreme in 

q.e.d. 

Henceforth, a P.O. extreme point u of F(t) is called a t-vertex, an extreme point ^ 
is called a corresponding node. 

CoroUary 1.3: For every t6K^_^,F(t)isa compact, convex polyhedron. 

For, the number of admissible systems 3 is finite and so is the number of vertices. 

Given an admissible system 3, consider for some t elR^_^ the system of linear equations 

in variables 77^ (p e R), (p € R, j 6 J) given by 

(4) d = 0 (p€R,jeJ^), + =-t d^^p€R.leL^), E f 

•' P P p^K 

we have 

Lemma 1.4: Let 3 be admissible. Then the system (4) has a unique solution if and only 
if (2) has a unique solution. 

"7P 

Proof: Clearly, C|, rj) is a solution of (4) if and only if (..., x^,. ...... ,u) = (...,|- is 

a solution of (2). Obviously, the square coefficient matrix of (4) does not depend on t € 
Hence, it is or is not nonsingular independently of t. 

Definition 1.5: Let us call a system 3 = (Jp...,Jj.,L^,...,L^) nonsingular if it is admiss- 
ible and the coefficient matrix of (4) is nonsingular. 

A nonsingular system obtained by a vertex u via some corresponding X node by means 
of the procedure indicated in the proof of Lemma 1.2 is also called corresponding (to u 
or to (X,u)). 




291 



Lemma 1.6: Let 0 be a nonsingular system. Then there exist an r x r matrix A = 
and r m X r matrices {p = l,...,r) such that, for t 6 the unique solution 

of (4) is given by 

(5) ^ = -At , JP = BPt,{peR) 

Thus, in particular, rj and are linear in t. Of course, the unique solution of (2) is 

equivalently given by 

(6) u = A t , x^ = |- B^ t, (p € R) 

Proof: This follows by Cramers rule (cf.[12] ). 

We denote by int T the interior of a subset T of and by CVCPH T its convex com- 
prehensive hvM 

Theorem 1.7: There exists a finite collection T = {T,...,T’ } of subsets of with the 
following property. 

1. T € T is a closed, convex polyhedral cone with apex 0 e R^ and nonempty 
interior. 

2. R^ = VJ {T I TGT}. 

3. For T, T’ €T the cone T fl T’ is lower-dimensional. 

4. For T G T there exists a (finite) family Q = Q,p of nonsingular systems such 
that, whenever t e T, t > 0 

(7) F(t) = CVCPH {A^t 1 3eQ} 

i.e., F(t) is the convex comprehensive hulloi vectors A'^t, 36 Q. 

Proof: 1st Step: Consider a nonsingular system 3 = L^,...,L^ and the 

corresponding solution (X, u) of (3). Clearly, u € F(t) if X > 0 and h^\x^) > h^^ (x^) 
(p G R, 1 6 L^, r G L-L^). This is equivalent to ^ > 0 and c^^ + t^ d^^ > c^^ + t^ 

d^^ (p G R, 1 G L^, r G L-L^,). Again, if A = A^ and B^ = B^^ {p G R) are determined 

by Lemma 1.6, we conclude that u G F(t) if 

(8) > 0 (p 6 R), ) B^t > ) (/) € R, 1 e L^, 1’ 6 L-L^). 

Now, let us regard the rows of this system (which is linear in t) as to be represented by 
a matrix D = - which depends on 3 only (and can be computed by means of the c^^ 

and B^). 

From this, it follows that 

(9) F(t) = CVCPH {A^t I 3 nonsingular, D^t > 0} 
holds true for t G R_^^* 




292 



2nd Step: The proof proceeds by omitting some of the nonsingular systems and 
rearranging the other ones until the requirements of the theorem are satisfied. See [12] 
for the details. q.e.d. 

SECTION 2 : t-Faces 

The previous section explains the behavior of the correspondence F(.): essentially it can 
be seen as the convex hull of finitely many linear functions in t - provided we restrict 

our observation to an appropriate subcone of We would like to eventually obtain 

the same picture with respect to a certain face (r-1 ~ dimensional subpolyhedron in the 
boundary) of F(t). If necessary, we shall of course restrict ourselves to further subcones, 
however, it would be desirable to obtain a boundary face again as the convex hull of a 
finite number of linear functions. 

First of all we shall have to clear up the relation of extremals in utility space and 

the polyhedral decomposition in - which is specified for each type p (cf. SEC. 1). 
We should expect that the nodes corresponding to extremals of the same face are loca- 
ted within the same 

Lemma 2.1.: Let t e and let (u^)q£Q be a finite set of vertices of F(t) belonging to 
the same P.E. face. Then, for p € R, there is /= /(p) € L with the following property: if, 
for each q € Q, is a corresponding node then, for p 6 R, x^^ 6 (q € Q). That is, 

any choice of extreme points of (a^ (q 6 Q) yields points situated in a common . 

It is sufficient to just sketch the following 

Proof: If, for some p € R, the x^^ are not adjacent to at least one then consider for 

a > 0, (q € Q), S a = 1, the utility vector u = E ck ^ which is an element of 
^ q€Q ^ qGQ ^ 

the face in consideration. Also, X := Sa x^ satisfies X G . However, since the x^ are 

q i 

not adjacent to a common we have 

t u^(x^) = t u^( E a x^^) > t S a u^(x^^) = E q u^ = u 
^ ^ qGQ ^ N€Q ^ ' qGQ ^ ^ 

while, for all remaining p’ G R we have at least >. This contradicts the fact that u is 
P.E. 

Let us use the notation t-faceioi a P.E. face of F(t) which has dimension r-1. A t-face 
is, of course, the convex hull of all the t-vertices it contains. F(t) is the comprehensive 

convex hull of all its t-vertices. We say that x, y G are adjacent (referring to some 
p G R) if there is some 1 G L such that contains both points. 

Corollary 2.2.: 1. If u is a t-vertex, then all corresponding nodes are adjacent. 




293 



2. Let (u^)qgQ (all) t-vertices belonging to the same face, say F^(t), and let be 

a corresponding node (q 6 Q). Suppose, for each q, C L is given by Lemma 1.1. {p = 
l,...,r). Then, for eveiy p eK, 

( 1 ) , Q ‘-2 

Thus, the 5c^ (p e R) are adjacent. 

3. of^ := {X 6 0!^ I u(X) € F^(t)} is a compact convex polyhedron; the extreme points 

— q 

are the extreme points of 0^ (q ^ Q), hence corresponding nodes. The p-coordinates of 
these nodes are adjacent. 

4. If, for p e R, we choose /= /(p) ^ > according to Lemma 2.1., then 

(2) rQ(t) = {t ® E(X): = (t^h^^^) (xl),...,t^h"^")(x^) I X € 

The choice of an index set Q in order to identify a face may seem to be an arbitrary 
one. We shall make an attempt to justify this procedure in SEC. 3. 

The next remark shortly describes the way, the normal of a face depends on the para- 
meter t. 

Remark 2.3.: Let t € a-nd let be a set of r t~vertices spanning a t~face, say 

F^(t), of F(t). In view of Lemma 1.1 and 1.5 there is, for any q 6 Q, an rxr-matrix 
such that u^ = A^t (q € Q) holds true. Let ^ denote the normal of the t-face 

F^(t); this normal can be chosen to be nonnegative and hence can be normalized to 
satisfy + ... + = 1. If so, then J may be obtained as the solution of the linear 

system of equations 

A(ii^ - u^) = 0 (q e Q, q p ) E A = 1 

pGR ^ 

where p is some fixed element of Q.That is 

(3) ACA*! - AP)t = 0, S A =1 

p€R ^ 

Using Cramers rule and expanding determinants we find at once r polynomials in 
tj,...,tj. of degree r-1, say (p = l,...,r) such that 

V‘) = Dj(t) .... .D^(t) 

Next we would like to discuss the behavior of price vectors corresponding to some face 
of F(t). To this end, pick t e and consider some face, say F^(t). Let u e F9(t) 

and let X e be a corresponding node, thus u = t ® u(X) holds true. Let J denote the 

normal at F^(t)in u. 




294 



Since u is Pareto efficient, we know that 'SJu = t u^(x^) = max { E^ t u^(x^) 
I X e , and by a standard Kuhn - Tucker - argument there is a joint tangential 

hyperplane for the graphs of all functions ^ _u^{ ) at 5E^ . (The constraints defining 01. 
include the t^ - therefore the t^ do not appear in the description of these hyperplanes.) 



As gradients of these hyperplanes all gradients of ) at x^ are feasible. With x^ in 
the interior of some ,they are uniquely defined (and equal A^c^^) - and at the boun- 
daries of the we may take the corresponding convex combinations. 

The joint tangential hyperplanes may be viewed as graphs of the linear functions 

+ p(y - x^) (y e satisfying A u^(x^) + p(y - x^) > u^(y) (y e R^) . 



Because of the above-mentioned situation, the price vector p for each p has to be a 
convex combination of certain A^c^^ - and the set of indices 1 depends on Q only. 

Because these reflections are still vague, we supply a more formal description in [12] . 



SECTION 3 : Q - Faces 

Consider a cone T as specified by Theorem 1.6. With T, for any t we know that F(t) is 
the convex, comprehensive hull of finitely many points of the form A t; here CT denotes 
a nonsingular system in the sense of SEC. 1. However, if Cl is nonsingular and A t is 
feasible then, nevertheless, the latter is not necessarily Pareto efficient. Also, if (^^)q^ 

is a family of nonsingular systems, then the points A^t = A^^ t do not necessarily 
constitute a t-Face. 

In order to specify a family of nonsingular systems Q C {CT [ CJ is nonsingular} we identi- 
fy indices q € Q and systems 3 = CI^. 

Theorem 3.1.: Let T C R^ be a cone as given by Theorem 1.6, and let Q be the 
corresponding family of nonsingular systems. Then there is a finite collation 
^ ={'?, '!'’,•••} of subsets of T with the following property. 

1. 'S' € 4 is a closed cone with apex 0 and nonempty interior. 

2. T= {S' I teT) 

o 

3. For T, S'’ € S the cone S fl S'’ is lower-dimensional. 

4. For any S’ G ‘S there exists a finite family § = {Q} of subsets of Q such that, 
for t € S', the sets 

F^(t) CVH {A*lt I q e Q} 
are exactly the t-faces of F(t). 




295 



In other words, within every 't the t-faces depend essentially only on the choice of a 
system Q C Q. Hence, if we pick Q € §, then for any t € 'f' we know that F^(t) is a 
t-face. 

The proof is slightly tedious and similar to the one given for Theorem 1.7. We refer the 
reader to [12] for the details. 

In the following we want to deal with those situations only that yield a unique price 
vector for every face of the correspondence F at varying t. Clearly, inside a cone T as 
specified by Theorem 3.1, faces may be identified with index-sets Q, and hence the 
required property is a property of the cone - and not of varying t. 

Definition 3.2: has finite character ii, for every T as given by 3.1. and every Q speci- 
fying a face F^(t) for all t c T, the price vector is uniquely defined, i.e., is a 

singleton, depending on Q only. 

This definition is very much *’ad hoc" and it will have to be enhanced by pointing to 

appropriate requirements for to satisfy such that finite character is ensured. This we 
will postpone to some other treatment. However, there are classes of markets satisfying 
this condition. An obvious candidate is the class of side-payment or TU— markets as the 
normal A equals e = (1,...,1) constantly - and indeed, this class has been shown to yield 
a finite convergence tneorem based on nondegeneracy - see [8] . 

Another class can be obtained by "generically" requiring that, for any T and Q the 

resulting face F^(t) enjoys a (relatively) interior point, say u, such that a 

corresponding element Y. e 01^ provides at least one within the interior of some H^^. 

Indeed, by counting equations, it is easily seen that mr > m -h r is sufficient to ensure 
that generically every face of F has dimension r-1. (The mapping 

{ X € R"'*' I X = ^ ^ ... ,tj.cV } must have full 

rank, which is generically ensured by mr - m > r ). 

On the other hand, if all elements of a face F^(t) yield corresponding X e 01^ such that 

every x^ in at least two H^^, then these elements satisfy r + m equations; thus if mr - 
fm + r) < r - 1 , then the Pareto face cannot have full dimension. Therefore, by asking 
lor r + m -h(r - 1) > mr > m + r we ensure the finite character of the market. 

With finite character we will essentially be able to demonstrate finite convergence of 
the core towards the Walrasian equilibrium - this is the topic of the next section. 



SECTION 4: Finite Convergence of the Core 

Within this section we are now going to collect our structural results in order to prove a 
"finite convergence" property of the core towards the Walrasian equilibrium. Presently, 
we can only succeed for core-payoffs that are within the relative c-interior of some 
Pareto-Face of F(t), say FQ(t). However, similar to the situation in [7] and [8] , it is 

possible to exhibit regions in (to be interpreted as areas of distributions of players 
over the types) such that the core and the Wdrasian equilibrium coincide. 




296 



Within some cone 'J' as defined by Theorem 3.1 a face F^(T) is essentially defined by a 
system Q (of index sets) and by the resulting extremals of the form A^t (q € Q). Denote 
by K^(t) the cone with vertex 0 that is spanned by A^t, i.e. 

(1) kQ(t) = { E a A^t I a >0(qeQ)}. 

qeQ ^ ^ 

If u € is a vector satisfying u € K^(t), u ( F(t), then u is located in front of F^(t), 

this cab be separated from F(t) by the hyperplane containing F^(t) - which is defined 
by the normal A. 

Lemma 4.1: Let u be of finite character. Fix Q and consider t € (cf. 3.1) such that 
F^(t) is a face of F(t), let p be the corresponding price vector (Definition 3.2). Let 
X 6 ^??^(u) be such that u = t ® u(X) € F^(t). Then, for any s 6 such that s ® u(X) 

€ Kp.(s), s ® u(X) i F(s), it follows that p E s (x^-a^) > 0 holds true. 

^ PER ^ ■ 

Proof: Let A be the normal to F^(t) and A the one to F^(s). Then, for suitable p €IR 
and c € K^_|_ the corresponding prices p and p satisfy p = A^ c, p = A^ c (Definition 
3.2), Hence 

(2) p = const p 

with a certain positive constant. From this we deduce the following line of inequalities 
for any X € ^g(h) with s ® u (X) 6 F^(s): 

^ A s u^(x^) > ^ A^ s u^(x^)... 

(since A separates s ® u(X) from F^(s)) 

• • • = ^ S (X u^(x^) - p (x^ - a^)) ... 

pGR ^ ^ 

(since X E Ol^{u)) 

(3) 

• •• > X S (A u^(x^) - p (x^ - a^)) 

pGR P P 

(by the Kuhn-Tucker-argument, see SECTION 2.) 

= X A s U^(x^) - P X S (x^ - a^) . 

pGR f P pGR ^ 

Hence, using (2) and (3) 

P s (x^- a^) = const p s^ (x^~ a^) > 0. 

PER ^ PER ^ ■ 



q.e.d. 




297 



Next, for small e > 0, let F^(t) denote the (closed) "e-interior” of F^(t); more 
precisely 

(4) FQ(t) = { S a A'lt I S a =l.a > £ (q e Q)} 

^ /?€Q ^ q€Q ^ ^ 

This is a compact convex polyhedron with extremals, say, 6 F^(t) (q € Q). 



Lemma 4.2: Fix 'i' according to Theorem 3.1. Let e > 0. For every t 6 't', there is a 
closed cone C^(t) such that for all u = t ® u(X) € F^(t) it follows that 

(а) t is located within the interior C^(t); (b) if s € C^(t), then s ® u(X) € K^(s). 

Proof: Let t € First of all, fix u = t ® u(X) 6 F^(t). Note that t ® u(X) = 

E a A^t with suitable "convexifying" coefficients a. The /?’th coordinate is 
q€Q 

t^ u^(x^) = S Q A^t . 

^ qGQ ^ 

(A^ is the pHh row of a matrix A); and hence the p’th coordinate of s ® u (X) is 
s 

(б) s u^(u^) = ^ S o A^^ t. (Note that this is linear in s!). Consider 

^ p q6Q ^ 

C := {s€lR^^ I s®u(X)eK^(s)}. 

Clearly, t G C and as t ® u(X) is in the interior of F^(t) (hence in the interior of 

rQ(t)), C is seen to be a (closed) cone containing t in it’s interior. (In fact, C is 
described by finitely many algebraic hypersurfaces.) 

Now, let C^ (q ^ Q) denote the cones generated by taking the extreme points u^^ 
(qGQ) of F^(t) and performing the above procedure. Put 

(7) cQ(t):=Qc<i. 

Then again, t is within the interior of the closed cone C^(t). 

Finally, let again u = t ® u(X) be arbitrary in F^(t); we know that 

t ® u(X) = u = E a = E t ® u(X^^) 

with suitable and (q G Q). The /?’th coordinate of s ® u(X) is, therefore, 

s u^(x^) = Ea t u(x^^^) = Ea s. u(x^^^), i.e. 

P ^ ^ Vq^^ q^^ 

(8) s ® u(X) = E a s ® u(X^^) (essentially the linearity of (6) in s is used!). 

qGQ ^ 




298 



Therefore, if s € C^(t), then s ® u(X^^)6K^(s) (by (7)), and K^(s) is convex, (8) 
implies that s ® u(X) G K^(s), q.e.d 

Remark 4.3: Note that C^(t) is "algebraic”, i.e., its boundary is constituted by a finite 
number of algebraic hypersurfaces. Also, C^(t) can be chosen to exhibit some property 
of being "positively homogeneous", i.e., if a > 0 then = C^(t). 

Definition 4.4: Let 9 k G 't. k generates a full cone (w.r.t. e, Q) if there are r linearly 
independent vectors ("profiles") s GW^ such that 

(a) s < k, (b) s e C^(k), (c) k - s 6 C^(k), 
is satisfied. 

Usinc profiles k G IN we now switch to markets u with a certain distribution 
k = (k^,...,k^) of players over the types: this setup we started out with in SECTION 0. 

k k 

We want to compare the Core C{u ) and the Walrasian allocations of u . 

Theorem 4.5: Let k G generate a full cone. If X G ^^(u^) is such that k ® u(X) G F^(k), 
then X is Walrasian. 

Proof: Fix X with the required properties. Since X G^(u^), we know that s ® u(X) is 
not in the interior of F(s) = V(s). If, in addition, s G C^(k) holds true, then we have all 
the conditions of Lemma 4.1. Therefore, there are r linearly independent and integer 
vectors s G IN^ satisfying the conclusion of Lemma 4.1, i.e., 

(10) p E s (x^-a^)>0, 

/?GR P 

here p is the price-vector corresponding to Q. Since for any such s, k - s has the same 
property ("full cone"-definition) we have as well 

(11) p S (k -s )(x^-a^)>0. 

However, as X € we have 

(12) Ek(x^-a^) = 0 

p6R P 

and hence it follows from (10) and (11) that 

(13) p E s (x^-a^) = 0. 

pGR P 

Since our assumption is that there are r linearly independent vectors s satisfying (13), 
we conclude that 

(14) p(x^-a^) = 0 (/?GR). 




299 



It is well known that this implies that (p, X) is a Walrasian equilibrium. q.e.d. 

Theorem 4.6: (An equivalence theorem). Let u® be of finite character and let 'f C 

be defined by Theorem 3.1. Then, for any e > 0 there exists a set C with the 
following properties: 




(a) If t € then there is a > 0 such that ot G H^. 

(b) is an algebraic hypersurface. 

(c) If k elN^ k €H^, and X £ e(u^) is such that k ® u(X) eF^(k), then X is 
Walrasian. 

Thus, for sufficiently large k G all core allocations that yield payoffs which are in the 
e~Pareto efficient face, are Walrasian. 

Proof: For t G 1 ', C^(t) is a closed cone with nonempty interior (and "algebraic”). In 
view of Remark 4.3, C^(t) increases ("linearly") with t. Therefore the set 

( 16 ) EQ(t):=cQ(t)n{s I s<t}ClR^^ 

contains a convex closed body, the volume of which increases ("linearly") in t. By 
MINKOWSKPs ("second") Theorem (see CASSELS [2] , and compare the argument in 

ROSENMULLER [7] , [8] ,) the set E^(t) described by (15) contains r linearly 

independent integer vectors s once the volume exceeds a certain constant, say Cq. 

Define = {t G*? | volume (E^) > Cq}. Then has properties (a) and (b). Property 
(c) follows from Theorem 4.5, since k G implies that k generates a full cone. q.e.d. 




300 



SECTION 5 : Examples 

1 2 

Example 5.1: Let m = 2, r = 2 and consider the utility functions u : 1R_^ — ^ K, 
u^(x) = min {x^+2x2> 2x^4- X 2 } (x € for type las well as u^ : — » CR, 

u^(x) = x^-fX2 (x e K^) for type 2. 

12 2 
Also, we fix initial allocations a = (2,1), a = (1,1) for both types; hence, for t 

the "ideal" initial endowment is 

( 1 ) t ® a = a‘ = ^ = ( 2 t^+t 2 . t^+tg)- 

The Pareto optimal allocations are represented by the following sketch 




Figure 2 




It follows that we have three allocations yielding extreme Pareto efficient points in 
utility space, to write: 

t ® X"" = (t^ t2 x'^ 2 ) = (a‘,0), t ® X“ = (t^ t2 x"^) = ((at^a^); (aJ-aJ.O)). 
t ® X^ = (t^ x^^, ^2 

((r,m,t) for "right, middle, top"); inserting (1) yields 
t ® X = ((2t^+t2, t^+t2), 0), t ® X = ((t|+t2, tj+t2); (t^jO)), 
t ® X’' = (0; (2t^+t2> t^+t2)) 

The corresponding points in utility space are easily computed; observe that 

1 2 

u(t®x) = t®u(x) (since u = (u ,u ) is homogeneous). Also, we may substitute 
u^(x) = h^^(x) = x^+2x 2 as any P.E. extreme point yields h^^(x) < h^^(x). Thus, we 
come up with the following three P.E. extreme points of V(t): 

= (4t^+3t2.0), u“ = (3tj+3t2,tj.), = (0,3t^+2t2). 



For t^ < 2t2, the situation is represented by the following sketch. Note that the normal 

T R 

vectors of the P.O. surface of V(t) are A = (2,3) and A = (1,1) (independent on t and 
without normalization.) 



301 




Figure 3 



The Walrasian equilibrium is obtained in a slightly modified "Edgeworth-Box” 
(featuring the aggregated allocations for types) 




Figure 4 

1 1 -1 
The equilibrium price is (2", j") corresponding allocations are t^^ x = (-yi -j") 

—2 ^1 ftW 9 

and t2 x = (t2,t2) + (2”) “"2“)’ corresponding utilities are ^u = 2t2) 





302 



For > 3t2 the situation is similar (although the "type-oriented Edgeworth Box" is 

—1 —2 
not quite suitable) and the Equilibrium is given by t^ x = (2t^--2t2, ^2 ^ 

= (3t2> 0 ) with price (j, |-) and utility vector ^^^u = ( 4 t^, 3t2). 



t 

Finally, if 2t2 < t^ < 3t2 then the price depends on t, p = (- 



1 ^ 2 ’^ 2 ^ 



and we have 



t^ x^ = (t^+t2, t^+t2) t2 = (tj^, 0) with utility vector ^^^u = ( 3 t^ + 3t2> t^). 
Note that (x^, x^) is a (rational) function of t while ^^^u is used for 



= (t^ u\x^), t2 u2(3e2)) = (ul(t^ x^). u2(t2 x^)). 



Now for the Core. We are to consider only ^^(V(t)) (i.e. a concept in utility space). 
Hence, we are interested in vectors u G K such that t ® u = (t ^ u^, t2 U2) C V(t) while 
s ® u is not in the relative interior of V(s). Let us first discuss the continuous case (s, t € 
Assume for the moment that t ® u is on the upper part of the P.O. -surface; thus 

t ® u = ^^^b where ^^^b is the "corner point" (3t^+3t2> t^) Now, if 0 < s < t 

satisfies A^ s ® u < A^ ^^^b then the profile 0 < t-s < satisfiesA^(t-s) ® u > A^(t-s) ® u. 

In a small neighborhood of ^we shall find continuously many such profiles s for which, 

in addition, A^ s ® u < A^ ^^^b is equivalent to A^ s ® u € V(t). It follows that, for such t 
we must have A s ® u = A -'b for continuously many s; i.e. A^^ s^ u^ + A2 S2 u = 

A^ ( 3 Sj^H- 3 s 2 ) + A2 



In view of the prevailing linearity we may actually insert s = ( 1 , 0 ) and s = ( 0 , 1 ) in 
order to find the unique solution 



u = 







9 t, etg 

Now, for t^ < 2t2, t ® u = (“5— > -y“) is in fact an element of V(t) and equals the equili- 
brium payoff. 

Now we switch to the discrete case. 

Let := {t 6 I t^ < 2t2>, E^ := {s G | s < t, s G t-s G t'^}. 

We should say that the payoff ^^^u is nondegenerate with respect to E^, if it is the 
unique solution of the linear system in variables y^, y2 given by A^ s ® y = A^ ^^^b (s 
G E^ n W^) that is, if there are 2 linear independent profiles in E^. 




303 



In this case, t^(V(t)) = {t ® u} and the core collapses towards the equilibrium. 

Clearly the set 

{t€lN^ I u n.d.w.r.t, E^} 

is the system of distributions such that 6 is true and, as the number of linearly 

T 

independent profiles depends on the volume of the compact convex polyhedron E^ we 
expect the region where C (and t^ < 2 t 2 ) to be of the following shape. 




For tj > 3t2, the argument is quite similar, except that u = (p, p) and 
t ® u = (4t^, 3tj| ) is a point on the south-east part of the Pareto surface of V(t). 

For 2t2 < t^ < St^ the argument is slightly different. 




304 



Literature 

[1] Aumann, R.J.: Markets With a Continuum of Traders. Econometrica 32, 

(1964),pp.39 -50. 

[2] Cassels, J. W. S.: An Introduction to the Geometry of Numbers. Springer 

Verlag, Berlin, Gottingen, Heidelberg (1959). 

[3] Debreu, G. and Scarf, H.: A Limit Theorem on the Core of an Economy. 

International Economic Review 4 (1963), pp.235 - 264. 

[4] Hildenbrand, W.: Core and Equilibria of a Large Economy. Princeton 

University Press, Princeton, N.J. (1974). 

[5] Mas-Collel, A.: The Theory of General Equilibrium: A Differentiable 

Approach. Cambridge University Press, (1985), 367 pp. 

[6] Peleg, B. and RosenmuUer, J.: The least-core, nucleolus, and kernel of 

homogeneous weighted majority games. Games and Economic Behavior, Vol.4, 
No.4, 1992, pp.588-^05 

[7] Rosenmiiller, J.: L.-P.-Games with Sufficiently Many Players. International 

Journal of Game Theory, Vol 11, (1982), pp. 129 - 149 

[8] Rosenmiiller, J.: The role of non-degeneracy and homogeneitv in n-person 

game theory: an equivalence theorem. Journal of Economics (Zeitschrift fiir 
Nationalokonomie) 47, Nr.4 (1987), pp. 367 - 389. 

[9] Rosenmiiller, J.: Discrete concepts in n-person game theory: nondegeneracy 

and homogeneity. Columbus, Ohio (1987). (Edited by T.Ichiishi, A.Neyman, 
Y.Tauman), Academic Press Inc. New York (1990), ISBN 0-12-370182-1. 

[10] Rosenmiiller, J. and Sudholter, P.: The Nucleolus of Homogeneous Games 

With Steps. Discrete Applied Mathematics, Vol. 50, 1994, pp. 53-76 

[11] Rosenmiiller, J: A Discrete Approach to General Equilibrium and 

NTU-Games, IMW-Working Paper No. 163, University of Bielefeld, 1991 

[12] Van der Waerden, B.L.: Algebra II . Springer Verlag, Berlin, Heidelberg, New 

York (1967), 300 pp. 




Credible Threats of Secession, Partnership, 
and Commonwealths* 

Philip J. Reny^ and Myrna Holtz Wooders^’** 

^ Department of Economics, University of Pittsburgh 
^ Department of Economics, University of Toronto 



Abstract. We define a concept of a credible threat to secede; a group of 
players can credibly secede if, by secession, the group can harm a subset 
of the complementary coalition while maintaining its own payoff and the 
payoff of the remainder of the complementary coalition. The set of efficient 
payoflFs that are immune to secession constitutes a subset of the core and 
may be empty even when the core is nonempty. To solve this problem and 
to explain the formation of not-necessarily-self-sufficient groups within larger 
organizations, we introduce the concept of a commonwealth - a payoff in the 
core and a partition of players into states so that all members of a state are 
mutually dependent upon each other and no state can credibly secede. We 
show that the commonwealth core is equivalent to the partnered core. That 
is, when secession is constrained to be carried out only by partnerships the 
two concepts coincide. As a corollary, the commonwealth core is nonempty 
whenever the core is nonempty. 



1. Commonwealths and Credible Threats of Secession 

There are many examples of organizations consisting of affiliations of smaller 
units within the organization. Such examples include the Y.M.C.A, the Girl 
Guides of America, the U.S., Canada, Spain, the U.N., etc. The units within 
the organization may be described by the terms branches, clubs, states, 
provinces, and member nations, to name a few. Typically, organizations have 
the property that exchanges and cooperation must take place between units 
to realize all gains to collective activities; units within the organization are 
not necessarily self-sufficient. To fix ideas, we shall refer to the units within 

* The authors gratefully acknowledge the financial support of the Faculty of Arts 
and Science at the University of Pittsburgh and the Social Sciences and Hu- 
manities Research Council of Canada. We are also grateful to the University of 
Bonn for having us both as visitors, which lead to our collaborative research. 
One author is especially grateful to Reinhard Selten for many discussions of 
the importance of credibility of threats. Finally, we thank Howard Petith for 
stimulating discussions on partnership, Deidre Herrick for proposing the term 
“commonwealth” and Werner Giith for helpful comments on an earlier version 
of this paper. 

** This author expresses her gratitude to the Autonoma University of Barcelona 
and the University of Alabama for support and hospitality during the period 
that this paper was written. 




306 



an organization as “states.” Organizations having certain stability properties 
will be called “commonwealths.” Although our analysis carries over to any 
organization made up of any number of units, these terms and their usual 
meanings may best reflect the ideas we have in mind. 

Certain options are available to states within an organization, including 
the option to secede. This paper studies the stability of an organization when 
the possibility of secession is available to the states within it. Indeed, part of 
our purpose is to explain the division of a population of players into distinct 
states, even when the states are not necessarily self-sufficient. Although the 
model we employ is that of a (cooperative) game with side payments, our 
notion of a credible threat to secede is derived from noncooperative game 
theory, and in particular the writings of Selten (1965, 1975). 

A payoff for a game with side payments is said to be immune to secession 
if it is efficient and if no group of players can credibly threaten to separate 
itself from the others. This means that no group of players can realize their 
part of the payoff for themselves and, by leaving, harm some or all of the 
remaining players. The requirement that the act of departing harms those 
who remain makes the possibility of leaving a threat and the fact that the 
departing group can realize its part of the payoff renders the threat credible.^ 
Recall that a payoff in the core of a game has the property that it is 
immune to improvement by coalitions; no group of players can withdraw and 
realize a payoff preferred by all members of the group. It is quite different 
to say that a payoff has the property that no coalition can secede and si- 
multaneously impose costs on some subset of the complementary coalition. 
If a group of players were in such a position, then that group, in the hope of 
achieving a larger share of the total payoff, may threaten to leave the total 
player set and cooperate only with some subset of players. Thus, a payoff in 
the core may not be immune to credible secession. This difficulty motivates 
our definition of a commonwealth and a commonwealth core. 

An elementary argument establishes that the set of payoffs that are im- 
mune to secession is contained in the core (Proposition 1). However, even 
for games with nonempty cores there may be no payoffs that are immune to 
secession. Consider, for example, the one buyer and two sellers game, where 
the buyer has a reservation price of one for an object and each seller has a 
reservation price of zero. The core of this game is nonempty, and gives all 
the surplus to the buyer. There are no feasible payoffs that are immune to 
secession. For any payoff giving the buyer less than all the surplus, there is 
a seller who, along with the buyer, can credibly threaten to secede. If the 
buyer receives all the surplus however, the two sellers can credibly threaten 

^ There are other possible definitions of credible threats. For example, one might 
say that a threat is credible if, were the threat carried out, it would impose 
a larger cost on the threatening group than on the group that initiated the 
threat. Such a notion of credibility, however, would be distinctly different than 
the standard notion of credibility of noncooperative games introduced by Selten 
and adopted here. 




307 



to secede. This motivates our definition of a commonwealth and the com- 
monwealth core. 

We define a payoff and a partition of the total set of players of a game into 
states to be a “commonwealth” if (a) the payoff is in the core of the game, (b) 
each state consists of a partnership, that is, the members of a state all “need” 
each other and (c) no state can credibly threaten to secede and by doing so, 
impose costs on another state. A payoff is in the commonwealth core if there 
is a commonwealth with that payoff. The motivation for the commonwealth 
core stems from the ideas that individuals who have a mutual need for one 
another naturally align themselves into decision-making states and that the 
commonwealth’s stability requires not only that the payoff be in the core but 
also that no state can credibly threaten to secede.^ 

In this paper we show that for games with side payments the common- 
wealth core coincides with the partnered core introduced in Reny, Winter, and 
Wooders (1993). A payoff for a game is partnered if it admits no asymmetric 
dependencies. An asymmetric dependency arises when one player needs the 
cooperation of another to realize his share of the payoff but the second player 
does not require the cooperation of the first. 

Finally, it is important to note that a state in a commonwealth is not 
necessarily self-sufficient. Achieving efficient outcomes may necessitate coop- 
eration among states. But this cooperation is more like the cooperation of 
an anonymous market (where there are many buyers and sellers) than the 
cooperation between members of the same state each of whom requires each 
of the others to obtain his part of the payoff. 



2. Games 

A game (in characteristic form) is a pair {N, v) where N = {1, ..., n} is a finite 
set of players and u is a function from 2^ to R^. with t;(0) = 0. A (possibly 
empty) subset 5 of N is called a coalition. We shall assume throughout that 
{N,v) is superadditive (i.e. t?(5)+u(T) < u(5UT),V5, T C A7'with5nT = 0). 

A payoff ioT a game (N,v) is a vector x in R^. Fix x in R^. For S C N, 
define x{S) = payoff x is feasible if x{N) < v{N) and efficient 

if x{N) = v{N). The payoff x is in the core of {N,v) if x is efficient and 
x{S) > v{S) for all coalitions S C N. 



^ The notion of the commonwealth core was stimulated by discussions of Quebec 
and Cataluna, neither of which would necessarily be better off after secession. 
That the secession of these states from their countries (Canada and Spain, resp.) 
may impose costs on other states may provide an explanation for the continuing 
discussions, in both cases, of secession. 




308 



3. Threats of Secession 

Let (iV, v) be a game with side payments and let x be an efficient payoff. We 
say that a coalition S G N can credibly threaten to secede if for some (possibly 
empty) coalition T, x(5 U T) < v{S UT) and for some coalition Q disjoint 
from S UT, v{Q U R) < x{Q U R) for all coalitions R disjoint from S. Thus, 
by seceding, S hurts coalition Q regardless of how players in iV — 5 might 
subsequently cooperate; therefore secession is a threat. The threat is credible 
since together with T (disjoint from Q), S (and T) can realize its payoff. 
If no coalition can credibly threaten to secede, we say that x is immune to 
secession, A payoff x is cohesive if it is efficient and immune to secession. 

Proposition 1. If (N,v) is a superadditive game, then every cohesive payoff 
is a core payoff. 

Proof. Let x be cohesive and suppose that x is not in the core. Since x is 
efficient there is then a coalition S with x{S) < v{S). By superadditivity 
and efficiency, x(5)-f x{N — S) = v{N) > v{S) v{N — S). Rearranging, we 
obtain x{S) — v{S) > v{N — S) — x{N — S). Since x{S) — v{S) < 0 it follows 
that v{N — 5) — x{N — 5) < 0. Taking T = 0 and Q = N — S, this contradicts 
the property that x is cohesive. ■ 

As illustrated in the introduction, there are games with nonempty cores 
having no payoffs that are cohesive. 



4. Partnership 

Let W be a finite set of players and let P be a collection of subsets of N. For 
each i in N let 

Vi = {SeV:ieS}. 

We say that V has the partnership property (for A^) if for each z in W the 
set Vi is nonempty and for each pair of players i and j in N the following 
requirement is satisfied: 

if Vi C Vj then Vj C Vi. 

That is, if all the coalitions in V that contain player i also contain player j 
then all the coalitions that contain j also contain i.^ 

The payoff x for the game (AT, v) is called a partnered payoff if the col- 
lection of coalitions supporting x (i.e. {S C N : x{S) < v{S)}) has the 
partnership property. The partnered core is the set of all partnered payoffs 
in the core of {N,v). Reny, Winter and Wooders (1993) show that the rel- 
ative interior of the core of a game with side payments is contained in the 
partnered core. 

^ For the reader familiar with these works, we note that partnered collections of 
coalitions are called ’’separating collections” in Maschler and Peleg (1966, 1967), 
and Maschler, Peleg, and Shapley (1971). See Section 7 for further discussion. 




309 



5- Commonwealths and the Commonwealth Core 

Let (N, v) be a game with side payments and let x be a feasible payoff. We’ll 
say that i needs j for x if for all 5 C W, x{S) < v{S) and i £ S together 
imply that j G S. If both i needs and j needs i for x, we’ll say that i 
and j are partners (for x) (or that i is partnered with j (for x)). Clearly, for 
a fixed payoff x, the relation “is partnered with (for x)” is an equivalence 
relation on the player set. Call each equivalence class a partnership (relative 
to x). Let Afx = {iVi, denote the (unique) partition of N into states 

in which each state Nk consists of a partnership relative to x. We say that 
state Nk can credibly threaten to secede from Mx if Nk can credibly threaten 
to secede in the sense of Section 3 but where the sets T and R are restricted 
to being unions of states (rather than arbitrary coalitions of players) , and Q 
is restricted to being a single state. If no state Nk can credibly threaten to 
secede, then Nx is called a commonwealth partition {relative to x). The pair 
(j\4,x) is a commonwealth (for the game {N,v)) if x is in the core of the 
game and Afx is a commonwealth partition. 

Let {N,v) be a game and let x be a payoff. We say that x is in the 
commonwealth core if there is a partition Mx such that the pair {Mx , x) is a 
commonwealth. 

Proposition 2. Let (N^v) be a superadditive game. Then a payoff x is in 
the commonwealth core if and only if it is in the partnered core. In addition, 
the commonwealth core is nonempty if and only if the core is nonempty. 

Proof: Let x be in the partnered core and suppose that {Nx-,x) is not a 
commonwealth. Since x is in the core, it must be the case that some state 
N\, say, can credibly threaten to secede from Mx- That is, (i) there is a union 
of states, T, such that x{N\ UT) < v{Ni UT); and (ii) there is another state, 
N 2 , say, disjoint from Ni and T, such that v{N 2 U R) < x{N 2 U R) for all 
unions of states R disjoint from N±. Now, suppose that i e N 2 . Choose 5 such 
that i £ S and x(5) < v{S) (possible since x is partnered). Since by definition 
of Afx each state is a partnership, S must be a union of states. Therefore we 
also have N 2 C S. Consequently, we have x{N 2 U 5') < v{N 2 U 5'), where 
S' = S—N 2 is a union of states. Hence by (ii) it must be the case that Ni C S. 
But since S was an arbitrary coalition containing i £ N 2 , this implies that 
i (indeed every player in N 2 ) needs every player in Ni. However by (i), no 
player in Ni needs any player in W 2 . But this contradicts the fact that x is 
partnered. Hence {Afx , x) is a commonwealth and so x is in the commonwealth 
core. 

Conversely, suppose that x is in the commonwealth core. Hence, {Afx^x) 
is a commonwealth. Suppose by way of contradiction that x is not partnered. 
Then there is a pair of players 1 and 2, say, such that 1 needs 2 (for x) but 2 
does not need 1. Consequently, 1 and 2 are in different states, say N\, and iV 2 , 
respectively. Moreover, since every state is a partnership, this implies that 
every player in state N\ needs every player in state W 2 , while no player in state 




310 



N 2 needs any player in state N\ . Thus we may say quite unambiguously that 
state N\ needs N 2 to realize its payoff x(iV’i), while N 2 does not need N\ to 
realize x{N 2 )- Therefore there exists a coalition S containing N 2 and disjoint 
from Ni such that x{S) < v{S). Moreover, since every state is a partnership, 
5 is a union of states. But this implies that N 2 can credibly threaten to 
secede from Afx (since x{N 2 U S') < v{N 2 U S') where S' = S — N 2 is a. union 
of states, and x{Ni U R) > v{Ni U R) for all states (in fact all coailitions) R 
disjoint from N 2 since Ni needs N 2 ) contradicting the fact that (A/^,a;) is a 
commonwealth. Since the relative interior of the core is in the partnered core, 
(see Albers (1974), Bennet (1983), or Reny, Winter and Wooders (1993)), it 
follows that the commonwealth core is nonempty whenever the core is. ■ 



6. Other Literature Relating to the Partnership 
Property 

The partnership property of collections of sets first appeared in Maschler and 
Peleg (1966,1967) and Maschler, Peleg, and Shapley (1971) in their study of 
the kernel of a cooperative game. The concern of Maschler and Peleg was 
“separating out” players, Maschler and Peleg call a partnered collection of 
sets a separating collection. Two players i and j are inseparable if, in our 
terminology, they are partners. Since the partnered core was motivated by 
a desire to focus on coalition formation we adopt the term “partnership” 
of Albers (1974,1979), Bennett (1983) and Bennett and Zame (1988).^ The 
terms introduced by Maschler and Peleg are also appealing, especially from 
the viewpoint of core solutions, since we may think of players who are partners 
as inseparable. 

It is important to note that a partnership is not at all the same concept 
as a coalition in a coalition structure. In the typical literature on coalition 
structure games, such ais Aumann and Dreze (1974) or Kaneko and Wooders 
(1982), each coalition is self-sufficient, that is, v{S) = x{S). The approach of 
this literature does not address why we may have organizations consisting of 
distinct sub-units, such as states, that are not necessarily self-sufficient. In 
Reny, Winter, and Wooders (1993) and Reny and Wooders (1993) we provide 
examples showing the differences between coalition structures and partner- 
ships. Another simple example is the one-buyer, two-seller game described 
in the introduction. The partnerships associated with the only payoff in the 
core, giving all the surplus to the buyer, are the 1-member partnerships. 

Other motivation for our work comes from the literature of noncoopera- 
tive games and competitive equilibrium. The importance of the credibility of 
threats was emphasized in Selten (1965), (1975). The concept of partnership 

Bennett (1983) is a rich source of examples of partnered payoffs for games with 

side payments. 




311 



in noncooperative theory first appeared in Selten (1981). There, a noncooper- 
ative foundation was given for partnered “expected” payoffs, as the outcome 
of a bargaining process. This research has been continued in Winter (1989), 
Bennett (1981) and other recent papers. As noted in the introduction, the 
fact that the competitive equilibrium is not partnered is a weakness of the 
concept. Bennett and Zame (1988), however, show that when preferences 
are strictly convex the competitive payoff is partnered^. Page and Wood- 
ers (1994) introduce the partnered core of an economy and show that, even 
though the competitive equilibrium may not be partnered, convexity of pref- 
erences ensures nonemptiness of the partnered core. In addition, Page and 
Wooders (1994) show that with additional conditions on the economic model, 
the Page- Werner condition of no unbounded arbitrage is necessary and suffi- 
cient for nonemptiness of the partnered core of an economy and the partnered 
competitive equilibrium. 

Finally, we remark that Reny and Wooders (1993) show that the partnered 
core of a balanced game without side payments is nonempty and thus provide 
a refinement of Scarf’s Theorem (1967) on the nonemptiness of the core. 
Ongoing research studies the notions of credible threats to secede and the 
commonwealth core of games without side payments. 



References 



Albers, W. (1974) “Zwei Losungskonzepte fur kooperative Mehrpersonspiele, die 
auf Anspruchsniveaus der Spieler basiern” OR-Verfahren (meth. Oper. Res) 21, 
1-13. 

Albers, W. (1979) “Core and Kernel Variants Based on Imputations and Demand 
Profiles” in Game Theory and Related Topics^ O. Moeschlin and D. Pallaschke, 
eds. North Holland, Amsterdam. 

Aumann, R. J. and J. Dreze (1974) “Cooperative Games with Coalition Structures,” 
International Journal of Game Theory 3, 217-237. 

Bennett, E. (1980) Coalition Formation and Payoff Distribution in Cooperative 
Games, Ph. D Dissertation, Northwestern University. 

Bennett, E., (1983) “The Aspiration Approach to Predicting Coalition Formation 
and Payoff Distribution in Sidepayment Games,” International Journal of Game 
Theory 12, 1-28. 

Bennett, E. (1991) Game Equilibrium Models III: Strategic Bargaining^ ed. R. Sel- 
ten, Springer- Verlag. 

Bennett, E. and W.R. Zame (1988) “Bargaining in Cooperative games”, Interna- 
tional Journal of Game Theory 17, 279-300. 

Bomze, I.M. (1988) “A Note on Aspirations in Non- Transfer able Utility Games”, 
International Journal of Game Theory 17, 193-200. 

Kaneko, M. and M.H. Wooders (1982) “Cores of Partitioning Games”, Mathematical 
Social Sciences <?, 313-327. 



^ In addition, Bennetand Zame (1988) show that the set of partnered aspirations 
is nonempty. See also Bomze (1988). 




312 



Maschler, M. and B. Peleg (1966) “A Characterization, Existence Proof and Di- 
mension Bounds for the Kernel of a Game,” Pacific Journal of Mathematics 18, 
289-328. 

Maschler, M. and B. Peleg (1967) “The Structure of the Kernel of a Cooperative 
Game,” SIAM Journal of Applied Mathematics 15, 569-604. 

Maschler, M., B. Peleg, and L.S. Shapley (1971) “The Kernel and Bargaining Set 
for Convex Games,” International Journal of Game Theory 1, 73-93. 

Page, F. H. Jr and M.H. Wooders (1994) “The Partnered Core of an Economy,” 
Autonoma University Discussion Paper no. W.P. 279.94. 

Reny, P.J., E. Winter, and M.H. Wooders (1993) “The Partnered Core of a Game 
with Side Payments,” University of Toronto Discussion Paper. 

Reny, P. and M.H. Wooders (1993) “The Partnered Core of a Game Without Side 
Payments,” Journal of Economic Theory (to appear). 

Scarf, H.E. (1967) “The Core of an n-person Game,” Econometrica 35, 50-67. 

Selten, R. (1965) “Spieltheoretische Behandlung eines Oligopolmodells mit Nach- 
fragetragheit,” Zeitschrift fur die gesamte Staatswissenschaft 121, 301-324. 

Selten, R. (1975) “Re-examination of the Perfectness Concept for Equilibrium 
Points in Extensive Games,” International Journal of Game Theory 4, 25-55. 

Winter, E. (1989) “An Axiomatization for the Stable and Semi-Stable Demand 
Vectors by the Reduced Game Property,” Discussion Paper No. A-254, The 
University of Bonn. 




Rules for Experimenting in Psychology and 
Economics, and Why They Differ 



Colin Camerer^ 

Division of Social Sciences 228-77, California Institute of Technology, Pasadena CA 91125 
USA. camerer@hss.caltech.edu. 



Abstract. This chapter discusses methodological differences in the way economists 
and psychologists typically conduct experiments. The main argument is that 
methodological differences spring from basic differences in the way knowledge is 
created and cumulated in the two fields— especially the important role of simple, 
formal theory in economics which is largely absent in psychology. 

Keywords. Experimental economics, psychology, bounded rationality. 



1 Introduction 

There are many ways to conduct experiments in social sciences. This chapter 
contrasts two philosophies or approaches, referred to as the "experimental 
economics" (E) and "experimental psychology" (P) approaches. The distinction is 
not sharp, of course, but even fuzzy distinctions can be helpful in framing debates 
and clarifying misunderstanding. The main focus in this chapter is the difference 
between E and P experiments in research in decision theory and game theory. 

There is substantial bickering between these camps about the right way to 
conduct experiments. My central argument in this chapter is that many crucial 
differences in experimental style are created by underlying differences in working 
assumptions about human behavior and about how knowledge is best expressed, not 
by underlying differences in areas of substantial interest. My hope is that 
understanding this point can sensitize both sides (and outsiders or newcomers) to 
differences in these philosophies, without particularly advocating one over the other, 
in order to defuse some sensitive aspects of the debate between E and P. 



Comments from participants at the Big Ten Accounting Doctoral Consortium, 
Minneapolis 5/13-15/94, Chip Heath, Wulf Albers, and Werner Guth, were helpful. Many 
conversations over the years -- particularly with Daniel Kahneman, Charlie Plott, and 
Richard Thaler, and numerous coauthors — have shaped the contents of this paper. 
Discussions at the Russell Sage Foundation during my 1991-92 visit there were especially 
formative. 





314 



The chapter has several sections. Next I argue that economics and psychology 
are more sharply distinguished by style than by substance. In section 2 we make the 
leap to experimental style, describing key differences in P and E experimental styles 
and showing how they follow from differing presumptions in the underlying 
disciplines. Section 3 gives a specific example to further illustrate the argument. In 
section 4 I express some preferences for certain stylistic features, add caveats, and 
mention the exceptional methodological style of Reinhard Selten. 

This paper draws clearly on many aspects of Smith's classic 1976 and 1982 
papers which guided experimental work in economics for the last twenty years. I 
was also inspired by a thoughtful unpublished 1988 paper by Daniel Kahneman 
which contrasted P and E methods. Similar ground is covered by Cox & Isaac 
(1986). Descriptions of now-standard methodology in experimental economics can 
be found in Hey (1991), Davis & Holt (1992), and Friedman & Sunder (1994). 
Chapters in Kagel & Roth (1995) review discoveries in experimental economics. 

1.1 Substance vs. Style: Economics and Psychology 

I claim differences in E and P experiments arise from differences in style of the two 
fields. By substance I mean, "what is the basic question being asked?" By style, I 
mean, "What constitutes an answer to the question? How are answers expressed?" 

As a starting point for discussion, let's suppose psychologists ask: 

P: How do people think & behave? 

and economists ask 

E: How are scarce resources allocated? 

At first blush, this difference in fundamental "why" questions seems to sharply 
divide the two fields along substantive lines-- e.g., psychologists study human 
thinking, and economists study resource allocation. 

But the two questions also express inherent methodological differences. 
Answering the P question demands observation of individual subjects in controlled 
settings (experiments). The E's "allocation of scarce resources" hints strongly at an 
underpinning from mathematical optimization. My claim is that the substantive 
difference is actually small, and the stylistic difference large. 

1.2 How Much Substantial Overlap is There? 

Take any one of several concrete examples: How do people decide how much time 
to spend with their families, instead of working? Why do people discriminate 
against others who are unlike them? Who do people vote for? When do people 
retire? How do people choose friends? Spouses? Why do people break the law? 
How do people invest their wealth? Choose Jobs or education? Why do people 
become addicted to drugs and acquired tastes? 




315 



Virtually all these questions can be posed as P or E questions, and have been 
studied by social scientists of both types (and also by sociologists, whose work is 
ignored in this chapter for brevity). To further illustrate that the differences between 
P and E are not entirely substantial. Table 1 illustrates some of the similarities in 
topics of substantial interest across the two fields. Many interesting topics-- 
altruism, learning, taste formation— have been studied by people in both fields. The 
key difference is perhaps not the substantive topics psychologists and economist 
study, but the way theories are expressed and data are gathered. 



Table 1: Similarities in substantial topics in P and E 



Psycholog y 

GUIDING QUESTIONS 
How do people think & behave. 
Why? 

SPECIFIC TOPICS 
reciprocity norms 
prisoner’s dilemma 
expectancy theory 
altruism 
impulsivity 
language & meaning 
learning 
taste formation 

symbolic consumption 
conformity/herd behavior 



Economics 

How are scarce resources allocated? 



"gift exchange" labor models 
externalities 

expected utility (EU & SEU) 
altruism (bequests to children) 
impatience (discounting) 
language & meaning (cheaptalk) 
learning (least-squares, population learning) 
taste modelling (addiction, 
participation externalities) 
signaling 

info, cascades, rational conformity 



1.3 Stylistic Differences in E and P 

I think the fundamental stylistic differences are two: (i) E's strongly prefer 

mathematical formalisms. And since formalisms become unwieldy and inelegant as 
they get too complex, E's like simple formal models, (ii) E's like predictions that are 
surprising, and testable using naturally-occurring ("field") data. Note how features 
(i) and (ii) work together. The desire for parsimony enables many E's to justifiably 
ignore whether assumptions are realistic (if alternative assumptions are "too" 
complicated), and hence place more value on naturally-occurring data than on 
experimental refutations. 

In contrast. P's tend to prefer verbal models or principles and eschew 
sophisticated mathematical expressions. Since these principles need not fit together 




316 



in mathematically coherent ways, models can resemble lists of effects or critical 
variables, and can be expressed (and discovered) by experimental demonstrations. 
Field tests or demonstrations are consequently rare. 

My argument for stylistic difference implies that a single substantive question 
can be translated from the P to E mode, or vice versa. 

For example, translating from P to E, it is a simple matter to translate a loose 
psychological account of, say, how financial incentives influence performance into 
economic terms: The "scarce resource" is attention or thinking effort, higher 

incentives produce more thought (the supply curve for thinking is upward-sloping) 
and more thought improves performance (thought is a factor or production in 
generating performance), as in Smith and Walker (1993). But doing so has no 
special interest to many Ps, because the resulting model will of course be 
oversimplified ("reductionist") and plainly at odds with at least some experimental 
observations. 

Similarly, one can ask an E question~do people play the subgame perfect 
solution to a sequential bargaining problem?— in a P way, by observing whether 
subjects in a laboratory search for information in the way that is necessary to 
calculate perfect equilibrium using backward induction (Camerer, Johnson, Sen & 
Rymon, 1993). The answer to that question is "No", but that answer does not deter 
an E from continuing to assume the answer is yes, if mathematical simplicity is 
sufficiently important and no equally parsimonious replacement assumption emerges 
from the experimental findings. 

A helpful and charitable way to think of the E and P camps is as competing 
cultures, religions, or schools of artistic expression. Psychologists are realists who 
paint detailed landscapes and strive to make them "lifelike". (A good psychology 
experiment is a drama that makes a point about human nature.) Economists are 
abstract expressionists who value pictures that expresses a feature of the world with 
minimal clutter. Naturally, the P's criticize the E's for clinging to unrealistic 
assumptions, and the E's criticize the P's for lacking formal theory, or parsimony. 



2 Experimental Styles in E and P 

My central claim is that general stylistic differences in how knowledge is generated 
in the E and P fields accounts for many of the major differences in how experiments 
are conducted. 

Table 2 lists some important differences in experimental styles. Most of the 
differences spring from basic differences in media for expression of knowledge and 
how the interplay of theory and data work in the two fields. 

For E's, the medium for expression of knowledge is, primarily, a body of 
formalism and modelling principles. A body of facts buttresses the formalism and 
sustains faith in it, but is clearly subservient because ample evidence against a formal 
principle is often insufficient to create widespread rejection and search for 
replacements. (E.g., the experimental evidence against expected-utility maximization 




317 



Table 2: Differences in P and E Experimental Style 

Psychology (P) Economics (E) 

MEDIUM FOR EXPRESSION OF KNOWLEDGE 
Body of facts ( 1 st) Formal model ( 1 st) 

Informal interpretation Body of facts 



INTERPLAY OF THEORY & DATA 
Experimental results are 
cloth from which theory 
is woven 

CONTEXT/LABELS 
Rich context preferred: 
engages subjects 
more lifelike 

SUBJECT POOLS 
Gender, age, etc. 
often recorded 



TYPES OF DATA 

Ratings, response times, 
subject self-reports, etc. 

REPEATED TRIALS? 

No need. Some of life 
is like first trial 

INCENTIVES OF SUBJECTS 

Volunteer subjects are 
well-motivated 
to "try hard" 

DATA ANALYSIS & REPORTING 

Assumes friendly reader. 

Reduced-form. 

Sometimes F's or p- values only. 

Don't report "failures". 

DECEPTION? 

Sometimes, to create unnatural 
situations or break confounds. 



Experiments operationalize and test 
general theory 
in specific (artificial) setting 

Rich context avoided: 

creates nonpecuniary utility 
irrelevant from theory's view 

Theory predicts no effects 

(or abstractness of setting swamps 
subject pool differences) 

Choices. 

Self-reports not informative. 

Yes. Theory predicts "last" 
(equilibrium) trial 

Financial motivation best. 



Assumes skeptical reader. 

More raw data. 

(Reader can reanalyze). 

No "failures" in a well-designed 
(competing-theory) experiment. 

Rarely. (Experimenter 

credibility is a public good) 





318 



is now forty years old and still piling up, but textbooks and most applied work 
continue to assume it.) The interplay of theory and data occurs as good experiments 
test theory or provoke its development by exploring behavior in an interesting 
domain where theoretical principles haven't been developed. For P's, theory is sheer 
cloth wrapped around a body of prominent experimentally-observed results. 
Explanations that don't fit new observations are jettisoned more quickly than old 
facts are. 

Several properties of good experimental design follow from these basic 
differences. 

Context or labels : In experimental E, an abstract context (or bland verbal labels) 
is sufficient to test theory; it is considered unnecessary to make the experimental 
context realistic since most theory does not predict that results will depend on verbal 
labels. (Indeed, labelling choices may even be undesirable since it may interfere with 
attempts to clearly control preferences by paying financial incentives.) By contrast, 
in experimental P a rich context is often used to motivate subjects to behave 
realistically and to provide a concrete illustration of an underlying (mathematical) 
principle. In addition, most psychologists feel that abstract, symbolic information is 
processed differently than concrete, lifelike stimuli. Hence, learning about subjects' 
intuitive grasp of statistics using only dice and coins could be misleading as a guide 
to how subjects reason about baseball statistics or categorize people into stereotypes. 

Subject pools : Economic theories usually do not predict any reason why different 
pools of subjects will behave differently, so E's rarely collect data on the 
demographics of subjects or test whether behavior varies with gender, age, 
education, etc. (though see Ball & Cech, 1990, for a review). Indeed, nonhuman 
subjects such as rats, pigeons, et al have been productively used in experiments on 
consumer choice theory, theories of labor supply, and expected utility (see Kagel, 
Battalio& Green, 1995). A long (now quiet?) tradition of interest in personal 
differences by some psychologists leads most psychologists to at least record 
personal information, if not theorize about differences due to age, gender, 
educational background, etc. 

Types of data: Since economic theory usually does not predict what cognitive 
process will lead to an outcome (say, competitive equilibrium in a market or a game- 
theoretic equilibrium), E's usually do not collect types of data other than choices. In 
contrast, data like Likert rating scales of attractiveness, response times, self-reports 
by subjects of what they were thinking, post-experiment Justifications for their 
choices, etc. are widely used in psychology and theories often make predictions 
about these variables. (Theory-minded E's would not know what to do with such 
data; though some E's interested in decision error have begun to use response times.) 

Repetition of trials : Economic theories are usually asserted to apply to 

equilibrium behavior, after initial "confusion" disappears and (unspecified) 
equilibrating forces have had time to operate. So E's generally conduct an 
experiment for several repeated trials under "stationary replication"— destroying old 
goods and refreshing endowments, in order to recreate the decision situation in 
exactly the same way each time (except for anything that was learned from the 




319 



2 

previous experience, in a "Groundhog Day" design ). Then special attention is paid 
to the last periods of the experiment (e.g., often the last half of the trials are used as 
data to test an hypothesis) or to the change in behavior across trials. Rarely is 
rejection of a theory using first-round data given much significance. P's, in contrast, 
sometimes run a single trial for each stimulus if learning is of no special interest. 

Incentives of subjects: A cmcial property of most E experiments is that the 
consequences of their choices are made clear (or "salient") to subjects, and they are 
always given financial incentives. The standard explanation for insisting on salience 
of this sort, and financial rewards (rather than say points, course credit, hypothetical 
dollars, etc.) is articulated by Smith (1976, p. 277): 

"Individuals may attach game value to experimental outcomes.. .Because of 
such game utilities it is often possible in simple-task experiments to get 
satisfactory results without monetary rewards by using instructions to induce 
value by role-playing behavior... But such game values are likely to be weak, 
erratic, and easily dominated by transactions costs, and subjects may be 
readily satiated with 'point' profits." 

The insistence on paying subjects is absolute among experimental E's even though 
most, in their hearts and in hallway discussions, are agnostic about whether results 

3 

would differ substantially if subjects' payments were hypothetical. The insistence 
on money seems to be a fetish arising from the frequently-made assumption that 
people are self-interested and largely motivated by money. My view is that more 
comparisons of high- and low-incentive conditions (including hypothetical payment) 
are needed to test Smith's assertions. Probably the result of such studies would be 
that in some situations, where the task is either particularly hard or simple (or 
intrinsic motivation is high), paying performance-based incentives makes little 
difference. In tasks which get boring quickly (e.g., "probability matching"), or 
where performance is of intermediate responsiveness to effort, paying subjects could 
make a lot of difference. Also, many studies comparing hypothetical and actual 



In the movie ’’Groundhog Day” a hapless reporter is forced to repeat the same 24- 
hour sequence of interactions over and over, with everything unchanged except his own 
memory of the outcomes of his actions in the previous day. The endless looping of 
experience, while boring, turns out to be ideal for learning since different actions can be 
tried while everything else is held equal. This is precisely the ’’stationary replication” 
design widely used in experimental economics. 

It is virtually impossible (I know of no examples in the last ten years or so) to publish 
experimental results in a leading economics journal without paying subjects according to 
their performance. Paying subjects often costs substantial sums of money, which acts like 
a very high submission fee, and may particularly limit the ability of younger investigators 
or those in less well-funded institutions to generate widely-read results. Also, paying 
subjects or using incentive-compatible mechanisms (like the "lottery ticket” procedure for 
inducing risk tastes, or the Becker-DeGroot-Marschak procedure for selling prices) can 
complicate an experiment or produce demand effects. 




320 



payments have found that the frequency of outliers is reduced by paying real money, 
which may be particularly important in outlier-sensitive situations, like minimum- 
action coordination games, or markets for long-lived assets which can exhibit price 
bubbles. 

In contrast, experimental P's usually do not pay subjects (though some do); 
instead they rely on the intrinsic motivation of subjects who volunteered out of 
curiosity, and whose attention is sustained by pride and eagerness to do a good job. 
(Unfortunately, in my view, many subjects in introductory P classes are forced to 
participate in experiments as part of course credit; they may indeed be readily 
satiated or only erratically motivated by points, as Smith feared.) 

Data reporting and analysis : Styles of data analysis are different in E and P. 
Experimental E's generally follow the useful convention in reporting other kinds of 
empirical economics (e.g., econometric results), report lots of raw data, using clever 
graphical methods and often testing hypotheses in statistically sophisticated ways 
(see, e.g., El-Gamal & Palfrey, in press, on Bayesian methods and "optimal design"). 
Then readers, knowing as much about the data as space permits, can form their own 
opinions about the results rather than blindly accepting the author's. 

In my opinion, data reporting and analysis standards are simply lower in many 
domains of experimental P (particularly social psychology). P's often report highly 
aggregated data, using embarassingly simple bar charts, with outliers trimmed 
capriciously and analyzed with crude statistical tests that are usually parametric and 
often badly misspecified. (F-tests are routinely used for differences in variances of 
grouped data that are clearly non-normal; tests assuming continuous variables are 
often applied to discrete data like 5- or 7-point scale ratings.) There also appears to 
be more of a "file drawer" problem in psychology— only surprisingly large effects 
are reported, ensuring that exact replications will show smaller effects due to 
regression-toward-the-mean. (Some psychologists publish only a third or fourth of 
the data they collect, which suggests the magnitude of the file drawer problem that 
can result if findings are not carefully replicated before publication.) While my 
description of experimental P data analysis methods is harsh, I think they can again 
be understood as partly forgivable in the light of other features of psychological 
research style. Experimental P's often have access to large subject pools at low cost 
(since subjects are often not paid at all), so free subject labor can substitute for 
expensive experimenter human capital in designing efficient statistical tests. And in 
experimental P, the experimenter writes a script (perhaps a short vignette, like a 
joke) or lets a drama unfold that illustrates an informal claim about human nature. 
Readers give the experimenter the benefit of the doubt (tmsting her results and not 
needing to see raw data), much as theatergoers trust the playwright. In this view, the 
file drawer effect results from refusing to stage bad plays. 

Deception: The use of deception is another important stylistic difference between 
E's and P's: E's hate it, and P's often don't mind. For the sake of discussion, define 
deception as actively claiming something to be true, when you know it to be false. 
For example, some experimenters have implicitly promised to pay subjects money in 
a way that (saliently) depended on choices, then told them at the end they would 




321 



earn a flat amount. In other cases a confederate subject, playing a role necessary to 
create a social illusion or treatment, is implicitly introduced as a fellow subject, 
experimenter, former subject, etc. 

Experimental E's hate deception because they feel it is often unnecessary, and 
more importantly, harms the credibility of all experimenters. Since American 
Psychological Association rules require experimenters to inform a subject about any 
deception in a post-experiment debriefing, veteran subjects always know they have 
been lied to before. Some studies also indicate that subjects who are told about a 
deception in a debriefing do not alway s believe the debriefing either. (Should they?) 
An example will help illustrate some features of this disagreement about deception. 

Many experimenters have studied "ultimatum" games in which one person offers 
a division of a sum X to a responder. Then the responder either takes it, or rejects it 
and leaves both with nothing (see Guth et al, 1982; Camerer & Thaler, 1995). 
Suppose you were interested in whether a sense of "entitlement" affected the kinds 
of divisions players offered and accepted. Hoffinan et al (1994) studied this by 
having pairs of subjects play a simple game of skill. Those who won got to make 
the offer in a subsequent ultimatum game; those who lost could accept or reject the 
winner's offer. They found that winners offered less (keeping more for themselves) 
and losers were willing to accept less. Winning created entitlement. 

Or did it? There is a subtle problem with this design: It perfectly correlates game- 
playing ability and entitlement generated by winning. Their results are consistent 
with the hypothesis that skilled game-players always demand more, unskilled 
players always accept less, and game-playing simply sorted players by their 
bargaining aggressiveness (or timidity) rather than created entitlement per se. For a 
psychologist, the natural way to break this two-variable confound is to play a game, 
then randomize feedback on whether players won or lost. That way, half the skilled 
game-players land in the low-entitlement group (they are told, deceptively, they 
lost); and half the unskilled players land in the high-entitlement group. Deception is 
required in this alternative design; it is hard to see how to break the skill-entitlement 
confound otherwise. One can see how experimental P's might criticize the Hoffman 
et al design for allowing a confound, and if one is eager enough to truly separate 
skill and entitlement, deception may be scientifically useful. 

This example illustrates the scientific benefit of deception. But it is important to 
take account of the possible costs of deception and weigh costs and benefits. Two 
costs loom largest: First, repeated deception probably harms overall experimenter 
credibility, but the extent of harm and whose experiments are jeopardized is not well 
understood. Second, since deception is generally used to create an unnatural social 
situation— such as a non-relation between skill and entitlement— one must ask: If the 
situation being created is sufficiently unnatural to require deception, will subjects 
belie\ e the deception? If subjects entertain the hypothesis that they are being 
deceived, then the most necessary deceptions— to produce the most unnatural 
situations— are least likely to be believed. This point can be illustrated by further 
ultimatum experiments. 




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Polzer, Neale & Glenn (1993) (PNG) used an ultimatum game design that did 
break the confound between skill and entitlement. After their subjects did a "word 
find" task, they were given deceptive feedback about actual performance, in order to 
sort half the high-skilled players into a low-entitlement condition (by telling them 
they performed more poorly than the subject they were paired with). PNG reported 
that entitled ("justified") players offered a modest $.32 less than nonentitled players 
(out of $10 being divided). More interestingly, the responder subjects who were 
nonentitled (were told they had low performance) demanded $.41 more than entitled 
players did. A natural possibility is that the deceptive feedback was not believed by 
some of the players who were told they were less skilled. Evidence for this 
possibility is the fact that the surprising increase in demands of nonentitled players is 
larger among groups of (self-selected) friends, who came to the experiment together 
and, knowing more about each other, were more likely to doubt the false feedback, 
than among groups of strangers. 



3 An Example: Fairness in Surveys and Markets 

An example will help illustrate some of the differences in experimental E and P 
styles summarized in Table 2. 

3.1 Surveys of Fairness 

Many social scientists have been interested in perceptions of fairness and their 
impact on social life. Kahneman, Knetsch & Thaler (KKT) (1986) set out to 
describe some of the rules people use to judge whether economic transactions are 
fair. They decided to use surveys in which short vignettes were described, and 
respondents could answer whether a transaction was fair or not. For example, 
subjects were told: 

A hardware store has been selling snow shovels for $15. The morning after a 
large snowstorm, the store raises the price to $20. Please rate this action as: 

Completely Fair Acceptable Unfair Very Unfair 

Of 107 respondents, 82% said the store's price increase was unfair or very unfair. 
From dozens of questions like this, they induced several basic propositions about 
perceptions of fairness. 

Their method illustrates some classic features of experimental P. The survey data 
are used to construct theory, not test it. The natural context makes the question more 
engaging than a drier, abstract version ("Is it fair to mark up prices after shortages?") 
would be, and perhaps easier to comprehend as well. The subjects were randomly 
chosen and telephoned, because KKT thought that average folks were obviously the 
people whose behavior they wanted to model, and students might be unfamiliar with 
many of the contexts they asked questions about (e.g., employers raising or cutting 




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wages). The data are answers to questions, rather than choices with financial 
consequences. No repetition is necessary (though many different questions were 
asked) since nothing much can be learned from asking the same question twice. The 
data analysis is simple (and in this case, need not be complicated). No deception is 
involved. 

From an experimental E view, these data tell us nothing about "real" behavior 
with substantial consequences for respondents. But the data are clearly useful for 
theory construction if one presumes only that (i) subjects are intrinsically motivated 
to answer accurately (i.e., to report what they really would feel about fairness if they 
rushed in from the snow and picked up a shovel marked $15), or have no reason not 
to; and (ii) their perceptions of fairness have something to do with their willingness 
to forego buying a shovel they think is priced unfairly. A critic who thinks the 
answers are worthless is implicitly assuming that either (i) or (ii) are false, which 
reflects a judgment about human behavior that itself seems empirically false to 
psychologists who have experience with survey design. 

3.2 Fairness in Markets 

Inspired by the KKT results, Kachehneier & Limberg & Schadewald (KLS 1991) 
designed a traditional E experiment to see whether fairness effects would affect 
prices and quantities in a market. Their clever design starts with the proposition, 
derived by KKT from their surveys, that a price increase is fair if it is accompanied 
by an increase in marginal costs, and is unfair otherwise. (For example, raising the 
snowshovel price to $15 after the snowstorm-induced demand shock is unfair- 
according to respondents to the question above— but subjects also said raising the 
retail price of lettuce is fair after a transportation mixup created a shortage and raised 
the wholesale price.) 

Their design began with several periods of posted-bid trading in which each of 
several buyers posted a bid at which they would buy a unit of an unspecified 
commodity from a seller. (This unusual design enabled direct observation of how 
buyers reacted to a visible cost change, the closest analogy to the KKT surveys.) 
Than partway through, a tax on seller revenues was imposed. The tax effectively 
shifts the marginal cost curve (since the price at which a seller must sell to earn zero 
after-tax profit is now higher). 

In a control condition, the buyers knew nothing about the tax; the fairness theory 
predicts they may object to price increases in this case. In the cost-disclosure 
condition, buyers were told about the revenue tax and it was explained why this 
might raise prices. Fairness theory predicts they will treat price increase as more fair 
in this case, and prices will rise more rapidly. In a third profit-disclosure condition, 
buyers were told the effect of the tax on the sellers' share of total surplus (at various 
possible prices). By design, the revenue tax also raised the sellers' share of total 
surplus at every price, so fairness theory predicts that buyers might resist paying 
higher prices in this case. 




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As the survey results implied, behavior in the three disclosure conditions was 
different. When the cost effect was disclosed, prices rose more rapidly to the new 
equilibrium price (buyers appeared to forgive price increases, or treated them as 
more fair). When the unfair profit effect was disclosed as well, price increases were 
slower. The no-disclosure control condition prices fell in between. Deng et al (1994) 
replicated the KLS findings using a more traditional posted-offer institution, and 
found similar convergence patterns but little difference in equilibrium prices across 
the information treatments. 

The ingenious KLS design takes the KKT results seriously but plays by 
experimental E rules. The experiment tests theory, in an abstract context (making 
the influence of fairness all the more remarkable and plain— it stems purely from 
attitudes about divisions of money, not about shovels or lettuce). Subjects make 
choices (bids and sales) with financial consequences, over groundhog-day repeated 
trials to see if fairness effects "wear off’ (they do, to some extent). The time series of 
data are reported, so readers can judge how quickly equilibration occurs, and tests 
for differences in conditions are sophisticated (and conservative). No deception is 
involved. 

The KKT-KLS interplay also shows how much can be learned by cross- 
fertilization of the E and P approaches. The KLS design would never have come 
about if many P-style survey questions did not reveal the texture of perceptions of 
fairness, and provide a clear hypothesis testable in later work. At the same time, 
survey questions are not well-designed, as the experimental market is, to test how 
much people will pay to punish unfair behavior, and how their behavior changes 
with repetition. So the two methods are productive complements in producing a 
fuller understanding of fairness in economics. 



4. Conclusions and Caveats 

4.1 Which Approach is Better? 

My own tastes find something of special value in each of the E and P approaches 
that could be usefully imported into the other. 

The modelling tradition in E, and guidance from theory, imposes useful 
discipline on economics experiments. Designing an experiment with a theory in 
mind forces you to be crystal clear about why all the pieces of the design are there. 
(It also clarifies theory, by forcing a theorist to be clear about at least one concrete 
domain in which the theory applies.) Designing to test theory has enabled us to 
create extraordinarily efficient experiments in domains where multiple theories 
compete. For example, Schotter, Weigelt & Wilson (1994) and Banks, Camerer, 
Porter (1994) each constructed a series of games to test several different equilibrium 
concepts. Designs of such efficiency are rare in psychology, at least partly because 




325 



specific theories which predict point estimates or different directional responses to 
variables are less common. 

Experimental E's are also extremely good about reporting raw data (many 
articles include raw data and instructions, to make reanalysis or replication by 
readers easy). 

On the other hand, many precepts in the E approach are restrictive. Ignoring 
almost all data but choices wastes a valuable opportunity to learn something more 
from a group of subjects who are often eager to explain their thinking processes and 
inferences. (Whether their thoughts are useful or not is difficult to answer, but it is 
surely less difficult to answer if we collect such data!) Also, many types of 
apparently non-choice behavior are choices and should be given equal or 
comparable status. For example, the time it takes a subject to respond in making a 
choice is itself a choice (of how much time to spend answering the question); 
theories of decision cost might make predictions about this time-choice. 

Insisting on paying subjects according to performance may miss the opportunity 
to learn something by varying incentives (see Smith & Walker, 1993), impose 
special burdens on poorly-funded investigators, and complicate some experiments 
unnecessarily. 

In addition, the distaste for "rich”, naturally-labelled contexts has inhibited the 
scope of problems people have studied (particularly those lying most squarely at the 
crossroads of psychology and economics); though happily, I think this stricture has 
weakened substantially in recent years. 

4.2 Some Caveats 

I end with some caveats. Obviously, there are many experimental E's and P's 
who do not fit well into the Table 2 columns, or fit in the "wrong" one. 
Mathematical psychologists are few in number but respect formalism and contribute 
substantially to it. Cognitive psychologists often study abstract tasks and offer 
modest incentives (perhaps to alleviate boredom). Some psychologists, like Amnon 
Rapoport, Ward Edwards, and Robyn Dawes, play by experimental economics rules 
for the same reasons E's do-- to force subjects to take tasks seriously, to make 
incentives (and hence performance) salient, to study the effect of learning over 
repeated trials, etc. 

And of course, some groups of experimental E's are more curious about 
psychological regularities, and less inclined to express findings in narrow 
formalisms. Prominent among such eclectic E's are members of the "German 
school". 

Which brings me to Reinhard Selten, who founded and inspired the German 
school along with Heinz Sauermann. I wrote this essay for a book honoring the 



There is a substantial literature in psychology on response times (for example, they 
are often used as a primary dependent variable in theories of memory). Among decision 
theorists, John Hey, Nat Wilcox, and others have found them useful to study. 




326 



remarkable Reinhard Selten because his own work combines the very best of both 
the E and P approaches. But how? 

Selten does it by having an open mind, reading and thinking voraciously, and by 
taking a "dualist" position. The dualist position is that formal theories of rational 
behavior are important for answering sharply-posed analytical questions, and 
possibly as normative bases for ideal living, but it is silly to think such theories are 
the best possible account of complicated behavior of actual humans. Instead, 
rational theories and behavioral (descriptive) theories must flow along separately. 

Consequently, Selten and others in the German school mix and match 
conventions of the experimental E and P approaches as they see fit. Selten's 
experimental papers reflect the P's characteristic attention to details of subjects' 
decision rules and individual differences. For example, Selten developed the 
"strategy method", a survey/choice hybrid in which subjects report what they would 
choose (and then are forced to) for each of several possible realizations of a random 
variable, like a private value in an auction, or a randomly-determined Harsanyi 
"type" in an incomplete-information game. For most purposes, the strategy method 
produces much richer data and enables within-subject tests which are much more 
statistically efficient than single-choice methods. 

To achieve mastery of a single mathematical approach or experimental style in a 
lifetime is difficult enough. To master two in a lifetime hardly seems possible, 
except that Reinhard Selten has done it. 



References 

Ball, Sheryl B. and Paula- Ann Cech. (1990), "The What, When, and Why of Picking a 
Subject Pool," Boston University working paper. 

Banks, Jeff, Colin F. Camerer, and David Porter. (1994), "Experimental Tests of Refinements 
of Nash equilibrium in signaling games," Games and Economic Behavior, 6: 1-31. 

Camerer, C. F., Eric J. Johnson, Talia Rymon, and Sankar Sen. (1994), "Cognition and 
Framing in Sequential Bargaining for Gains and Losses," in K. Binmore, A. Kirman, 
and P. Tani (Eds.), Frontiers of Game Theory, Cambridge: MIT Press, 11 -Al. 

Camerer, Colin F,. and Richard H. Thaler. (1995), "Anomalies: Ultimatums, dictators, and 
manners," Journal of Economic Perspectives, 9: 209-219. 

Cox, James C., and R. Mark Isaac. (1986), “Experimental economics and experimental 
psychology: Ever the twain shall meet?” in A.J. MacFadyen and H. W. MacFadyen 
(Eds.), Economic Psychology: Intersections in Theory and Application, Amsterdam: 
North-Holland. 

Davis, Doug L., and Charles Holt. (1992), Experimental economics. Princeton: Princeton 
University Press. 

Deng, Gang, Robert Franciosi, Praveen Kujal, Roland Michelitsch, and Vernon Smith. (1994), 
"Fairness: Effects on temporary and equilibrium prices in posted offer markets," 
unpublished. 

El-Gamal, Mahmoud, and Thomas R. Palfrey. “Economical experiments: Bayesian efficient 
experimental designs,” International Journal of Game Theory, in press. 




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Friedman, Daniel and Shayam Sunder. (1994), Experimental Methods: A Primer for 
Economists. Cambridge: Cambridge University Press. 

Giith, Werner, Rolf Schmittberger, and Bemd Schwarze. (1982), "An experimental analysis of 
ultimatum bargaining," Journal of Economic Behavior and Organization, 3: 367-388. 

Hey, John D. (1991), Experiments in Economics. Cambridge, UK: Blackwell. 

Hoffman, Elizabeth, Kevin McCabe,Keith Shachat, and Vernon L. Smith (1994), 
"Preferences, property rights and anonymity in bargaining games," Games & Economic 
Behavior, 1: 346-380. 

Kachelmeier, Steven, S. Limberg ,and M. Schadewald. (1991), "A laboratory market 
examination of the consumer price response to information about producers' costs and 
profits," The Accounting Review, 66: 694-717. 

Kahneman, Daniel, Jack Knetsch, and Richard Thaler. (1986), "Fairness as a constraint on 
profit seeking: Entitlements in the market", American Economic Review, 76: 728-741. 

Kagel, John H., Battalio, Ray C., and Leonard Green. (1995), Economic Choice Theory: An 
Experimental Analysis of Animal Behavior. Cambridge: Cambridge University Press. 

Kagel, John L. and Alvin E. Roth (Eds.). (1995), Handbook of Experimental Economics, 
Princeton: Princeton University Press. 

Polzer, Jeff T., Margaret A. Neale, and Patrick O. Glenn. (1993), "The effects of relationships 
and justification in an interdependent allocation task," Group Decision and Negotiation, 
2: 135-148. 

Schotter, Andrew, Keith Weigelt, and Charles Wilson. (1994), "A laboratory investigation of 
multiperson rationality and presentation effects," Games and Economic Behavior, 6: 445- 
468. 

Smith, Vernon L. (1976), "Experimental economics: Induced value theory," American 
Economic Review Papers and Proceedings, 66: 274-279. 

Smith, Vernon L. (1982), "Microeconomic systems as an experimental science," American 
Economic Review,. 12: 923-955. 

Smith, Vernon L., and James M. Walker. (1993), "Monetary rewards and decision cost in 
experimental economics, " Economic Inquiry, 31: 245-261. 




Reciprocity: The Behavioral Foundations of 
Socio-Economic Games 



Elizabeth Hoffinan/ Kevin McCabe,^ and Vernon Smith^ 

* College of Liberal Arts and Sciences, 208 Carver Hall, Iowa State University, 

Ames, lA 50011 

^ Carlson School of Management, Department of Accounting, University of Minnesota, 
Minneapolis, MN 55455 

^ Economic Science Laboratory, McClelland Hall 1 16, University of Arizona, 

Tucson, AZ 85721 



Abstract. Growing experimental evidence and field research results on small group 
behavior suggest that the assumption of self interest must be expanded to allow more 
general forms of commitment in the form of reciprocity, defined as a response to 
another person's action that forgoes immediate benefits or incurs immediate costs. 
Negative and positive reciprocity have emerged as solutions to two important socio- 
economic activities: distribution and investment. The development of social 

institutions can be viewed as attempts to encourage reciprocity. Evolutionary 
psychologists hypothesize that the widespread development of reciprocity in human 
society results from the evolution of specialize mental algorithms for developing and 
punishing cheaters who behave noncooperatively in social exchange. We build on 
this work, and extend it as an organizing principle to examine subject behavior in 
ultimatum, dictator, and more general extensive form bargaining games. 

Keywords. Experimental Games, Reciprocity, Evolutionary Psychology 



1. Introduction 

Why do people cooperate? In many 'commons'-type environments economists 
theorize that individual incentives to 'free ride' will interfere with attempts to 
increase social efficiency. While there exist numerous examples of noncooperative 
behavior, particularly under private information, the puzzle for economists is the 
extensive existence of small group cooperative behavior. Anthropological and 
archeological evidence suggests that sharing behavior is ubiquitous in primitive 
cultures that lack modem economic institutions for storing and redistributing wealth 
(see Cosmides and Tooby, 1987, 1989; Isaac, 1978; Kaplin and Hill, 1985; Lee and 
De Vore, 1968; Tooby and De Vore, 1987; and Trivers, 1971). Extensive field 
research by Ostrom et. al. at Indiana University have identified many contemporary 
and historical cases of decentralized cooperative solutions to commons problems 
(see Ostrom, Gardner, and Walker, 1994). 




329 



Experimental evidence on small group behavior suggests that the assumption of 
self interest must be expanded to allow more general forms of commitment in the 
form of reciprocity (see Axelrod, 1984). Reciprocity is defined as a response to 
another person’s action that forgoes immediate benefits or incurs immediate costs. 
In addition we will maintain the perspective that the use of reciprocity can be either a 
conscious, or an unconscious attempt to achieve greater individual fitness. 
Reciprocity can be divided into negative reciprocity, a form of commitment, such 
that individuals are willing to punish inappropriate behavior, and positive 
reciprocity, a form of obligation such that individuals are willing to reward 
appropriate behavior. 

From a rational choice perspective, subjects who deviate from the subgame 
perfect strategies must believe this will increase their expected return; but why 
would they believe this? One approach is to look at the role of reciprocity in a 
sequence of games; either self-enforcing equilibria or reputational equilibria (see for 
example Kreps, 1990). But the literature on folk theorems for repeated games tell us 
the following: while equilibria with efficient social exchange can emerge in repeated 
games, they represent only a small subset of the possible equilibria that may occur. 
While the theory of repeated games is useful for explaining how reciprocity may be 
supported by noncooperative behavior, it does not explain the origin of this 
behavior. 

One promising approach to explain the existence of reciprocity is to derive this 
behavior as an evolutionary stable strategy, as for example, in the work by Robert 
Frank (1988) on the strategic role of the emotions. The evolutionary approach is 
currently being studied in economics (see Giith and Kliemt, 1994) evolutionary 
psychology (Cosmides and Tooby 1992, herein denoted CT), and evolutionary 
biology (Smith, 1989). 

The thesis of this paper is that in addition to extending the gains from exchange 
to nonmarket settings, negative and positive reciprocity have emerged as solutions to 
two important socio-economic activities: distribution and investment. Furthermore, 
the emergence of some social institutions can then be explained as attempts to 
encourage reciprocity. In developing this thesis we note that opportunities for 
reciprocity, together with the possibility of bounded rationality (Simon, 1979 and 
Selten, 1989) and the endogenous development of common expectations, suggest 
many potential, and often subtle, interactions between players' expectations, 
preferences, messages, and environmental factors. These complexities make it 
difficult to propose positive theories without experimental testing. One experimental 
approach, developed by evolutionary psychologists (CT, 1992), has been to study 
possible specialized mental algorithms for social exchange. 

In the next sections we outline the approach developed in evolutionary 
psychology and relate the outcomes of that research to results in experimental 
economics on cooperation in bargaining games. 




330 



2. Experimental Detection of Specialized Mental Algorithms 
for Social Exchange 

Neural science has developed a large body of knowledge on the specialized 
architecture of the brain. For example, we know that the brain has a dedicated 
architecture for the input and processing of visual sense data. The idea that the brain 
is specialized has led to the development of computational theory by evolutionary 
biologists to explain the development of brain architecture as the solution to 
particular adaptive problems. An example is the work by Marr (1982) on the 
adaptive problem of visual perception. Also see Wallman (1994) on the evidence 
that what the eye sees affects how it grows. Thus myopia and hyperopia, while 
having a genetic basis, are also influenced by growth factors affected by reading (the 
eye focuses on the page) and outdoor living (the eye focuses on infinity). Similarly, 
the language ability of the human brain is linked to the development of the human 
vocal tract. Up to the age of three months the human infant's larynx engages the 
nasal passage, making it impossible to breathe and drink simultaneously; it is also 
impossible to form words. This throat structure is similar to the structure of all other 
adult nonhuman mammals who, likewise, cannot speak. After the age of three 
months the larynx descends deep enough into the throat to allow the tongue to move 
back and forth, producing the vowel sounds emitted by adult human beings. This 
adaptation illustrates the biological basis of human language behavior, which in turn 
is linked inseparably with human social behavior. 

While sensory input may depend on specialized mental algorithms, it has long 
been thought that the analysis of this information involves general logic-driven 
mental algorithms. However, experimental evidence questions the validity of this 
conjecture. For example Wason (1966) developed a task to study how efficient 
people are at detecting violations of material implication. The logic of falsifying a 
statement of the form "if P, then Q," requires people to understand that it is violated 
only under the condition that P is tme and Q is false. In the Wason task, four cards 
each contain an antecedent on one side and a consequent on the other. A subject is 
shown one side of each of four cards, displaying a tme antecedent (P), a false 
antecedent (not P), a tme consequent (Q), and a false consequent (not Q). 

Subjects are asked to indicate only those card(s) that definitely need to be turned 
over in order to see if any cases violate the mle. The correct answer is to select the 
cards showing P (to see if there is a not-Q on the other side) and not-Q (to see if 
there is a P on the other side). Here is an example of such a task. A subject is 
informed that part of your new clerical job at the local high school is to make sure 
that student documents have been processed correctly. Your job is to make sure the 
documents conform to the following alphanumeric mle: 

If a person has a 'D' rating, then his document must be marked code '3'. 

(If P[D] then Q(3) )* 




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You suspect the secretary you replaced did not categorize the students' documents 
correctly. The cards below have information about the documents of four people 
who are enrolled at this high school. Each card represents one person. One side of a 
card tells a person's letter rating and the other side of the card tells that person's 
number code. Indicate only those card(s) you definitely need to turn over to see if 
the documents of any of these people violate this rule. 

|D| |F| |3| |7| 

(P) (not-P) (Q) (not-Q) 

♦Parenthetical entries were not included in actual instructions. 

The correct responses are to indicate the cards showing the letter D (P) and the 
numeral 7 (not-Q). The results of this experiment show that less than 25% of college 
students choose both of these cards correctly. 

Now consider an example that concerns a law which states that "If a person is 
drinking beer, then he must be over 20 years old." The cards read: drinking beer, 16 
years old, not drinking beer, 25 years old. The correct response is to choose the card 
"drinking beer" and the card "16 years old." In this experiment about 75% of college 
students get it right. There are many differences between this example and the 
previous one, but note that one difference is that the second rule involves a social 
contract. 

Another interpretation of the difference in results between the example of the 
students' documents problem and the underage drinking problem might be that the 
underage drinking problem refers to a more familiar situation. In that case we might 
say that there are content-effects; the familiar content is responsible for the improved 
rate of positive responses. However, in a survey of this literature (Cosmides, 1985) 
suggests that "Robust and replicable content effects were found only for rules that 
related terms that are recognizable as benefits and cost/requirements in the format of 
a standard social contract ..." (CT, 1992, p. 183). Sixteen out of sixteen experiments 
using social contracts showed large content effects. For non-social contract rules, of 
19 experiments 14 produced no effect, 2 produced a weak effect and 3 produced a 
substantial effect. The bottom line from several such studies was "Familiarity ... 
cannot account for the pattern of reasoning elicited by social contract problems" 
(CT, 1992, p. 187). 

In all of the above cases, the correct adaptive response is identical to the correct 
logical response. Can more crucial experiments be designed to separate this 
confounding effect? The evolutionary psychology literature has produced two 
distinct groups of experiments that confront this issue: (i) experiments that switch 
the P and Q in the standard social exchange task thereby breaking the identity 
between cheating detection and the correct logical response; (ii) experiments that 
change the subjects' perspective from own to other, thereby changing the 
interpretation of what constitutes cheating. For example "If an employee gets a 
pension, then the employee must have worked for the firm for at least 10 years." 




332 



What constitutes cheating is different from the perspective of the employee than 
from that of the employer. But the correct Popperian response is independent of 
perspective. The experiments show that subjects' answers switch with perspective, 
thereby supporting the adaptive hypothesis. 



3. Constituent Games For Social Exchange 

If humans have specialized mental algorithms that assist them in achieving 
cooperative outcomes in social exchange environments, then factors that facilitate 
the operation of these natural mechanisms should increase both cooperative behavior 
and cooperative outcomes. For example, cooperative behavior should increase if 
individuals can observe and monitor one anothers behaviors, even if there are no 
direct mechanisms for enforcement. If it is possible for agents to directly punish 
cheating by other agents, cooperative behavior should increase even further. 
Similarly, if agents can communicate with one another, they can frame a group 
decision as a social exchange problem and activate natural inclinations to cooperate. 

If, in the absence of direct communication, agents can signal other agents of their 
intentions, they can create shared expectations of social exchange and cooperative 
behavior. 

If reciprocity has evolved in response to particular adaptive problems, then this 
should have implications for behavior in basic constituent games for social 
exchange: the ultimatum (dictator) game and the investment game. Ultimatum 
(dictator) games have been used to study the effect of negative reciprocity on 
bilateral bargaining. From these studies we learn that negative reciprocity moderates 
selfish behavior through norms which place limits on noncooperative behavior. This 
moderation has a direct consequence for terms of trade and ultimately the 
distribution of goods and services. Investment games have been used to study the 
effects of positive reciprocity on investment decisions. From these studies we have 
learned that positive reciprocity can increase investment opportunities, thus having a 
direct consequence for economic growth. 

3.1. The Ultimatum Game 

In an ultimatum game, player 1 offers $X to player 2 from a total of $M. (The 
amount M has varied in experiments: $5, $10 and $100). If player 2 accepts the 
offer, then player 1 is paid $(M-X) and player 2 receives $X; if player 2 rejects the 
offer, each gets $0. In the dictator game player 1 makes an offer that player 2 must 
accept. 

If we assume subjects come to an ultimatum or dictator game with no 
preconceived norms of behavior in such an environment, that all subjects prefer 
more money to less, and that all subjects have common knowledge of these 
characteristics of all other subjects, then the noncooperative equilibrium of the 
ultimatum game is for player 1 to offer player 2 the smallest unit of account, and for 




333 



player 2 to accept this offer. However, player 2 can punish player 1 for "cheating" 
on an implied social norm of sharing across time, learned in social exchange 
experience, by rejecting player I's offer. In the absence of common knowledge, the 
possibility of punishment may change player I's equilibrium strategy. In the dictator 
game, noncooperative game theory predicts that player I’s should keep all the 
money, i.e. send nothing to player 2's. 

Brewer and Crano (1994), a recent social psychology textbook, identifies three 
important norms of social exchange that may apply in ultimatum games. The norm 
of equality implies that gains should be shared equally in the absence of any 
objective differences between individuals that would suggest some other sharing 
rule. The norm of equity implies that individuals who contribute more to a social 
exchange should gain a larger share of the returns. The norm of reciprocity implies 
that if one individual offers a share to another individual, the second individual 
reciprocates as soon as possible. 

The ultimatum game was first studied by GUth, Schmittberger, and Schwarze 
(1982, herein GSS) and has now been replicated a large number of times. Figure 1 A 

% Frequency 




% Offer 



m Offers j I Rejections 



Figure lA. Percentage frequency of percentage of pie offered to player 2 in ultimatum game 
experiments. (A) GSS, variable pie ultimatum, inexperienced subjects, N = 21. Offers are 
rounded to the nearest 10%. 





334 



graphs the percentage frequency of percentage pie offered to player 2 (rounded to 
the nearest 10%) in the GSS ultimatum games with inexperienced subjects (M is 
variable and the number of pairs is N = 21). This figure is typical of ultimatum 
games in that subjects often make proposals which offer receivers much more than 
the smallest $ unit of account. The modal proposal is often 50%. Furthermore 
player 2s will often reject asymmetric proposals even though these proposals offer 
them as much as 40% of M.In a direct test of the importance of negative reciprocity 
in explaining proposals in ultimatum games, Forsythe, Horowitz, Savin and Sefton 
(1994, herein FHSS) compare ultimatum game results to dictator game results. 
Figure IB shows the FHSS replication of the ultimatum game using M = $10 (N = 
24). Comparing Figure 1 A with IB we observe that the FHSS replication has led to 
even stronger sharing. Hoffinan, McCabe, Smith (1994a) suggest that this is due to 
the strengthening of norms of reciprocity found in the FHSS instructions and 
procedures. FHSS also find that removing the threat of punishment, as in the 



% Frequency 




0 10 20 30 40 50 60 



% Offer 

^ Offers H Rejections 



Figure IB. Percentage frequency of percentage of pie offered to player 2 in ultimatum game 
experiments. (B) FHSS, $10 ultimatum, inexperienced subjects, N = 24. Offers are rounded to 
the nearest 10%. 





335 



dictator game, reduces the amounts given to receivers, but not by as much as 
noncooperative game theory predicts. This is seen by comparing Figure 2A with 
Figure IB. Hoffinan, McCabe, Shachat, and Smith (1994, herein HMSS) design a 
version of the dictator experiment, which provides dictators with complete 
anonymity through double-blind controls. 




01 23456789 



Dollars 

I Offers 



Figure 2 A. Comparison of percentage frequency of offers to player 2's in Dictator 
Experiments. (A) FHSS, Dictator, Divide $10 experiment, N = 24. 



The distinguishing characteristic of the double-blind design is that dictators’ 
decisions cannot be identified by the experimenter, any other person who might 
observe the experiment or study the results, or the dictators' counterparts. Dictators 
and counterparts are recruited to separate rooms. Each dictator randomly draws an 
opaque envelope from a box. Each envelope contains twenty pieces of paper. 
Twelve contain ten $1 bills and ten white slips of paper. Two contain twenty white 
slips of paper. Each dictator takes the envelope to a private place removes ten pieces 
of paper, seals the envelope, and leaves the room without identification. When all 
dictators have finished, the box is delivered to the room where the counterparts are 
located. Each counterpart randomly selects an envelope. A subject monitor opens it 



336 



and records the number of $1 bills. The HMSS results are strikingly different from 
the dictator results summarized in FHSS, in which subjects were observed by the 
experimenter (compare Figures 2A and 2B). In the double-blind dictator 
experiments, 64% of players take all the money. 



% Frequency 

80 




0123456789 

Dollars 



Offers 



Figure 2B. Comparison of percentage frequency of offers to player 2's in Dictator 
Experiments. (B) HMSS, Dictator, Double-Blind 1 experiment, N = 36. An Epps-Singelton 
test statistic of 23.6 is significant at the .001 level. 



Hoffinan, McCabe and Smith (1994b) then vary each of the elements of the 
double-blind dictator experiment in ways intended to reduce the degree of "social 
distance" between the experimenter and others who might see the data, and the 
subject dictators while preserving complete anonymity between the dictator and their 
counterpart. Comparing the results on social distance to the authors' replication of 
FHSS's dictator experiments, they find that the experimental results form a predicted 
ordered set of distributions. As the social distance between the subject and others 
increases, the Player Is offers to Player 2s decreases. These results demonstrate quite 



337 



Strongly the power of observability in invoking innate social norms of reciprocity, 
even where direct punishment is not possible. 

Recognizing that the threat of punishment might create expectations that change 
player I's behavior, HMSS also consider experimental treatments explicitly designed 
to manipulate subject expectations about the social acceptability of proposals in 
ultimatum games. FHSS make no role distinction between the two individuals 
"provisionally allocated" $10, and they are told to "divide" the money. Not 
surprisingly, perhaps, in such an environment, more Player Is fear deviations from 
equal division will be punished as "cheating" on the implied social exchange. 
HMSS replicate the FHSS results in a slightly different experimental environment. 
The key elements they maintain are: (1) players 1 and 2 are randomly assigned to 
those positions; and (2) the task is described as proposing a "division" of $10 
"provisionally allocated to each pair." They refer to this treatment as random/divide 
$ 10 . 

HMSS then explore two treatment changes, in a 2x2 experimental design. First, 
without changing the reduced form of the game, HMSS describe it as a market in 
which the "seller" (player 1) chooses a price (division of $10) and the "buyer" 
(player 2) indicates whether he or she will buy or not buy (accept or not accept). 
From the perspective of social exchange, a seller might equitably earn a higher 
return than a buyer. Treatments with buyers and sellers are referred to as exchange 
treatments. Second, they make player Is earn the right to be a seller by scoring 
higher on a general knowledge quiz. Winners are then told they have "earned the 
right" to be sellers. Going back to social exchange, equity theory predicts that 
individuals who have earned the right to a higher return will be socially Justified in 
receiving that higher return. Treatments using a general knowledge quiz to assign 
property rights in the position of seller or player 1 are referred to as contest 
treatments. 

In a situation in which it is equitable for player 1 to receive a larger 
compensation than player 2 (i.e., contest/exchange) player 1 offers significantly less 
to player 2, and player 2 accepts with the same probability (no significant difference 
in the rejection frequencies) (compare Figures 3 A and IB). These results suggest 
that the change from (random/divide) to (contest/exchange) alters the shared 
expectations of the two players regarding the social exchange norm operating to 
determine an appropriate sharing rule. Moreover, Hoffrnan, McCabe, and Smith 
(1994a) find no significant reduction in offers when the amount to be divided is 
increased from $10 to $100 in these two treatments (compare Figures 3 A and 3B). 
This result suggests that attitudes toward socially acceptable proposals are very 
strong; as the stakes are increased, so is the opportunity cost of proposing an 
improper division and getting punished. Finally, the difference between 
(random/divide) and (contest/exchange) carries over to a non-double-blind dictator 
experiment as well. Thus, the change in expectations takes place even when there is 
no threat of punishment from player 2. 




338 



% Frequency 




Dollars 



m Offers Bejections 



Figure 3 A. Comparison of percentage frequency of offers to player 2s in (A) HMSS $10 
contest/exchange ultimatum game, N = 24. 



3.2. The Investment Game 

The investment game allows us to isolate the effect of positive reciprocity on 
investment behavior. Player 1 is given $M ($10) as a show up fee. In stage one, 
player 1 in room A decides how much of the $M (denoted $S) to send to an 
anonymous counterpart called player 2 in room B. It is common information that 
whatever amount is sent will be multiplied by some factor (f = 3) by the time it 
reaches player 2. In stage two, player 2 then decides how much of the money (f x 
$S) to return to player 1 . Note that stage two is a dictator game where the non- 
cooperative prediction is for player 2 to return nothing. But this in turn implies that 
player 1 should send nothing. Tmst occurs in the investment game whenever a 
positive amount is sent in stage one. Positive reciprocity occurs when subject 2 
sends back an amount greater than the amount sent. 





339 



% Frequency 




Dollars 



Offers m Rejectbns 



Figure 3B. Comparison of percentage frequency of offers to player 2s in (B) HMSa, $100 
contest/exchange ultimatum game, N = 23. An Epps-Singleton statistic of 1.37 leads to the 
conclusion that offer distributions are the same in the $10 and $100 treatments. However, the 
increase in rejection rates in the $100 ultimatum games is significant at the .001 level. 



In implementing the investment game, first studied by Berg, Dickhaut and 
McCabe (1995, hereinafter BDM), BDM modify the double-blind procedures used 
by HMSS in order to control for the possibility of repeat game effects and thus 
protect any observed results from being attributed to reputation, collusion, or the 
threat of punishment. Figure 4A graphs the amount sent from room A (shown as 
open circles; data has been sorted on amount sent), the tripled amount which reached 
room B (shown as grey bars), and the amount returned to room A (shown as solid 
circles). BDM observe a large (although variable) degree of trust and a significant 
(compared to double blind dictator games) degree of positive reciprocity (compare 
Figure 4A to 2B). 





340 



Dollars No History 




Figure 4A. Comparison of amounts sent by player I's ($S), total return to the pair (3 x $S), 
and payback decisions ($ amount returned to player I's by player 2's), in the investment game: 
(A) BDM, Treatment 1, No History, M = $10, f = 3, N = 32. 



There are many instances within organizations where public information reflects 
an organization’s social history. Since social history provides common information 
about the use of trust within an organization such a history may reinforce 
individuals' predisposition towards trust. Thus, Coleman (1990) argues that norms 
are more likely to be internalized when an individual clearly identifies with a 
particular group. The process of having an individual identify with a group is 
termed socialization. In the initial BDM experiment subjects were all University of 
Minnesota undergraduates. BDM run a second treatment with subjects from the 
same pool, where subjects were provided a social history about the behavior of 
subjects in the first treatment. By providing social history in a double-blind, -one- 




341 



shot setting, BDM are able to focus on the internalization of social norms, as 
opposed to other potential mechanisms for reciprocity such as reputation building. 



Dollars Social History 




Figure 4B. Comparison of amounts sent by player I's ($S), total return to the pair (3 x $S), 
and payback decisions ($ amount returned to player I's by player 2's), in the investment game: 
(B) BDM, Treatment 2, Social History, M = $10, f = 3, N = 28. 



Comparing Figure 4B to Figure 4A we observe an increase in both trust and 
reciprocity. BDM calculate Spearman's rank correlation coefficient, between the 
paired room A and room B decisions. Since absolute amounts sent and amounts 
returned will bias this statistic upwards, i.e., low amounts sent preclude some high 
returns, we compare amount sent to percentaged returned. An r^ = .01 suggests no 
correlation between amounts sent and payback decisions in Figure 4A. An r^ = .34 
was found to show a significant (at the .06 level using resampling methods) increase 
in correlation between amounts sent and payback decisions in Figure 4B where 
social history is provided. 




342 



In settings characterized by repeat interactions people can build a reputation for 
positive reciprocity. To study this behavior we will extend the basic investment 
setting studied in BDM to two periods. Subjects will be assigned partners at the 
beginning of the experiment and will keep the same partner for both periods. We 
will extend the double blind controls to this setting to allow direct comparisons of 
the data with both the no history and social history treatments in BDM. Pilots for 
this design have been run by Dickhaut, Hubbard, and McCabe (1995). Preliminary 
results suggest an increase in reciprocity in period one followed by a significant drop 
in reciprocity in period two. 



4. Discussion 

New modeling efforts have incorporated reciprocity into the utility assumptions of 
game theory. Bolton (1991), incorporates a loss to be 'unfairly' treated into subject's 
utility functions in order to explain a willingness to reject unacceptably low offers in 
ultimatum games. But his model is contradicted by results in the dictator game. 
Rabin (1993) more generally captures reciprocity by incorporating a 'kindness' 
function into subjects' utility in such a way as to capture the following behavior: as 
one's counterpart increases his or her 'kindness', the utility maximizing response is to 
be kinder in return. Putting norms directly into subjects utility functions is simply a 
shortcut for modeling a predisposition towards reciprocity; i.e. expectations are 
represented as utilities. An alternative might look at individuals learning strategies. 
How do individuals form initial expectations towards reciprocity? How do 
individuals confirm or modify these expectations? 



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Adaptation of Aspiration Levels 
- Theory and Experiment - 



Reinhard Tietz 

J.W. Goethe-University, 60054 Frankfurt a.M.^ 



1. Introduction 

Stimulated by the work of Herbert A, Simon, Reinhard Selten has concerned him- 
self with aspiration levels and bounded rationality since the beginning of his 
academic career.^ His joint paper with Heinz Sauermann "Anspruchsanpassungs- 
theorie der Unternehmung" was published already 1962.^ 

After presenting the most important components of this theory I will show how 
it has influenced my own work. Thirty years ago the two authors suggested to me 
to investigate also macroeconomic relations experimentally."^ For this purpose a 
sufficiently complex macroeconomic model had to be constructed which could be 
used as an experimental environment for negotiations with macroeconomic 
impact. For this model Reinhard Selten proposed the name KRESKO. I will show 
in which way the model is governed by multilateral adaptation processes of 
aspiration levels. 1 will then discuss bargaining theories controlled by aspiration 
levels which were developed exploratively based on these experiments. Finally I 
will touch on some open research questions. 



^ Translation of a paper presented at a colloquium in honor of Reinhard Selten in Bonn, 
January 27th, 1995. 1 hope \ha\. Reinhard Selten and myself will be able to continue and 
to complete the work on aspiration levels and bounded ratoionality started 30 years ago. 

^ Cf R. Selten, In search of better understanding of economic behavior, in: A. Heertje 
(Ed.), The Makers of Modem Economics, New York et al. 1994, pp. 115-139; H. A. 
Simon, Theories of Decision Making in Economics and Behavioral Science, The Ameri- 
can Economic Review, Volume 49 (1959), pp. 253-283; H. A. Simon, New Develop- 
ments in the Theory of the Firm, The American Economic Review, Volume 52 (1962), 
Papers and Proceedings, pp. 1-15. 

^ H. Sauermann and R. Selten, Anspmchsanpassungstheorie der Untemehmung, 
Zeitschrift ftir die Gesamte Staatswissenschaft, Volume 118 (1962), pp. 577-597. 

^ H Sauermann, Preface of the Editor to: R. Tietz, Ein anspmchsanpassungsorientiertes 
Wachstums- und Konjunkturmodell (KRESKO), Tubingen 1973, p. V. 




346 



2. The Unilateral Adaptation Theory of Aspiration Levels of the 
Firm 

The term aspiration level comes from experimental psychology where it was first 
used by Lewin et al.^. Sauermann and Selten define the aspiration level as the 
bundle of all aspirations that must be fulfilled in order to satisfy the individual fi 
An aspiration level is considered realistic if the aspirations can be expected to be 
reached.^ Only realistic aspiration levels are of importance in a theory of bounded 
rationality. An aspiration level is adapted upwards when possibilities of action are 
seen which allow a better solution. An aspiration level which no longer seems to 
be attainable has to be adapted downwards. 

The theory of adaptation of aspiration levels, presented by Sauermann and 
Selten for the situation of the firm, is a general theory of decision behavior in 
unilateral decision situations. 










Correction of 
decision rules 



Formation of 
expectations 



Adaptation of 
aspiration levels 



Figure 1: Flow Chart of the Decision Process^ 



As shown in Figure 1, this theory can be seen as a dynamic process consisting 
of 6 overlapping phases with feedback: Correction of decision rules, formation of 
expectations, adaptation of aspiration levels, plaiming, decision, and finally, 
realization. 



^ K. Lewin, R. Dembo, L. Festinger, andP. P. Sears, Level of Aspiration, in: J. McV. Hunt 
(Eds.), Personality and the Behavior Disorders, New York 1944, pp. 333-378. 

^ H. Sauermann andP. Selten, l.c., p. 579. 

*7 Ibid. 

^ Figures 1 to 6 are taken from Sauermann Selten, l.c. 




347 



The two most important components of the theory are the scheme of adaptation 
of aspirations and the scheme of influences. The scheme of adaptation of aspira- 
tions is defined as the grid of discrete aspiration values in the space of the goal 




variables. Each point of the grid represents a potential aspiration level. For each 
such point an order of importance determines which of the respective goal varia- 
bles is to be adapted upwards first. 

For an adaptation downwards a sacrifice variable is defined in which a sacri- 
fice, a reduction of an aspiration, has to be made. For two goal variables, for 
instance rentability and market share, the aspiration scheme could have the form 
shown in Figure 2. At each grid point the arrows indicate the upward and down- 
ward directions of adaptations. 

Figure 3 shows an upward adaptation from an earlier aspiration level (circle) to 
a new one (square) in accordance with the scheme of adaptation of aspirations of 
Figure 2. The realization area as expected hy the decision maker is regarded here 
as given and is indicated by hatching. The old aspiration level was not Pareto- 
optimal, thus inducing an upward adaptation. The last adaptation step switches to 
the second most important variable Z 2 , since an increase in the most important 
variable Z\ would lead out of the realization area and therefore is not allowed. 




348 



4 * 










///// 

///// 




t 






rrff? 

///// 


7777) 

///// 

'//// 


1 






T7T77 

///// 

W//i 


7777/ 

///// 








///// 

///// 

ai/1 


V/// 

///// 

///// 


77777 

///// 

am 




— > 



Figure 3: Upward Adaptation of Aspirations 



In contrast, Figure 4 shows in the first step a downward adaptation since the 
last aspiration level lies outside the realization area relevant here. The subsequent 
upward adaptation in the last step has to use the second most important variable, 
here Zj, tmtil a Pareto-optimal point is reached as an actual aspiration level. 









1 






. . r 


\ 


■ 








) 










1 


1 n - - 




m 


7T777 

///// 

am 


m 


J 


> 



> 

Zq 



Figure 4: Downward Adaptation of Aspirations with Subsequent Upward Adaptation 



349 



As the theory has been formulated rather generally, Sauermann and Selten have 
presented two more concrete models, the routine model and the plaiming model. 
The routine model disregards the first phase of Figure 7 , the correction of deci- 
sion rules. The search for new possibilities of action is thus omitted. The forma- 
tion of expectations is restricted to changes of the scheme of influence. 



^2 






o 

+ - 
o + 



Figure 5: The Scheme of Influence 



The scheme of influence {Figure 5 ) represents the decision maker's subjective 
ideas of causal influences only in a qualitative way. In a matrix a positive influ- 
ence of a possible action (Aj, A2, or A3) on a goal variable (Zj or Z2) is repre- 
sented by a negative one by and missing influence by ‘o’. A possible 
action is a plan in which instrumental variables are changed. Possible actions can 
also be the objects of a search process governed by planning rules. The scheme of 
influence is modified by means of expectation-forming rules according to the 
decision maker's experience. 




Figure 6: Construction of the Expected Realization Area 




350 



As an example. Figure 6 shows how the expected realization area is derived 
from the scheme of influence in combination with the last realization of the goal 
variable R. The scheme of influence from Figure 5 as transferred to the scheme of 
aspiration adaptation is represented by arrows indicating the direction of reaction. 
Potential aspiration levels that are still attainable are indicated by asterisks. 

In the planning model the rules of forming expectations are corrected in accord- 
ance with the realization. The phases of the routine model are transposed to the 
planning level. Calculation by which the effects of possible actions are computed 
takes the place of the realization. The scheme of influence is changed in accord- 
ance with this calculation. The determination of the realization area in the routine 
model is replaced by the derivation of the calculation area at the planning level. 

The work of Sauermann and Selten in the form of the models sketched here is 
not to be seen only as a descriptive approach to boundedly rational behavior. In its 
general form the theory is at the same time a program for research which in some 
aspects is still awaiting concretization. It poses many questions which should be 
clarified by empirical, experimental, and theoretical investigations: Which rules 
prevail or are expedient in the formation of expectations? How is the order of 
importance derived? How many potential aspiration levels are useful? How are 
the potential aspiration levels formed and modified? In which way is the adapta- 
tion of aspiration levels effected in bilateral or multilateral situations? Which 
normative or, in a weaker form, semi-normative properties should be required for 
a dynamic theory of boundedly rational decision behavior controlled by aspiration 
levels? 

The idea of aspiration-guided decision behavior has also influenoed explanatory 
approaches of Selten to the behavior in oligopoly experiments.^ There, new aspi- 
rations are formed in each period. As modelled in a simulation study, the aspi- 
rations are not directly concerned with the goal variables; on the contrary, the 
adaptation is accomplished between various criteria which deliver aspirations in a 
more natural way. For instance, the principle of sustaining the net profit is 
replaced by the weaker principle of sustaining the same gross profit as before, if 
the aspiration of sustaining the net profit appears to be unrealizable. 



^ R. Selten, Investitionsverhalten im Oligopolexperiment, in: H. Sauermann (Ed.), Contri- 
butions to Experimental Economics (Volume 1), Tubingen 1967, pp. 60-102; R. Selten., 
Ein Oligopolexperiment mit Preisvariation und Investition, ibid., pp. 103-135; R. Selten, 
Die Strategiemethode zur Erforschung des eingeschrUnkt rationalen Verhaltens im 
Rahmen eines Oligopolexperimentes, ibid., pp. 136-168. 

R. Tietz, Simulation eingeschrankt rationaler Investitionsstrategien in einer dyna- 
mischen Oligopolsituation, in: H. Sauermann (Ed.), Contributions to Experimental 
Economics (Volume 1), Tubingen 1967, pp. 169-225. Cf. also: R. Selten and R. Tietz, 
Zum SelbstverstUndnis der experimentellen Wirtschaftsforschung im Umkreis von 
Heinz Sauermann, Zeitschrift fiir die gesamte Staatswissenschafl, Volume 136 (1980), 
pp. 12-27. 




351 



3. Multilateral Processes of Adaptation of Aspirations as a 
Macroeconomic Concept of Coordination of Decisions 

The fascination arising from the dynamic approach of adaptation of aspiration 
levels led me to construct a complex macroeconomic model which could be used 
as an experimental environment for bargaining experiments with macroeconomic 
relevance. In accordance with principles of bounded rationality this model is 
solved in each period by markets and by mutual processes of adaptation of aspira- 
tion levels. 




Figure 7: The Monetary Circulation of the KRESKO Economy 



This model of a closed national economy includes four sectors: households, 
industry, credit banks, and the central bank, which are interpreted as representa- 
tive decision units {Figure 7). The government is represented only rudimentarily. 
The sectors are connected with each other by markets for labor, consumer goods, 
capital goods, short-term credit, and long-term credit. The production is organ- 
ized by a putty-clay production function with endogenous technical progress. By 
means of dimensional analysis principles of construction were derived in order to 



R. Tietz, Ein anspruchsanpassungsorientiertes Wachstums- und Konjunkturmodell 
(KRESKO), Tubingen 1973. 

Figure 7 is taken from R. Tietz, The Macroeconomic Experimental Game KRESKO, in 
H. Sauermann (Ed.), Contributions to Experimental Economics, Volume 3, Tubingen 
1972, pp. 267-288, esp. p. 269. 







352 



analyze the relations between about 300 variables and to ensure the existence of 
equilibrium paths for this growth and business cycle system. 

Here, we will restrict detailed consideration to the individual decision and 
coordination process, which is ruled by aspiration levels in a way similar to the 
theory of Sauermann and Selten. In each period the aspiration levels were formed 
by means of expectation and plaiming variables {Figure 8^^). 

The univariate formation of expectation in field El uses in the first step only 
information on the past of the regarded variable (II) by means of an adaptive 
exponential smoothing procedure, which is controlled by the prediction quality. 
The multivariate formation of expectations in E2 uses in addition information on 
the assumed causal structure from 12. Planning variables are formed in P3 using 
expectations on the behavior of other decision makers (E3). If these planning 
variables also have control and regulation functions they also assume the charac- 
ter of aspiration variables (A3). On the basis of these plans preliminary decisions 
are made in D4 which are tentatively coordinated in M5 with preliminary deci- 
sions of other decision makers (D4’), e.g. in markets. If necessary, additional 
expectations, plans, and decisions are made for other variables (fields E6, P7, A7, 
and D8). If the aspiration level formed by the aspirations defined so far can be 
reached (W9), the decision process for this topic is finished. Failure to reach an 
aspiration level triggers a hierarchically organized process of adaptation of aspi- 
rations. 

Besides the aspirations formed in the fields A3 and A7 which are concerned 
with economic variables, there are aspirations that limit the number of adapta- 
tions of aspirations and govern the selection of variables to be adapted. In the 
KRESKO model these aspirations are represented by counting-variables in hier- 
archical order. Thus, field WIO examines whether the ’’strong" adaptations over 
K13 and Z13 have been performed less than vlO* times. If this is the case a 
similar test for the variable vll controls the "weaker” (or iimer) adaptation via 
K12 and Z12. In K12 smaller revisions are made only for the variables formed in 
E6, P7, and A7. Based on these corrected data, new decisions are made in D8. 
After the limit vll* for the counting-variable for weak adaptations is reached, a 
strong (or outer) adaptation occurs via K13. This leads to corrections of expecta- 
tions from E2 and of plans and aspirations from P3 and A3. Thereby new deci- 
sions have to be made in D4 which again necessitate coordination on the market 
(M5). While the weak adaptations are made only internally by the individual, the 
strong adaptations also require revisions by other decision makers. 

If the limit for the maximum of strong adaptations would be violated in WIO by 
an additional strong adaptation, after some final corrections in K14 and D14, 
other decision makers also have to make adaptations in D14*. The revised deci- 



Figures 8 and 9 are taken from/?. Tietz 1973, l.c., pp. 105, 1 12, resp. 465. 




353 




I 

I 

I 



Figure 8: The Individual Decision Process 























354 



sions of both sides are again coordinated in M15. Thereafter the burden of adap- 
tation switches over to other decision makers in K16' and D16'. 

The macroeconomic process of coordination between the sectors regarded as 
representative decision units corresponds in its main features to the individual 
process. (Therefore, it may be sufficient to reproduce Figure 9 only in German.) 
For instance. Figure 9 shows that, because of violated liquidity aspirations of the 
firms (Liquiditdtsanspruche der Untemehmen) in field W1.13, a weak adaptation 
occms in K1.17. In this case the firms adapt their liquidity aspirations in the form 
of time deposits downwards, and their demand for short-term credit (Nachfiage 
nach kurzfristigen Krediten), upwards. Subsequently the market for short-term 
credit (Geldmarkt) M5. 1 is again passed through. On the other hand, the strong 
adaptation in K 1.1 9 concerns the planning of production (Produktionsplammg) 
with feedback via Z1.20 and Llll to P1.3, where decisions on pr^uction, 
investment, and employment (Produktion, Investition, Beschdftigung) again have 
to be made. 

Finally the adaptation process shifts via M5.2 or M5.3 to the sector of the credit 
banks. By the so-called "through financing" (Durchfinanzierung) the reduced 
liquidity aspirations of the firms are preliminarily fulfilled. In K3.10, for exam- 
ple, the banks revise their plans for refinancing and credit (Refinanziemng, 
Kreditplanung). 

The - in a sense - hierarchical order of the sectors and the concept of "through 
financing" guarantee a solution of the system: The central bank, via the through 
financing in M6.5, in the end has to fulfill the liquidity aspirations of the banks 
that were greatly reduced several times. 

In the given form, KRESKO is one possible model of a multilateral macro- 
economic process of adaptation of aspirations. One may criticize the relatively 
rigid hierarchical stmctxne between and within the sectors. Possibly, it favors the 
tendency of the model to exhibit depressions rather than inflations. The question 
could also be asked whether the formation of aspiration levels has to take place on 
an equilibrium path in the same way as in a disequilibrium, meaning whether the 
process shall always start with the highest aspiration level. We will return to these 
questions after discussing the experimental results which have been obtained 
later. 



4. Bilateral Theories of Bargaining Based on the Adaptation of 
Aspiration Levels 

The great importance of qualitatively distinguished aspiration levels, derived in a 
seemingly natural way, was especially evident in bargaining experiments 




355 




I 10. S| PeriodenabschluB; Bewertung, Ermittiung deflnitorischer GrSBen, Statlstik 





356 



designed to explain bargaining process and bargaining result by means of proc- 
esses of adaptation of aspiration levels. Such potential aspiration levels may be 
recorded with the so-called planning-report method. By means of questionnaires 
the test persons are induced to concern themselves intensively with the forth- 
coming decision situation. 

In the experiments concerning labor market negotiations between the parties of 
a wage agreement, for instance, questions regarding expectations about the devel- 
opment of the general economic situation were asked. Subsequent questions were 
concerned with the three bargaining variables; standard working horns, increase 
of the standard wage rate, and the period of notice, with the first of these ques- 
tions regarding their order of importance. Next, subjects were asked for their 
expectations about plans (or better, aspiration levels) of the bargaiiung partner, 
that is to say about his first demand D and the respective conflict limit L . 



bargaining variable level 

(wage increase) units 




sequence of questions (time) 



Figure 10: Planning Report and Aspiration Levels (Labor Union) 




357 



Figure 10 shows the aspiration values for one bargaining variable (left ordi- 
nate) in the respective order of preference and in the chronological order of ques- 
tions (abscissa). (Questions about expectations ( D and L ) are followed by the 
question about the true aspiration level, namely the planned or target bargaining 
result P. The next lower potential aspiration level is derived from the question 
about the value regarded as attainable A. In a certain sense it represents the 
expected willingness-to-yield of the bargaining partner and is perhaps derived 
from an estimate of the opponent's P-value. The lowest aspiration level is the 
planned conflict limit L at which the bargaining is broken off. From these values 
more tactical variables can be derived, such as the planned threat to break off 
negotiations T - mostly situated between A and L - and finally one's own first 
demand D®, which is the actual aspiration level in the beginning of the nego- 
tiation process. Marlies Klemisch-Ahlert could show experimentally that the 
aspiration levels discussed here are of great importance before and during bar- 
gaining, even in situations in which no written planning report is asked for.^^ 

Besides the original five aspiration levels described above, the midpoints of 
adjacent levels can become potential aspiration levels also. Values in the ten areas 
outside and between these nine aspiration levels are seen as respective equiva- 
lents. This results in a 19-level rating scale in aspiration or level units (right 
ordinate). 

A comparison of the aspiration grids of both bargaining partners yields, e.g. by 
means of the static aspiration balancing principle, an agreement value or range 
at which both partners reach, as far as possible, equally high aspiration levels. 
The basically two-stage "planning-difference theory", first developed with Hans- 
Juergen Weber, determines the degree of toughness of a negotiation according to 
the differences between the mutual aspiration levels P.^^ Then it is determined 
which partner has to concede first; we call this partner player 1. In hard bargain- 
ing situations the partner with the greater concession reserve, the difference 
between the first demand and the level regarded as attainable A, expresses the 
greater willingness to concede and therefore is predicted to concede first. In more 
relaxed situations in which concessions below the value P are not necessary for an 
agreement, the so-called "tacit concession" gives a better explanation. It is the 
difference between the expected and the actual first demand of the opponent. The 
partner with the smaller tacit concession has to concede first, since the other one 



Both of these values (T and D®) also have the character of potential aspiration levels, 
but they may be varied more easily by tactical considerations. 

M Klemisch-Ahlert, Bargaining in Economic and Ethical Environments - An experi- 
mental study and noraiative solution concepts, Berlin-Heidelberg-New York-London- 
Paris-Tokyo 1995, p. 20. 

R. Tietz and H.-J. Weber, On the Nature of the Bargaining Process in the KRESKO- 
Game, in: H. Sauermann (Ed.), Contributions to Experimental Economics, Volume 3, 
Tubingen 1972, pp. 305-334. 




358 



has implicitly made the greater concession in advance (as related to the planning 
phase). In that case the agreement occurs at the P-value of player 2, the player 
who does not concede first; in hard bargaining situations player 1 reaches his A- 
value unless the respective value of player 2 is better for him.^^ We also call 
attention to the case in which the two planned aspiration values P coincide. This 
situation can be interpreted as equilibrium, since according to both theories 
discussed above, agreement likewise coincides with the P-values. 

Finally, by means of a third theory, the dynamic aspiration-balance theory, all 
individual steps of the bargaining process are represented. Here the important 
aspiration-securing principle is added. It compares the aspiration levels secured 
by the opponent’s actual bid. Besides the compatibility test of the conflict limits, 
the aspiration-securing principle is the first so-called decision filter for the first 
concession (cf Figure 11). The partner with the higher secured aspiration level 
has an aspiration-securing advantage measured in level units and therefore is 
predicted to concede first (I = 1, resp. J = 1). 

Subsequently, the same filters as in the planning-difference theory, the conces- 
sion reserves and the tacit concessions, are compared. If a filter does not lead to a 
decision by virtue of the equality of the compared variables, the next filter is 
tested. For completeness, the three filters are followed by a random filter which 
selects the first concession maker with equal probability. 

In addition, the statements of the filters determine the relative strength of 
player 1. He is regarded as strong if a postpositioned filter contradicts the decid- 
ing filter; his strength in this case is rated with the value 3 (FO = 3). If player 1 is 
selected only by the third filter, he has the moderate strength of 2 (FO = 2). If the 
decision occurs by the first or second filter without contradictions, player 1 is 
regarded as weak, with a strength value of 1 (FO = 1). This rating of strength 
influences the extent of the concessions of both players in the subsequent stage of 
the process iff bargaining, which we will not discuss here in detail. 



R. Tietz and H.-J. Weber, Decision Behavior in Multivariable Negotiations, in: H. 
Sauermann (Ed.), Bargaining Behavior, Contributions to Experimental Economics, Vol- 
ume 7, Tubingen 1978, pp. 60-87, esp. p. 66. 

Cf. e.g., R. Tietz, Der Anspruchsausgleich in experimentellen 2-Personen-Verhand- 
lungen mit verbaler Kommunikation, in: H. Brandstdtter and H. Schuler (Eds.): Ent- 
scheidungsprozesse in Gruppen, Beihefl 2 der Zeitschrift ftlr Sozialpsychologie, Bem- 
Stuttgart-Wien 1976, pp. 123-141; reprinted in: H. W. Crott and G. F. Muller (Eds.): 
Wirtschafts- und Sozialpsychologie, Hamburg 1978, pp. 140-159; R. Tietz, Anspnichs- 
anpassimg und Verhandlungsverhalten, Frankfurter Arbeiten zur experimentellen Wirt- 
schaflsforschung. Discussion Paper No. A 25, Frankfurt 1987. 




359 




Figure 11: Who Makes the First Concession? 



It is important to note that under certain conditions a weaker player 1 has to 
make comparatively greater concessions, whereas his partner can limit himself to 
smaller ones. In this manner the strength of the player contributes to an adverse 
effect of excessive demands. The aspiration-securing principle works in a similar 
way by modifying the principle of alternating concessions. It prevents concessions 
which clearly would reinforce an aspiration-securing disadvantage. 

As could be shown by simulation studies, the dynamic aspiration-balance theory 
imder semi-normative aspects has the desirable property of a concave payoff 






360 



function over the tactical variables first demand and conflict threat. Therefore, 
aggressiveness pays only up to a certain degree. This means that the best replies 
lie in a moderate range, which also favors moderate equilibria. The theory pos- 
sesses a certain inner stability against itself Escalations are avoided. The theory 
can be used semi-normatively for recommendations. This characteristic is not 
shown in other, empirically comparable, bargaining theories, e.g. the one pre- 
dicting an agreement at the middpoint between the first offers. Reinhard Selten 
has always demanded that a good bargaining theory should have this irmer stabil- 
ity as a prerequisite, even under bounded rationality. 

The dynamic aspiration-balance theory has, in a way similar to the two simpler 
theories, delivered good predictions for various experiments with a total of over 
300 negotiations. Essential components of this theory were also used by Jurgen 
CrOjimann in order to represent combined quantity and price negotiations.^® The 
market experiment he used was originally developed by Reinhard Selten and 
represents a negotiation market with eligible partners.^^ The phenomenon of 
demand inertia observed by CrOfiman was previously detected by Selten?^ It can 
also be characterized as relation equilibrium and was later explained by the prin- 
ciples of fulfillment and balancing of aspiration levels.^^ 

Various concepts have effects on the formation of aspirations, meaning the 
adaptation of aspirations between negotiations.^"^ Besides the prominence princi- 
ple and changing conditions of the experimental environment, the success meas- 
ured in aspiration levels explains only about half of the cases: the aspiration level 



Cf i?. Tietz, W. Daus, J. Lautsch, andP. Lotz, Semi-Normative Properties of Bounded 
Rational Bargaining Theories, in: R. Tietz, W. Albers, and R. Selten (Eds.), Bounded 
Rational Behavior in Experimental Games and Markets, Berlin-Heidelberg-New York- 
London-Paris-Tokyo 1988, pp. 142-159; R. Tietz, Semi-normative Theories Based on 
Bounded Rationality, Journal of Economic Psychology, Volume 13 (1992), pp. 297-314. 
H. J. CrOfimann, Entscheidungsverhalten auf unvollkommenen M^kten, Frankfurt 
1 982; H. J. CrOfimann and R. Tietz, Market Behavior Based on Aspiration Levels, in. R. 
Tietz (Ed.), Aspiration Levels in Bargaining and Economic Decision Making, Berlin- 
Heidelberg-New York-London-Paris-Tokyo 1983, pp. 171-185. 

R. Selten, Ein Marktexperiment, in: H. Sauermann (Ed.), Contributions to Experimental 
Economics, Volume 2, Tubingen 1970, pp. 33-98. 

22 Ibid., pp. 70-72. 

23 R. Tietz, 1992, l.c. 

2^* H.-J. Weber, Theory of Adaptation of Aspiration Levels in a Bilateral Decision Setting, 
Zeitschrift fiir die gesamte Staatswissenschafl, Volume 132, 1976, pp. 582-591; H.-J. 
Weber, Zur Theorie der Anspruchsanpassung in repetitiven Entscheidungssituationen, 
Zeitschrift ftlr experimentelle und angewandte Psychologic, Volume 24, 1977, pp. 649- 
670; R. Tietz, H.-J. Weber, U. Vidmajer, and C. Wentzel: On Aspiration-Forming Behav- 
ior in Repetetive Negotiations, in: H. Sauermann (Ed.), Bargaining Behavior, Contribu- 
tions to Experimental Economics, Volume 7, Tubingen 1978, pp. 88-102. 




361 



is raised after success, and lowered after failure}^ But our investigations have 
also shown that in repetitive negotiations contrary behavior may be in the interest 
of cooperation and may favor relation equilibria. In this modification it is expe- 
dient for the more successful player to show more willingness to yield by expand- 
ing his aspiration grid, whereas the less successful player compresses his aspira- 
tion grid, thereby expressing less willingness to concede. 

In negotiations with two or more variables the position of a variable in the 
order of importance influences the degree of differentiation of the respective aspi- 
ration grid and thereby also the number of concessions in this variable.^^ Like- 
wise in these negotiations, the predictions computed by the above mentioned 
univariate theories show relatively high hit rates at least in one variable. The most 
successful theory here is the dynamic aspiration-balance theory since deviations 
lie outside a meaningful compensation area in only 15 % of the cases; this means 
that disadvantages for a player are cumulated across all variables in only a few 
cases.27 



5. Final Remarks 

I tried to show how the theory of adaptation of aspiration levels by Sauermann 
and Selten could be further developed and extended to bilateral and multilateral 
problems. The research program postulated by this theory is still not completed. 
On the basis of the realized experiments we have a certain knowledge of the 
boundedly rational decision behavior guided by aspiration levels. Some questions 
still await detailed investigation: for example, under which conditions in multi- 
variate bargaining do the partners continuously switch between individual vari- 
ables, and when are concessions made in bundles?^^ 

Likewise the macroeconomic approach may be improved in the light of experi- 
mental results:^^ The aspiration-adaptation processes complementing the coordi- 
nation by markets, such as between industry and credit banks, could be modeled 
explicitly as bilateral negotiations in accordance with the dynamic aspiration- 



Cf e.g. F. Hoppe, Erfolg und MiBerfolg, Psychologische Forschung, Volume 14 (1931), 
pp. 1-62; K. Lewin et at. l.c.; H. Heckhausen, Motivanalyse der Anspruchssetzung, Psy- 
chologische Forschung, Volume 25 (1955), pp. 118-154. 

R. Tietz and H-J. Weber, 1978, l.c., esp., p. 76 f. 

27 Ibid. p. 82. 

2^ Other questions are: How is the order of importance adapted? Is it possible to predict 
the forming of adaptations more precisely? 

2^ Since the various bargaining experiments were performed after the completion of the 
macroeconomic model, we could not take into consideration the experimental results. 
One may ask, which features should be reflected - beside an international version - in a 
revision of the KRESKO-model, especially for educational purposes. 




362 



balance theory. This might perhaps make it possible to do without the rigid hier- 
archy of sectors.^^ Depending on the constellation of the aspiration grids, some- 
times the industry, sometimes the banks would have to concede first. Perhaps, this 
change would reduce the tendency of the model to yield depression and increase 
the tendency for inflation under a loose monetary policy. 

Of course, the five aspiration levels should then be formed explicitly in a more 
natural way. It is not suflBcient to start from a high level which is adapted down- 
wards by small proportional reductions. Some considerations in this direction: 
The desired and planned bargaining result P is the level that will, according to 
the bargaining theories, be reached by both partners if their plans coincide. This 
fulfillment of plans characterizes a state of equilibrium. On an equilibriiun path, 
at which all expectations are fulfilled and therefore are extrapolated, the aspira- 
tion level P must be derived directly from the expectations. In an equilibrium, the 
aspiration level D®, usually positioned higher, should converge to P and coincide 
with it at least in the long run. The difference between D® and P, the tactical 
reserve, could depend, e.g., on the uncertainty about the economic situation, 
which may be derived from the difference between the short-run and the long-run 
trend estimations of the univariate expectation-forming procedure. Likewise, 
more natural points of reference could be found for the other three aspiration 
levels.^ ^ This modified concept of forming aspirations would presumably lead to 
more realistic behavior of the model. 



The hierarchy of sectors is reflected especially in the pragmatic concept of counting 
variables. 

The "information bridges” often positioned before the bilateral relations (cf "IB 5.1” in 
Fig. 9) could be used for forming the third aspiration level A, the value regarded as 
attainable. The conflict limit L could be derived from minimal requirements. For the 
conflict threat T the experience from earlier negotiations with the same partner could be 
used. 




363 



References 

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CrOfimann, H. J., and R. Tietz, Market Behavior Based on Aspiration Levels, in. R. Tietz 
(Ed.), Aspiration Levels in Bargaining and Economic Decision Making, Berlin-Heidel- 
berg-New York-London-Paris-Tokyo 1983, pp. 171-185. 

Heckhausen, i/., Motivanalyse der Anspruchssetzung, Psychologische Forschung, Volume 
25 (1955), pp. 118-154. 

Hoppe y F., Erfolg und Milkrfolg, Psychologische Forschung, Volume 14 (1931), pp. 1-62. 

Klemisch-Ahlert, M, Bargaining in Economic and Ethical Environments - An experimen- 
tal study and normative solution concepts, Berlin-Heidelberg-New York-London-Paris- 
Tokyo 1995. 

Lewin, AT., R. Dembo, L. Festinger, and P, P. Sears, Level of Aspiration, in: J. McV. Hunt 
(Ed.), Personality and the Behavior Disorders, New York 1944, pp. 333-378. 

Sauermann, H., Vorwort des Herausgebers zu: R. Tietz, Ein anspruchsanpassungsorientier- 
tes Wachstums- und Konjunkturmodell (KRESKO), Tubingen 1973, p. V-VI. 

Sauermann, H., andR. Selten, Anspruchsanpassungstheorie der Untemehmung, Zeitschrift 
flir die Gesamte Staatswissenschaft, Volume 118 (1962), pp. 577-597. 

Selten, R., Investitionsverhalten im Oligopolexperiment, in: H. Sauermann (Ed.), Contri- 
butions to Experimental Economics (Volume 1), Tubingen 1967, pp. 60-102. 

Selten, R. , Ein Oligopolexperiment mit Preisvariation und Investition, in: H. Sauermann 
(Ed.), Contributions to Experimental Economics (Volume 1), Tubingen 1967, pp. 103- 
135. 

Selten, R., Die Strategiemethode zur Erforschung des eingeschrankt rationalen Verhaltens 
im Rahmen eines Oligopolexperimentes, in: H. Sauermann (Ed.), Contributions to 
Experimental Economics (Volume 1), Tubingen 1967, pp. 136-168. 

Selten, R,, Ein Marktexperiment, in: H. Sauermann (Ed.), Contributions to Experimental 
Economics, Volume 2, Tubingen 1970, pp. 33-98. 

Selten, R., In search of better understanding of economic behavior, in: A. Heertje (Ed.), 
The Makers of Modem Economics, New York et al. 1994, pp. 115-139. 

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im Umkreis von Heinz Sauermann, Zeitschrift filr die gesamte Staatswissenschaft, Vol- 
ume 136 (1980), pp. 12-27. 

Simon, H.A. , Theories of Decision Making in Economics and Behavioral Science, The 
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Simon, HA., New Developments in the Theory of the Firm, The American Economic 
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Tietz, R., Simulation eingeschrankt rationaler Investitionsstrategien in einer dynamischen 
Oligopolsituation, in: H. Sauermann, Contributions to Experimental Economics 
(Volume 1), Tubingen 1967, pp. 169-225. 

Tietz, R., The Macroeconomic Experimental Game KRESKO, in H. Sauermann (Ed.), 
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Tietz, R., Ein anspmchsanpassungsorientiertes Wachstums- und Konjunkturmodell 
(KRESKO), Tubingen 1973. 




364 



Tietz^ Ky Der Anspnichsausgleich in experimentellen 2-Personen-Verhandlungen mit 
verbaler Kommunikation, in: H. Brandstatter and H. Schuler (Eds.): Entscheidungspro- 
zesse in Gruppen, Beihefl 2 der Zeitschrift fUr Sozialpsychologie, Bem-Stuttgart-Wien 
1976, pp. 123-141; reprinted in: H. W, Crott and G. F, Muller (Eds.): Wirtschafls- imd 
Sozialpsychologie, Hamburg 1978, pp. 140-159. 

TietZy R.y Anspruchsanpassung imd Verhandlungsverhalten, Frankfurter Arbeiten zur 
experimentellen Wirtschaflsforschung, Discussion Paper No. A 25, Frankfurt 1987. 

TietZy R.y Semi-normative Theories Based on Bounded Rationality, Journal of Economic 
Psychology, Volume 13 (1992), pp. 297-314. 

TietZy R.y and H.-J. Webery On the Nature of the Bargaining Process in the KRESKO- 
Game, in: H. Sauermann (Ed.), Contributions to Experimental Economics, Volume 3, 
Ttlbingen 1972, pp. 305-334. 

TietZy Ry and i/.-J. Webery Decision Behavior in Multivariable Negotiations, in: H. Sau~ 
ermann (Ed.), Bargaining Behavior, Contributions to Experimental Economics, Volume 
7, Tubingen 1978, pp. 60-87. 

TietZy R.y H.-J, Webery U. Vidmajer, and C. Wentzel. On Aspiration-Forming Behavior in 
Repetetive Negotiations, in: H. Sauermann (Ed.), Bargaining Behavior, Contributions to 
Experimental Economics, Volume 7, Tubingen 1978, pp. 88-102. 

TietZy R.y W. DauSy J. Lautsch, and P. LotZy Semi-Normative Properties of Bounded 
Rational Bargaining Theories, in: R. Tietz, W. AlberSy and R. Selten (Eds.), Bounded 
Rational Behavior in Experimental Games and Markets, Berlin-Heidelberg-New York- 
London-Paris-Tokyo 1988, pp. 142-159. 

Webery H,-J,y Theory of Adaptation of Aspiration Levels in a Bilateral Decision Setting, 
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Webery //.-%/., Zur Theorie der Anspruchsanpassung in repetitiven Entscheidungssitua- 
tionen, Zeitschrift fiir experimentelle und angewandte Psychologic, Volume 24, 1977, 
pp. 649-670. 




A Model of Boundedly Rational Experienced 
Bargaining in Characteristic Function Games 

Wulf Albers 

Institute of Mathematical Economics, University of Bielefeld 



Abstract. The approach models the negotiations process in characteristic 
function games. The aim is to give a model of the behavior of idealized 
experienced players. They do not make mistakes, but they have all properties 
of a boundedly rational player that do not disappear with experience. The 
negotiation process and negotiation motives of such ideal players are modeled, 
with the idea to characterize the ’limit’ of subjects’ behavior who developed 
growing experience with a given game. - The approach models the phenomena 
prominence, structure of proposals, dominance, loyalty, attributed (revealed) 
demands, stepwise coalition formation, and foresight, and presents a solution 
concept for the negotiation process. Possible modifications and refinements 
are given. - A final section contains several examples, and comments on 
relations to Bargaining Set, Demand Equilibrium, and Equal Share Analysis. 



1. The Model 

The concept is given by a set of definitions, which model the negotiation 
process of experienced boundedly rational players. Ba^ic elements of the ap- 
proach are proposals, i.e. coalition structures, and corresponding payoff ar- 
rangements (1.1). Negotiation processes are sequences of proposals, each of 
which dominates the preceding one. The exactness of perception of numer- 
ical payoffs is restricted by a smallest perceived money unit. Accordingly, 
only such proposals are considered which can be generated on this level of 
exactness (1.2). Dominance is modeled in the usual way, the minimal im- 
provement in a normal dominance is one smallest money unit. Loyalty is the 
phenomenon that ’similar’ players spontaneously prefer to coalesce with one 
another, and do not leave a coalition with each other as easily as nonsimilar 
players. According to this phenomenon the usual definiton of dominance is 
modified (1.3). Every payoff, a given player received in an earlier stage of the 
negotiation process, generates the assumption that this player will thereafter 
enter a coalition only if she gets at leaist the same amount as before (at- 
tributed demands). These demands depend on the coalition structure (1.4). 
By the condition of attributed demands possible dominances are essentially 
reduced, all negotiation processes end after a finite number of steps. This per- 
mits rollback analysis of possible future dominances. Strategies on the space 
of possible negotiation chains are defined. Stable strategies are specified as 




366 



solutions in a way which models the foresight of the players (1.5). An addi- 
tional condition is given concerning the preference for agreements with high 
loyalty index (1.6) Negotiations on the strategy level are modeled in (1.7). 

The model is based on the observations of more than 1000 experimental 
games. It addresses the (special) negotiation situation, where the formation 
of a coalition has to be announced to the experimenter, and becomes a final 
result when it has not been replaced by a new announcement within a certain 
time (usually 20-30 minutes). This procedure excludes ultimative elements, 
and forces agreements to refiect the power structure of the game, it has been 
introduced by SELTEN,SCHUSTER (1964). The model addresses that type 
of experience that is generated, when subjects play the same game repeatedly 
for several rounds with changing partners, and communication is allowed. 

A problem of the development of the concept from the observed data 
was that experienced players do not show processes. They apply foresight, 
and immediately select stable states. Experiments with unexperienced players 
produce observable processes, but at least every second performed dominance 
is a mistake, which experienced players would not make, since they would 
apply foresight, and stop the process. Nevertheless, the observable processes 
admit insight into general patterns, motives, and the structure of actions. 
The results of games with experienced players permit to decide, whether the 
predictions of the model concerning the observable part of the negotiation 
process are correct. 

As far as we know, there are no experimental results, that contradict 
the basic points of the model, although there are experimental results in 
which deviations can be observed for unexperienced subjects. Nevertheless, 
we expect that future experimental results will make it reasonable to refine 
the model in one and the other detail. Suggestions for such refinements are 
given in some of the remarks. 

1.0 Basic Notations 

A game is (N,v) is a set of players N={l,...,n}, and a function v, which assigns 
a value v(C) to every coalition CcN, where v(0)=O, and v({i}=0 for all iGN. 
P(N) is the set of all subsets of N. Subsets of P(N) are denoted as A,C,L,P,.. 
For given P, AgP, BgP: AcB denotes that A is a strict subset of B. ACiB 
denotes that AcB and there is no CgP such that AcCcB. Pcb:= {AgP: 
AcB}, and Pcib:= {AgP: ACiB}. #A is the number of elements of A. R is 
the set of real numbers. R^ is the real N-space. For xGR^: x(S):= S{ x(i): 
iGS ). 



1.1 Proposals 

Definition: PCP(N) is a coalition structure, iff (1) {i}EP for all iGN, 
and (2) A,BgP =4^ ADB=0, or ACB, or ADB. 




367 



Notation: For p: P(N) -> R^, ACCCN let 
pC:= p(C) (gR^) (i.e. the image of C under p), 
p+C~p+(C):= pA: AeP(N), ACC ) (gR^), 
pC(A):= U{ pC(i): IgA ), 
ptot:= re pC: CCN ). 

For a characteristic function v: P(N) R^, and a given coalition struc- 
ture P, CgP let v’(C):- v(C) - U{ v(A): AgPcic). 

Definition: Let P a coalition structure, and p: P(N) — >■ R^. (P,p) is a 
proposal, iff for all CCN: (1) pC(C)=v’(C) for all CgP, and (2) pC(i)=0 
if i^C or C^P. 

Coalition structures are not defined as partitions (as for instance in the bar- 
gaining set), but refiect the whole structure of possible subcoalitions as they 
can be formed in a process of stepwise coalition formation. - Proposals dis- 
tribute for every formed coalition C its marginal value v’(C) (obtained by 
forming C) among the members of C. A proposal contains the information 
of all prior agreements of the process of coalition formation for all coalitions 
that have been formed, and not been broken afterwards. 

1.2 Prominence 

Definitions: The prominent numbers are PRO:= {a lO^: aG{l,2,5}, i 
integer}. - U (a(p)-p: pGPRO) is called a presentation of a real number 
X, iff (1) x= Z'(a(p)-p: pGPRO), and (2) all coefficients a(p) are +1,-1, or 
0. - The exactness of a presentation is the smallest prominent number 
p of the presentation with coefficient a(p) unequal zero. - The exactness 
of a number x is the crudest exactness over all presentations of x. - A 
number has prominence d, iff its exactness is > d. 

A proposal (P,p) has prominence d iff for all Dg PU{N} there is a 
coalition structure (be. Q finer than P), such that for all CgQcd 

the partition Qcic of C fulfills 

(1) (’numerical prominence’) pD(A) (or p*^D(A)) has prominence d for 
all but one AgQcic, or 

(2) (’equality’) pD(A)=pD(B) for all A,Bg Qcic (or p+D(A)=p+D(B) 
for all A,BgQcic). 

The definition of prominent numbers and presentations follows ALBERS, 
ALBERS (1984), see also ALBERS (1996). Every (real) number can be pre- 
sented as a sum of prominent numbers with coefficients 0,-1, +1 (where no 
prominent number is used more than once), for instance 16=20-5+1. This 
presentation need not be unique, for instance 3= 5-2=2+ 1. The exactness 
of a number is the smallest prominent number necessary for its presenta- 
tion. If several presentations are possible, then that presentation defines the 
exactness, for which the exactness is maximal. Examples of presentations 
(exactnesses in brackets) are: 1=1(1), 2=2(2), 3=5-2(2), 4=5-l(l), 5=5(5), 




368 



6=5+l(l), 7=5+2(2), 8=10-2(2), 9=10-1(1), 10=10(10), 11=10+1(1), 12= 
10+2(2), 13=10+5-2(2), 14=10+5-1(1), 15=10+5(5), 16=20-5+1(1). 

A subject who ’operates with exactness 5’ considers numbers as 5,10,15,.. 
i.e. the numbers with prominence 5 (exactness >5). To generalize the notation 
of prominence to proposals, three ways of generating a numerical payoff p(A) 
(or p*^(A)) for a coalition A are specified: (la) to select p(A) as a prominent 
number, or (lb) to compute a remainder v(C)-i7(p(A): AgPcic) (where all 
p(A) are generated according to (la)), or (2) to devide a joint payoff equally 
among the coalitions of a partition Pcic. 

Assumption: In the following it is assumed that the prominence with 

which a game is played, is given and greater zero, and that all proposals 

have this prominence. 

Remarks: 1. To avoid to be misunderstood: The prominence is not imple- 
mented by the experimenter, for instance by saying that only multiples of 
certain amounts are permitted as payoffs. The prominence structure of pro- 
posals is spontaneously selected by the subjects, similar as in the response 
on the question ’how many inhabitants has Cairo’, which is usually given 
as one number with prominence 500.000. — 2. On the linearity of percep- 
tion: The theory of prominence is developed as a theory of perception, and 
subconscious construction of numerical responses in individual decision situa- 
tions. The result is that perception of differences has a logarithmic character: 
the relative, and not the absolute differences of numbers are perceived. For 
distributional situations as here, the problem is that a high outcome of one 
player (generating a low sensitivity for the payoffs of this player) is related to 
a low payoff of another player (generating a high sensitivity for her payoffs) . 
Since usually all players attend the payoffs of all others, it seems reasonable 
to assume constant sensitivity over the whole range of reasonable alterna- 
tives. — 3. There is not yet a rule which predicts the prominence of a game. 
But experimental results indicate that the prominence is determined by the 
game, and the conditions of communication. As a rule of thumb we use: (a) 
for face to face communication the prominence is as fine as possible, but not 
finer than 5% of the maximal payoff, (b) for communication via terminals 
experienced players reach a prominence of 1, when the total payoff is 100. 

A remark concerning the prediction of prominence: The exactness of a nu- 
merical response depends on the exactness of the stimulus. In certain situ- 
ations, practitioners characterize the exactness of a signal by its ’range of 
reasonable alternatives’, characterized by its extremes ’worst case’, and ’best 
case’. (As a rule of thumb we use that worst case and best case are the 
cuts of the 10% tales, for measurable distributions.) The prominence can be 
predicted from the range of reasonable alternatives as follows: 

Prominence Selection Rule: For a numerical stimulus with a given 

range of reasonable alternatives the prominence d of the response is se- 




369 



lected in a way that there is a set X of at least 3, and at most 5 numbers 
such that (1) X has prominence d (for all xGX), (2) y-x>d (for all x<yGX), 
(3) X is maximal with (1),(2). 

In the bargaining situations considered here, the extension of the range of 
reasonable alternatives seems mainly to be given by the inpreciseness of the 
predictions of reactions of others. Part of the inpreciseness is induced by the 
problem to predict loyalty (see below). It seems that for games with loyalty 
under certain conditions the range of reasonable alternatives coincides with 
the range of outcomes predicted by Equal Share Analysis, or Equal Division 
Bounds (SELTEN 1972,1982). 



1.3 Loyalty, and Dominance 

In the following it is assumed that the smallest money unit, d, also defines 
the minimal incentive under which a player (or a jointly acting group of play- 
ers) changes a coalition. The incentives, under which a coalition is changed, 
essentially depend on the question, if a player belongs to, forms, or leaves a 
loyal group. 

Definition: Let (P,p) a proposal, CgPU{N}, and LcPcic). L is a 
loyal group of (P,p) below C, iff (1) (’symmetry’) pE(L)=pE(M) (or 
p+E(L)=p+E(M)) for all Eg Pdc, L,MgL, and (2) (’maximality’) L is a 
maximal subset of Pcic) that fulfills (1). 

Notation: For a given proposal (P,p) and iGN let LLi(P,p) the union 
of all loyal groups that contain i. 

Predominance rules which parts of an old proposal are distroyed, and which 
are kept, when a new coalition is formed. Dominance in addition defines which 
incentives are necessary that the players agree to perform the dominance. 

Definition: A proposal (Q,q) predominates a proposal (P,p) via a 
coalition D, iff 

(1) Q= {AgP: AcD} U {AgP: AnD=0} U {D}, and 

(2) qA=pA for all AgPHQ. 

Definition: Let (Q,q) predominate (P,p) via D, and for every iGD let Li 
the unique loyal group of (Q,q) below D that contains i. (Q,q) dominates 
(P,p) via D [ notation: (P,p) — >D (Q,q) ], iff for all iGD: (1) qtot(i)>ptot(i), 
and (2): 

(a) (’deletion of a loyal group’) if there is LGLLi(P,p) such that #L>1, 
and Dn(UL) C UL, then 

qtot(ULi)-ptot(ULi) > d+l-d, or qtot(ULi)/ptot(ULi)>5/3 (where 1 denotes 
the degree of loyalty), (b) (’extension of loyal groups’) otherwise, if 
LLi(P,p)cLLi(Q,q) then qtot(i)-ptot(i) > 0, 

(c) (’neither deletion nor extension’) otherwise 
qtot(ULi)-ptot(ULi) > d, or qtot(ULi)/ptot(ULi)>5/3. 




370 



Ad (a): The degree of loyalty, 1, depends on the conditions of communication. 
Under free communication, the degree of loyalty is 1=1 (sometimes 1=2). 
When subjects communicate via terminals, then there is usually no loyalty 
( 1 = 0 ). 

A loyal group L is a maximal set of sub coalitions of a larger coalition, 
where each subcoalition receives the same payoff. The larger coalition may 
be formed recently, or in some former dominance of the process of coalition 
formation. The implicit assumption of the model is, that the symmetry among 
players (or groups of players), given by the fact that they receive equal payoffs, 
creates a feeling of solidarity which causes them not to leave a coalition with 
one another, except for substantial additional payoffs. (Compared to former 
definitions of a loyal group, we here omitted the condition that the game 
must be symmetric with respect to the coalitions of L.) 

A new coalition is formed by a dominance (P,p) — >D (Q,q). Coalitions of 
the preceding proposal are broken, if they intersect D. Old coalitions which 
are subsets of D, or do not intersect D, are kept, their payoff agreements 
remain unchanged (see the definition of a predominance) . Thereby a proposal 
keeps track of those coalitions and agreements which have not been broken 
in later dominances. 

Every member of a new coalition D is assumed to agree to form D only, 
when she thereby improves her payoff by at least d. If she is in a coalition L 
of a loyal group, then it is sufficient that the sum of the improvements of the 
members of L add up to at least d (condition (c)). If in (Q,q) a new loyal 
group is generated, or an old loyal group is extended, and this group includes 
i, then it is sufficient, that the payoff of i does not decrease (condition (b)). 
A player leaves a loyal group only, if she improves her payoff by an extra 
amount 1-d, which may be interpreted as the amount necessary to corrupt 
the player to betray her partners of the old loyal group. This amount has 
to be paid to every player of the old loyal group, who enters D. In addition, 
she needs the ’regular’ incentive d. to change the coalition, as in the normal 
case (condition (a)). For low payoffs, the additive improvement d can be 
replaced by a relative improvement of 5/3. This relative improvement is also 
sufficient, to compensate the bad feeling connected with leaving partners of 
a loyal group. 

Remarks: 1. The described phenomenon of loyalty has been verified in sev- 
eral experiments (see for instance esamples 2. 5-2. 7). A detailed experimental 
analysis on loyalty in Apex Games is given in HAVENITH (1991). — 2. 
Experimental results indicate that the amount of loyalty, 1-d, and the promi- 
nence of bargaining are of about the same size. It seems that, in situations 
with positive loyalty, the amount of loyalty can only be roughly predicted. 
According to the diffuseness of the prediction of loyalty, all predictions of 
future moves become unprecise. However, future moves can be analyzed in a 
nonprobabilistic way, if the prominence of analysis is made sufficiently crude. 
Accordingly, the prominence should be at least as large as the amount of 




371 



loyalty, 1-d. — 3. In the approach here, loyalty is generated by symmetry of 
the players with respect to payoffs. A question is, whether players can feel 
loyalty even in situations, where their payoffs are not quite equal, or when 
their payoffs are given by some other rule of fairness. — 4. The model predicts 
that players switch into a socially desired coalition even, if they thereby do 
not increase their payoff. This observation has been made in games with free 
communication (face to face, and via terminals). We are not sure, if the same 
kind of pattern establishes, when communication is restricted to very formal 
pieces of information. — 6. That a relative improvement of payoff greater 
than 5/3 is evaluated as essential, seems to be related to the structure of 
’spontaneous numbers’. The spontaneous numbers are {e-a-10^: 1}, 

aG{l,1.5,2,3,5,7}, i integer}. Under ’ordinary’ conditions, they seem to be the 
finest perceived numerical stimuli (see ALBERS 1996). x=5/3 is the mini- 
mal number with the property that any two spontaneous numbers a>b are 
necessarily more than one step apart, if a/b>x. 

1.4 Negotiation Chains, and Attributed Demands 

Definition: A negotiation chain is a sequence of proposals (Po,po) 
— >Di ... — >Dr (Pr,pr) of whlch oach dominates the preceding one, and 
which starts with Po={{l},...,{n}}, po=0. 

Notation: Two situations [~^T> (P,p),A], AgPcd, and [— >>D’ (P’,p’), A’], 
A’gP’cd- are comparable, iff there is a permutation tt of the set of play- 
ers, such that (1) v(7tS)— v(S) for all ScN, and which (2) maps [D,(P,p),A] 
to [D’,(P’,p’),A’], and (3) maps the loyal groups of (P,p) below D to the 
loyal groups of (P’,p’) below D’. (Note that p and p’ are only used to get 
the loyal groups.) 

Notation: Every dominance .. -^D (P,p) defines an attributed de- 
mand a([D,(P,p),A]= qtotD(A) for every coalition AgPcd. 

Definition: A negotiation chain fulfills the attributed demands con- 
dition, iff the attributed demands of comparable situations are not less 
for later than for earlier proposals. 

A negotiation chain describes the complete history of a process of coalition 
formation. In every step of the process one coalition is formed according to 
the conditions of dominance. The respective last proposal of a negotiation 
chain gives a protocol of all agreements that have not been cancelled by later 
dominances. 

It is assumed that players, who have to decide whether to perform a 
next dominance use the demands others made during the preceding part of 
the process, to predict, whether these others will be content with certain 
payoffs in a new coalition, or whether they will leave the coalition later to 
improve their payoff. It seems to be reasonable to assume that they will 
be content with a given amount x(A), if they did not leave ’comparable’ 
previous coalitions, where they received the same amount x(A) (or less than 




372 



that). It seems also reasonable to assume that they will leave a coalition, in 
which they get x(A), in a future dominance, if they left another coalition in 
a comparable situation, where they received the same payoff x(A) (or more). 
Moreover, players assume that a coplayer’s demand is the same, in symmetric 
situations (generated by a permutation of the set of players). For a group A 
getting a share from the joint payoff of a coalition D a situation is defined to 
be comparable, if the majcimal coalitions outside D, the maximal coalitions 
inside D, that do not cut A, and the corresponding payoffs, are the same, 
and if the loyal groups below the coalition D formed in the dominance, are 
identical. 

Remarks: 1. An interesting question is, who generates these demands. It 
seems that in experimental situations it is rather the others who deduce rea- 
sonable demands from the behavior, than that oneself has the demand. It is 
possible, that the central issue of the learning process happening in subjects 
playing a given game (repeatedly) is not that much the logic of the procedure, 
but to learn the adequate demands of the players in different situations. — 2. 
It even happens, that subjects transfer ideas of adequate demands from one 
game (they played repeatedly) to a similar but different next one. — 3. At- 
tributed demands, as modeled here, do not permit that subjects reduce their 
demand during the bargaining process. Accordingly, the concept can only 
model situations, where the adequate demands have already been learned. 
(It may be remarked that subjects apply principles of attributed demands 
also during the learning phase, but in a modified way: they forgive a too high 
demand, as long as the demand setting player did not actively change the 
coalition to improve her payoff.) — 4. It is not clear, whether attributed de- 
mands can also be transferred (by permutation of players) to situations that 
are less symmetric. Conditions of ’tolerated’ asymmetry have to be explored. 
They may depend on the number of players. — 5. There is some experimen- 
tal evidence for the attributed demand condition. As an example we mention 
spatial games which were arranged by a strongly formalized nonverbal pro- 
cedure. Coalitions which violated the attributed demand condition were in 
more than 90% of the cases thereafter broken by the player whose attributed 
demand was not satisfied. This happened only in the games of the first round. 
In games of later rounds, coalitions were only entered, when the attributed 
demands of all members were fulfilled. 

A remark concerning the linearity of negotiation chains: An interesting ques- 
tion is, if negotiation processes do really follow a linear structure as modeled 
by a sequence of dominations. Another scenario might be for instance, that 
players build up a tree-shaped structure of alternatives, where they nego- 
tiate through subtrees, redecide early moves in the tree, etc. Experimental 
results support the following model: Proposals have the character to suggest 
1. coalitions, or 2. payoffs, or 3. payoffs in coalitions. Define a proposal to 
be complete, if it gives a coalition and a payoff distribution within the coali- 
tion. And define a complete proposal to be essential, if it is dominated by 




373 



the very next proposal. The following statements can be made: (1) Every 
complete proposal dominates the last essential proposal. (2) The essential 
proposals define a chain of which each dominates the preceding one. Exper- 
imental results support that (1) is correct in more than 95% of the complete 
proposals, and that (2) is correct in more than 90% of the essential proposals. 
This strongly suggests, that the assumption of a mainly linear structure of 
negotiation chains is reasonable. (But the approach here permits to model 
different structures as well.) 

1-5 Strategies and Solution Concept 

An important consequence which follows from prominence structure and at- 
tributed demands condition, is that all negotiation chains have finite length 
(which can be estimated by a universal upper bound). This permits roll back 
analysis of future moves, and keeps the complexity of strategic analysis within 
certain bounds. 

Definition: For every game (N,v) the corresponding negotiation game 
is defined as a generalized extensive game, with a tree-shaped ordered set 
of states, and a payoff function, which assigns a payoff to every maximal 
state: The nonmaximal states are the negotiation chains that fulfill 
the attributed demand condition. They are ordered in a natural way by 
’inclusion’: X<Y iff there is rGN such that X consists of the first r domi- 
nances of Y. (X<iY denotes that there is no state Z such that X<Z<Y.) - 
For every nonmaximal state X (= negotiation chain), with a last proposal 
(P,p), there is a corresponding maximal state X*>iX defined as X*::= 
(X (P,p)). The payoff in X* is defined by ptot. - The unique minimal 
state is is (Po,po):= ({{l},...,{n}}. 

Definition: A strategy Si of a Player i is a set of states, such that 
for every nonmaximal state X there is exactly one state Y>iX in Si (i.e. 
Si assigns to every nonmaximal state X a coalition D and a proposal 
(P,p), and thereby a next state Y=(X ->>D (P,p)) ). - If for a given 
nonmaximal state X there is a unique state Y>iX, Y=(X-^D (P,p)), 
such that Y is contained in the strategies Si of all iGD, then the move 
X — > Y is performed. If there is no such Y (including the case that Y is 
not unique), then the move X X* is performed. - Every n-tuple S= 
(Si,...,Sn) of strategies defines a path path(S), i.e. a unique sequence of 
moves through the game tree starting in the unique minimal state, and 
ending in a maximal state. The payoff of this maximal state defines the 
payoff of S. - The payoff of the restriction of an n-tuple of strategies to 
a ’subgame after X’ is the payoff in the endpoint of the unique path 
path(S>x starting in X, and defined by the strategy. 

Definition: An n-tuple of strategies is stable, if for no state X there 
is a coalition CcN which can increase the payoff of all of its members 
in the subgame above X by changing their strategies in their decisions 




374 



Y>iX in such a way, that the sequence (old follower after X) ~^C (new 
follower after X) fulfills the conditions of dominance. — A proposal (P,p) 
is stable, iff there is a stable strategy such that its path ends with a state 
X*, of which (P,p) is the last proposal. 

The solution concept works on the tree-shaped ordered set of possible his- 
tories, which define the states (= negotiation chains). Thereby every sub- 
decision of a strategy in every given state depends on the special history 
in which the state is reached. Strategies are stable if there is no suffi- 
cient motive to change them, when they are once established (by a social 
norm or joint agreement). Examples of stable proposals are (80,40,-,-,-), 
(-,30,30,30,30), and (24,24,24,24,24) in the Apex Game with payoff 120 (see 
example 2.5). If Player 2 decides between these alternatives by her payoff, she 
will prefer (80,40,-,-,-). The condition of maximal loyalty (see 1.6) selects 
(24,24,24,24,24). There are also reasons to select (-,30,30,30,30). 

Remarks: 1. We suggest, that every characteristic function game with 

finitely many players has at least one stable strategy, and that on the path 
of a stable strategy no coalition is ’broken’. — 2. The approach does not 
permit that two disjoint coalitions are formed in the same move. This makes 
sense, since thereby the players can only support a dominance into a state 
they know in advance. However, there are also games, for which it makes 
sense to support a dominance independently from decision of others to form 
a coalition at the same time. In the model here, two coalitions can only be 
formed in two steps. 

The concept as an instrument to decide about stability: As long as subjects 
are still learning, players do not assume that every payoff agreed upon in an 
essential proposal really settles an unreducable demand. The 3-person game 
(n=3, v(l,2)=v(l,3)=100, v(S)— 0 otherwise) may serve as an example. Here 
the following chain of arguments may happen within an experimental game 
(with 5 as smallest money unit): (50,50,-) -> (55,-,45) (60,40,-) ... -> 

(90,10,-) — > (95,-,5) (and thereafter (100,0,-) (-,0,0) and no further 

dominance possible). There is no reason for Player 1, not to follow this chain 
up to (95,-, 5). However, the solution concept stops the chain after (50,50,-) 
(55,-,45) with the argument, that Player 2 cannot enter (60,40,-), since 
he revealed a demand of 50 before. Neither the total sequence, nor the chain 
(50,50,-) (55,-,45) is stable. Stable states consist of only one proposal 

(except for (Po,po)), namely (95,5,-), or (95,-,5). - The example shows that 
the solution concept is an instrument to decide about stability, or to predict 
negotiation chains of experienced players, but not a tool that describes the 
learning process of unexperienced players. 

1.6 Preference of States with Higher Loyalty-Index 

Notation: Let (P,p) — (Q,q), and iGD. The loyalty-index of this 
dominance for a player iGD is zero, if there is a loyal group of (P,p) which 




375 



is not loyal group of (Q,q). Otherwise the loyalty index of the dominance 
is given by the number of sets of the unique loyal group of (Q,q) below D 
that contains i. 

Definition: A stable strategy S is loyalty-mELximal, iff it is loyalty- 
maximal in all states X; it is loyalty-maximal in a state X, iff (1) it is 
loyalty-maximal in all states Y>X, and (2) for every stable strategy T 
which is loyalty-maximal for all Y>X, holds: 

if (P,p) is the maximal element of X, (P,p) ->D (Q,q) the move after 
X selected by S, (P,p) ->D (Q’,q’) the move after X selected by T, then 
either (a) there is a player iGDflD’ for whom the loyalty index of (P,p) 
— >D (Q,q) is greater than the loyalty index of (P,p) (Q or (b) 
the loyalty indices of the two dominances are equal for all iEDfiD’. 

It is assumed that - whenever coalitions unite - subjects prefer arrangements, 
where the uniting coalitions share equally, as long as these alternatives gen- 
erate solution strategies. Moreover, in every state X they prefer such a stable 
strategy which brings them (or their group) into a loyal group with a maxi- 
mal number of coalitions. This principle selects among the stable strategies. 
The idea is that - as long as there are no different forces - the precommitment 
of the subjects is, to select a coalition with maximal loyalty index, and that 
they deviate from this only, when the obtained state is not stable. 

Remark: Up to now there have been only very few experiments, that address 
the condition of maximal loyalty. The approach here is guided by experimen- 
tal observations (in characteristic function and spatial games) which strongly 
indicate that - if possible - players like to enter states with high loyalty index. 
Moreover, it seems to be true that higher payoffs cannot compensate a lower 
loyalty index, as long as both alternatives are stable. (See for instance the 
Apex Game with payoff 120 in example 2.5.) 

1.7 Negotiations on the Strategy-Level 

On a higher level of reflection, players negotiate about possible strategies 
in the negotiation game. Accordingly, an n-tuple of strategies may not only 
be changed in one move above one state X by one coalition, but for several 
moves, by possibly different coalitions. 

Notation: For any state X, and any strategy S let (Px,px) the last pro- 
posal of X, and (Ps>x,ps>x) the last proposal of path(S>x). 

Definition: An n-tuple of strategies T=(Ti,...,Tn) dominates an n- 
tuple of strategies S=(Si,...,Sn), iff for every state X with {Z} = T>ix 7^ 
{Y} = S>ix, Zt^X*, and the coalition D with X — Z holds (pT>x)tot(A) 
> (ps>x)tot(A) for all Ag(Pz)cd. 

Notation: Every dominance of states X — >D Y with Y^^X defines at- 
tributed demands via the corresponding dominance (Px,px — )>D (Py,py) 
(see section 1.4). — Every n-tuple of strategies S defines for every state X a 




376 



follower Y, and thereby an attributed demands via the corresponding dom- 
inance X ->D Y, {Y}=S>ix (if Yt^X*). — Every dominance of n-tuples 
of strategies S — T defines attributed demands for every state X with 
{Z}=T>ixy^{Y}=S>ix, Z^^X*, and the coalition D which enacts the dom- 
inance X -^D Z, namely a[D,(Pz,pz),A]= (pT>x)tot(A) for all Ag(Pz)cd. 

Definition: A sequence of n-tuples of strategies (of which each 

dominates the preceding one) fulfills the attributed demands condi- 
tion, iff (1) every strategy fulfills the attributed demands condition, 
and (2) the attributed demands of are not less than the attributed 
demands of the comparable dominances of all l<k. 

We restrict the considerations to sequences of dominating strategies, 
that fulfill the attributed demands condition. In the same way, as for 
the sequences of dominating proposals (in section 1.5), we now consider 
sequences of strategies as negotiation chains, define such sequences as 
states (of higher order), obtain moves, and strategies (of higher order). 
Undominated strategies aredefined as stable. - For a given game (N,v), 
the obtained game is called the negotiation game on the strategy level, or 
shortly the strategy game of (N,v). 

When a player changes her strategy above a state X, she needs not meet 
her attributed demands immediately in the first move after X, but in the 
last of move of the path of the new strategy after X. This implies a kind 
of foresight on the strategy level. Nevertheless every dominance on every 
path(S>x) defines attributed demands. 

In the new approach, too high demands of dominated strategies of the 
bargaining history can be punished. Thereby a kind of force is implemented 
that drives the game to a solution. (In the approach of section 1.5, strategies 
were simply replaced by new ones without generating attributed demands, 
the ’force of attributed demands’ was not implemented on the strategy level.) 
Remarks: 1. The analysis via the strategy game seems necessary for games, 
in which nested coalitions are formed. — 2. Solutions of the strategy game 
seem to select such sequences of strategies which have length 1, i.e. contain 
only one strategy. Moreover, we suppose that the path of such a strategy 
is a sequence of proposals, in which no coalition is broken. — 3. The ap- 
proach implicitly models the blocking power of coalitions, and does not need 
to replace the characteristic function of the game by a power function (see 
MASCHLER 1963, or THRALL,LUCAS 1963). 

2. Solutions of Selected Games 

In the following, solutions of some selected games are presented. The cor- 
responding main arguments for the stability of the results are sketched by 
selected negotiation chains, which present the essential ideas of the corre- 
sponding proofs. It may be remarked, that for most of these games, the 




377 



negotiation chains of the essential ideas can be joined to one large sequence. 
This makes it reasonable, that subjects can learn the reaisoning in limited 
time. 

2.1 The 3-person game with quotas 80,40,40: The Loyalty 
Paradox 

Game: N={1,2,3}, v(l,2)=v(l,3)=120, v(2,3)=80, v(S)=0 otherwise, exact- 
ness 10, loyalty 10. - Stable: (70,50,-), (70,-,50), (-,40,40). - Arguments for 
the stability of (70, 50,-), (-,40,40), instability of (80,40,-): (’ — ’ denotes that, 

using foresight, this player will not perform the dominance. ’ ’ denotes 

that no further dominance is possible): 



70 50 — 70 50 — 80 40 — — 40 40 

— 40 40 

80 — 40 — 60 20 60 60 — 

— 40 40 70 — 50 70 — 50 



Selecting between the stable alternatives (70,50,-),(70,-,50),(-,40,40), Play- 
ers 2 and 3 prefer (70,50,-), (70,-,50) by payoff, while (-,40,40) is loyalty 
maximal. 

Modification: If in the same game the loyalty is 0, then (-,40,40) is no 
longer stable, since (-,40,40) (70,50,-), and (70,50,-) is stable. 

Remarks: The modified example shows the ’loyalty paradox’, namely that by 
the incentives of loyalty, (70,50,-) becomes stable, while (-,40,40) is outruled. 
Experimental data support this phenomenon (see ALBERS, CROTT,MUR- 
NIGHAN 1985). 



2.2 The 3-Person Game with Quotas 80,80,40: Loyalty Paradox, 
and Results Outside the Range Predicted by Equal Share Analysis 

Game: N={1,2,3}, v(l,2)=160, v(l,3)=v(2,3)=120, v(S)=0 otherwise, ex- 
actness 10, loyalty 10. - Stable: (90,-,30), (-,90,30), (80,80,-). - Arguments 
for the stability of (70,50,-), (-,40,40), instability of (80,-,40): 



90 — 30 90 — 30 80 — 40 80 80 — 

80 80 ~ 

100 60 — — 80 40 100 — 20 

— 90 30 80 80 ~ — 90 30 



Modification: If the same game is played with exactness 1, loyalty 0, then 
(81,-,39), (-,81,39) are stable. - Arguments for the stability of (81,39,-), 
instability of (80,-,40), (80,80,-): 




378 



81 — 39 

82 78 — 
— 81 39 



81 — 39 

— 80 40 
80 80 — 



80 — 40 
80 80 — 



80 80 -- - 
81 — 39 

— 80 40 
80 80 — 



Remarks: The example shows, that the outcome of a game can be outside the 
range predicted by equal share analysis. - The predictions of the modification 
are supported by experiments via terminals with experienced players. 

2.3 The 3-Person Game with Quotas 100,0,0: An Example of 
Blocking 

Game: N={1,2,3}, v(l,2)=v(l,3)=100, v(S)=0 otherwise, exactness 5, loy- 
alty arbitrary. - Stable: (95, 5,-), (95,-, 5). - Arguments for the stability of 
(95, 5,-), instability of (100, 0,-): 

95 5 — 100 0 — 

— 0 0 

100 — 0 

— 0 0 



Modification: If v(N)=100, then coalition 2,3 gets blocking power. Sta- 
ble are (75,25,-), (75,-,25), and, in addition, (50,25,25) on the strategy 
level. - Arguments for the stability of (75,25,-), (50,25,25) and instability 
of (80,20,-), (70,30,-): 



75 25 — 80 20 -- 70 30 -- - (-- 0 0 

(— 0 0 75 — 25 (50 25 25 

80 — 20 (50 25 25 

(— 0 0 65 35 — 

(50 25 25 70 — 30 



(50,25,25) is no solution, of the negotiation game, since (-,0,0) is dominated 
by (75,25,-). It is a solution of the strategy game, since (-,0,0) -> (50,25,25) 
can only be dominated by giving Player 1 less than 75, what is dominated 
thereafter. 

Remarks: The example demonstrates the dominance of strategies. - The 
preference of (75,25,-) and (75,-,25) in the modified game is supported by 
experiments of MURNIGHAN,ROTH (1977). 

2.4 A Simple Symmetric 4-Person Game: Blocking, and 
Instability of (40,40,40,-) 

Game: N=(l, 2,3,4), v(S)=120 if #S=3, v(S)=0 otherwise, exactness 10, loy- 
alty 0. - Stable (up to symmetry): (55,55,10,-), (57,57,5,-). - Arguments for 




379 



the stability of (55,55,10,-), instability of (60,60,0,-), (40,40,40,-) (using the 
maximal loyalty condition of 1.6): 



55 55 10 ~ 55 55 10 — 60 60 0 — 40 40 40 — 

— ( 0 0 55 55 — 10 

60 60 — 0 70 — 25 25 (10 — 55 55 (stable, s. a. ) 

( 0 0 — 40 40 40 

(10 — 55 55 — 0 60 60 

(00 

— 0 60 60 (40 40 — 40 

( 0 0 

(40 40 40 “ 



Modification: If the same game is played with v(N)=120 (instead of 
v(N)=0), then the 2-person coalitions get blocking power. - Stable (up to 
symmetry): (45,45,30,-). - Arguments for the stability of (45,45,30,-), insta- 
bility of (40,40,40,-): 

45 45 30 — 45 45 30 — 40 40 40 

— 45 45 — 30 

50 50 — 20 60 — 30 30 (stable ,s . a. ) 

0 0 — 40 40 40 

30 30 30 30 



Remarks: The example shows that the solutions can be essentially different 
from the predictions of the demand equilibrium and equal share analysis 
(both predict (40,40,40,-), ..., (-,40,40,40)). - The predictions concerning 
the modified game are supported by experimental results. 



2.5 The 5-Person Apex-Games with Payoffs 100, and 
120: Relevance of the Numerical Payoff 



Game: N=(l,2,3,4,5), v(S)=100 if (IgS and #S>1) or if S=:(2,3,4,5), v(S)=0 
otherwise, exactness 5, loyalty 5. - Stable: (65,35,-,-,-), ..., (65 ,-,-,-,35). 
- Arguments for the stability of (65,35,-,-,-), instability of (70,30,-,-,-), 
(60,40,-,-,-), (-,25,25,25,25), (20,20,20,20,20): 



65 35 

70 — 30 

— 22 35 22 22 



65 35 

— 40 20 20 20 

65 — 35 

70 30 -- 

— 22 22 35 22 



70 30 

— 35 22 22 22 



60 40 

65 — 35 

(stable, s .a. ) 



— 25 25 25 25 

65 35 

(stable , s . a. ) 



20 20 20 20 20 

65 35 

(stable , s . a. ) 



Modification: If the same game is played with payoff 120 (instead of 100), 
the subjects spontaneously select exactness 10, loyalty 10. - Stable are (up 
to symmetry) (80,40,-,-,-), (-,30,30,30,30), (-40,27,27,27), (24,24,24,24,24). 




380 



- Solutions are of type (80,40 - The main arguments for the sta- 
bility of (80,40,-,-,-), (-,30,30,30,30), (-,40,27,27,27), and the instability of 
(-,30,30,30,30) are: 



80 40 

90 — 30 

— 27 40 27 27 



— 40 27 27 27 

70 — 50 

80 40 — 

(stable , s . a. ) 



80 40 

— 50 23 23 23 

80 — 40 

90 30 ~ 

— 27 27 40 27 



— 40 27 27 27 

70 50 

80 — 40 

(stable , s . a. ) 



90 30 

— 40 27 27 27 



70 50 

80 — 40 

(stable, s. a.) 



— 50 23 23 23 

80 — 40 

(stable, s .a. ) 



— 30 30 30 30 

70 50 

80 — 40 

(stable ,s .a. ) 



24 24 24 24 24 

70 50 

80 — 40 

(stable , s . a. ) 



Remarks: The example shows, that the result of a game can essentially 
depend on the numerical structure generated by payoff, prominence, and 
loyalty. - Experiments showed surprising differences in the results of the two 
games (the first game was played with payoff 1000, prominence of 100 and 
loyalty 100, this is structurally equal to the game with payoff 100, prominence 
10 and loyalty 10): 

type of result (x,y, (-,y,z,z,z) (-,a,a,a,a) (b,b,b,b,b) 



v(N)=1000 19 (13) 3 ( 2) 2 ( -) - ( -) 

v(N)=120 3 ( -) 1 ( -) 1 ( -) 18 (13) 



Table 2.5: Frequencies of final results over all rounds (rounds 3-5) 

The Apex Game with payoff 100 has been played on the Jerusalem Conference 
(1964) by game theoreticians. The result waus (65,35) in all but one cases. 
The deviating result (62.5,37.5) was proposed by a player who proportionally 
transformed his experimental result (30,10) from the game with payoff 40. 



2.6 Stability of (—,40,20,20,20) after (70,30,—,—,—) in the Apex 
Game 

A crucial point of the new approach is the instability of (70,30,-,-,-). The 
approach here says that (70,30,-,-,-) -> (-,40,20,20,20), and that thereafter 
no further dominance is possible, by loyalty of 3,4,5, and by the fact that 
Player 1 revealed a demand of 70. The Bargaining Set approach says that 
the argument is countered by (70,30,-,-,-) -> (-,40,20,20,20) -> (70,-,30,-,-). 

Therefore we ran an experiment with confederates in the position of Player 
2, who had the following orders: step 1: immediately contact Player 1, and 
form a coalition with payoffs (70,30,-,-,-), step 2: leave this coalition, and 
enter (-,40,20,20,20), step 3: be quiet, and wait for a 20 minutes time. The 




381 



prediction of our concept is, that (-,40,20,20,20) remains stable. - The result 
was as predicted in 12 of 12 cases. 

2.7 Stability of (70,30,-,-,—) in the Apex Game, if (2, 3, 4, 5) Can 
only Split Equally 

The result above can be reformulated as ’(70,30,-,-,-) is not stable for the 
Apex Game with payoff 100’. The crucial point for the instability is, that it 
is dominated by the uneven split (-,40,20,20,20). But what happens, if only 
even splits are permitted for coalition (2,3,4,5)? The prediction is the result 
(70,30,-,-,-). (Since (75,25,-,-,-) -> (-,25,25,25,25) (end). (65,35,-,-,-) 
is unstable, since [65,35,-,-,-) -> (70,-,30,-,-) (^stable as before).) - This 
prediction was the outcome of all 5 experimental games of this type. 



3. Relations to Other Solution Concepts 

3.1 Bargaining Set 

The Bargaining set was proposed by AUMANN,MASCHLER (1964). There 
are several points in which the bargaining set deviates from the concept here, 
and in all points experimental evidence clearly supports our approach. 

1. In our approach the coalition structures are not only partitions. Step- 
wise nested coalitions are permitted. The bargaining set models only the first 
step of coalition formation. 

2. Different from the Bargaining set, the approach here does not only 
consider argument-counterargument chains of length three, but of arbitrary 
length. By the model here the stability is not decided by the existence of a 
counterargument for any argument. In addition the stability properties of the 
counterargument are checked. - The Apex Game may serve as an example. 
Here the bargaining set argues that (60,40,-,-,-) is stable. The complete 
analysis of the approach here shows, however, that (60,40,-,-,-) is not stable: 



60 


40 


— 


— 


— 


65 


— 


35 


— 


— 


— 


40 


40 


10 


10 


65 


— 


— 


35 


— 


70 


— 


— 


— 


30 


— 


22 


22 


22 


35 



*) not ( 70 , — , — , 30 , — ), since thereafter ( — , 22 , 22 , 35 , 22 ), and no 
further dominEince possible 

In a similar way, the negotiation chain related to the loyalty paradox 2.1 
confirms, that counterarguments do not count, if they do not have sufficient 
stability properties: (-,40,40) ->• (70,50,-) — /-> (80 ,-,40) (-,40,40) (end). 




382 



3. The implicit assumption of the bargaining set is, that every player has a 
demand which applies for every coalition he enters. This is not supported by 
experimental evidence. Contrary to that, demands are functions of the coali- 
tion structure. For instance in the Apex Game, the observed counterargument 
after (60,40,-,-,-) -> (65,-,35,-,-) is not (-,40,40,10,10), but (-,20,40,20,20). 
When the role of the selected partner of Player 1 has shifted to Player 3, and 
after Players 3,4,5 formed a block, the attributed demand of Player 2 can 
change (and does change) essentially. Experimental results show, that the 
result of type (-,20,40,20,20) can become stable (see 2.6), while proposals of 
type (40,40,10,10) were never observed (not even mentioned in any argument 
of any player). 

4. Social preferences for certain coalitions are not modeled. For exam- 
ple, experiments show that in the Apex Game the proposal (75,25,-,-,-) is 
dominated by (-,25,25,25,25) (with equal payoff for Player 2). This type of 
dominance is not permitted in the Bargaining Set. (With a modified defini- 
tion of dominance, the Bargaining Set would capture this phenomenon.) 

3.2 Demand Equilibrium 

The demand-equilibrium has been defined by ALBERS (1974), see also AL- 
BERS (1979,1981), or TURBAY (1979), BENNETT (1980). The approach 
generates a generalization of the main simple solution of VON NEUMANN 
and MORGENSTERN (1944). It is related to the competitive bargaining set 
of HOROWITZ (1973). 

In this concept demands again only depend on the person, and not on the 
coalition structure. Again only the respective next step of coalition formation 
is considered. 

The concept consideres demand-profiles dGR'^, and uses the notations 
coa(d):= {SCN: d(S)<v(S)}. coa(i,d):== {SGcoa(d): iGS}. 

Definition: A demand profile dGR^ is a demand-equilibrium, iff 

(1) d(S)>v(S) for all SCN (’no slack’), 

(2) coa(i,d)^0 for all iGN (’coalition forming ability’), and 

(3) coa(i,d) coa(j,d) for no pair i,jGN (’no player depends on another’) 

To obtain a better description of the process of coalition formation, one would 
have to introduce demands as functions of coalition structures. But even then 
the ’no slack’ condition (condition (1)) would not fit with observed behavior 
in cases, where players form a certain kind of block. An example is given by 
the block of Players 1,2 in the outcome (45,45,30,-) of the 4-person game 2.4. 
The interesting feature of this kind of block is, that it is stabilized by the fact 
that, whenever a Player i of the block leaves the other(s) by a dominance 
(here for instance to (50,-,35,35)), then she can be replaced by one of the 
other players of the block, (here via (-,45,37,37)), where the replacing player 
receives as much as before. Accordingly, there may be slack, as long as the 




383 



block players are together and get the same payoffs. - It seems that this kind 
of block gets increasing importance with increasing number of players (see 
also ALBERS 1996). 



3.3 Equal Share Analysis, and Stages of Learning 

There is empirical evidence that unexperienced players start their first in- 
vestigations in a new game with equal share considerations, and select that 
coalition as a first candidate, which has the highest equal share value. They 
revise their demands according to the respective strength tested in the ne- 
gotiation process. For instance, they reduce their demands, when they are 
not feasible (there is no coalition that can fulfill their demands), or when 
they depend on other players (see condition (3) of the definition of demand 
equilibrium in 3.2). 

This process of learning the structure of the game is described for spa- 
tial games in ALBERS (1984), ALBERS,BRUNWINKEL (1983). The first 
two steps of reflection of this learning process correspond to the predictions of 
Equal Share Analysis (SELTEN 1972), and Equal Division Bounds (SELTEN 
1982). (For a detailed definition of steps of reflection see ALBERS 1984.) Ex- 
perienced players, as they are modeled here, will usually not remain in these 
early stages of reflection, so that more detailed or even deviating predictions 
are possible (see for instance 2.2). (Experimental results indicate that a pro- 
cedure where subjects play a series of different games, the game changes in 
every round, and communication is not permitted, reduces the learned level 
of reflection, and tends to support a behavior as predicted by Equal Share 
Analysis.) 



References 



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auf Anspruchsniveaus der Spieler basieren. Operations Research Verfahren XXI 
Meisenheim, 1-23 

ALBERS ,W. (1979) Core- and Kernel- variants based on imputations and demand 
profiles, in O. Moeschlin, D. Pallaschke (eds.): Game Theory and Related Top- 
ics, Amsterdam New York Oxford, 3-16 

ALBERS,W. (1981) Some solution concepts based on power potentials, in O. 
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ALBERS,W. (1983) Bloc forming tendencies as characteristics of the bargaining 
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ALBERS ,W., ALBERS, G. (1984) On the prominence structure of the decimal sys- 
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ALBERS, W., CROTT,H., MURNIGHAN,K. (1985) The formation of blocs in an 
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ALBERS ,W.,BRUNWINKEL, A. (1988) Equal share analysis for location games, in 
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ALBERS,W. (1988) Aspirations and aspiration adjustment in location games. 
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ALBERS,W. (1993): Ten rules of bargaining sequences, in U. Schulz, W.Albers, 
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ALBERS,W. (1996) Foundations of a theory of prominence - present state and 
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BENNETT, E. (1983) The aspiration approach to predicting coalition formation 
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BINMORE,K. (1985) Bargaining and Coalitions, in A.E.Roth (ed.): Game Theo- 
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KAHAN,J.P., RAPOPORT,A. (1984) Theories of coalition formation. Hillsdale, 
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MASCHLER,M. (1963) The power of a coalition. Management Science 10, 8-29 
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OSTMANN,A. (1993) Aspiration processing in multilateral bargaining: Experi- 
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Theory, Field, and Laboratory: 
The Continuing Dialogue 



Elinor Ostrom^ Roy Gardner^, and James Walker^ 

* Workshop in Political Theory and Policy Analysis, Indiana University, 513 North Park, 
Bloomington, IN 47408-3895, USA 

^ Department of Economics, Indiana University, Ballantine Hall, Bloomington, IN 47405, 
USA 



Abstract. Reinhard Selten has proposed a four-step schema for policy analysis, 
consisting of (1) focus on a section of reality, (2) formal modelling of that reality, (3) 
comparison of outcomes under different institutional arrangements, and (4) pursuit 
of empirical research based on those comparisons. In this paper, we apply Selten's 
schema to common-pool resources and the problems that they face. We show how the 
last decade of research has conformed to Selten's pattern. In particular, we stress the 
role field studies and controlled laboratory experiments have played in building 
behavioral theories for these resources - a role we expect to see expand in the future. 



In Guidance, Control, and Evaluation in the Public Sector, Reinhard Selten 
concludes his analysis of institutional utilitarianism with the following observations: 

Generally, a section of reality like the provision of health care is shaped by a 
multitude of legal and organizational facts which are constraints on individual 
behavior and creates incentives and disincentives. Thereby, individual behavior is 
limited and guided by a system of rules. The use of the words institutional 
arrangement refers to a rule system of this kind. 

It is important to compare the social desirability of different institutional 
arrangements which can be proposed for the same section of social reality. . 

. . In order to do this, one has to construct formal models of the institutional 
arrangements. . . . Hypothetical exercises in the framework of abstract models 
without immediate empirical application enhance the understanding of reality by 
a clarification of conceptual issues. Moreover, the results of purely theoretical 
investigations may inspire the improvement of empirical research techniques 
(1986: 258-59). 

Selten's advice on how to conduct research that contributes to policy analysis can be 
summarized as follows: (1) focus on one section of reality at a time, (2) develop 
formal models of institutional arrangements and incentives for that reality, (3) 
compare the social desirability of outcomes from alternative institutional 
arrangements, and (4) pursue empirical research based on these comparisons. 

The research program that we have pursued over the past decade is strongly 




387 



influenced by Selten's recommendations.^ We have attempted to engage in a dialogue 
between theory and two kinds of empirical research - field studies and laboratory 
experiments. This research has been driven by an effort to predict and explain 
complex empirical phenomena. Here we describe the highlights of that decade-long 
dialogue. 



1 A Focus on Common-Pool Resources 

The section of reality that we have chosen as our focus is a class of empirical and 
analytical problems related to the joint use of common-pool resources (CPRs). CPRs 
are natural or man-made resources in which (1) exclusion is nontrivial (but not 
necessarily impossible) and (2) yield is subtractable (Gardner, Ostrom, and Walker, 
1990). Problems related to these resources have frequently been treated as if they 
were pure public goods, since exclusion is difficult in both domains. Unlike pure 
public goods, however, the subtractability of the units in a CPR makes them subject 
to congestion and even destruction. Examples of CPRs abound in the natural world, 
including inshore and ocean fisheries, grazing lands, forests, lakes, and the 
stratosphere. Examples of man-made CPRs include irrigation systems, parking lots, 
and computer networks. Families, firms, and legislatures are social arrangements that 
create pools of resources that can also suffer from similar problems of overuse. 

Individuals jointly using a CPR are thought of as facing an inevitable social 
dilemma, referred to as the tragedy of the commons (Hardin, 1968). In this tragic 
scenario, individual users ignore the externalities they impose on others, leading to 
suboptimal outcomes. There are several ways out of such dilemmas, all of which are 
themselves problematic. For instance, it may be possible to create a set of institutional 
arrangements to transform the incentives of a CPR problem; this transformation can 
create a second-order public good problem, which is again a social dilemma. 
Monitoring rules of a transformed situation so that users conform to them is likely to 
become a third-order public good problem, yet another social dilemma. Since the 
users of the resource appear to be stuck in a social dilemma no matter what they do, 
analysts tend to propose reform imposed from the outside. Thus, one has sweeping 
calls for privatization or central government control of CPRs, as well as calls for 
institutional arrangements that do not fit neatly into either category of state or market 
solution (Ostrom, 1990). 

The impetus for focusing on CPRs was Ostrom's participation in an 
interdisciplinary panel of the National Academy of Sciences (NAS) of the United 
States to examine various forms of common-property arrangements (National 



Ostrom had the privilege of participating with Selten in the ZiF Research Group that 
produced Guidance, Control, and Evaluation in the Public Sector (Kaufmann, Majone, and 
Ostrom, 1986). Both Geirdner and Ostrom had the privilege of participating in Selten's ZiF 
Research Group on Game Theory and the Behavioral Sciences, 1987-88. All three of us had 
the privilege of participating in the Bonn Summer Workshop on Experimental Economics and 
Bounded Rationality, organized by Selten in 1991. All these experiences have had a profound 
effect on the research discussed here. The authors wish to acknowledge the support of the 
National Science Foundation (grant SBR-93 19835). 




388 



Research Council, 1986). The case studies brought together by NAS revealed that the 
users of many CPRs had escaped from the tragedy of the commons. These users, after 
suffering from overuse of the resource, recognized the problem and the need for 
institutional reform. They crafted institutional arrangements of their own to cope with 
the problem. Further, they monitored their own agreements and sanctioned those who 
did not conform. In this way, they overcame all three orders of social dilemma. 
Tragic outcomes were not inexorable. At the same time, these success stories were 
not the only ones observed. Sufficient evidence existed of major failures, including 
species extinctions, to pose a genuine intellectual puzzle. How and why did some 
participants in difficult social dilemmas find a way out, while others did not? With 
this question inspired by field studies, the dialogue between theory and field begins. 



2 Formal Models of CPR Situations 

One way to approach the above question is to model a CPR situation as a game. 
Suppose the game has only inefficient equilibrium outcomes, so that even the most 
optimistic equilibrium selection theory will predict a suboptimal outcome. Then one 
can treat a restructuring of the rules of the game as a way for players to achieve better 
outcomes through better equilibria. This approach we call the technique of rules and 
games (Gardner and Ostrom, 1991; Ostrom, Gardner, and Walker, 1994), and it 
applies with considerable success to CPRs. 

The first question one must answer in any game modeling attempt is what game to 
use. Hardin had presented a story that could easily be translated into a Prisoners' 
Dilemma (PD) as the basis of his gloomy predictions, and the PD has been used 
extensively as a game model of CPRs (Dawes, 1973; Stevenson, 1991). Having a 
large number of case studies forced us to recognize the prevalence of many different 
game structures. These structures included assignment and provision games (see 
Gardner and Ostrom, 1991), monitoring games (Weissing and Ostrom, 1991 , 1993), 
all coexisting within games involving appropriation externalities (see Ostrom, 
Gardner, and Walker, 1994: ch. 3 for details). 

Whether any of these games lead to a social dilemma depends on particular 
parameter values. Consider the number of users, n. Harvesting from a fishery for 
one's own consumption (n = 1) has a very different equilibrium outcome from 
harvesting from the same fishery to sell in competition with other fishers in a market 
(n > 1). For values of n below some critical value n*, optimal outcomes of some sort 
are achievable. For large values of n (n > n*), however, outcomes are increasingly 
inefficient (see Gardner, 1995: ch. 4). The resource may be physically the same, but 
changes in the parameter in question change the predicted equilibrium. 

Based on the above two considerations, we decided not to rely on the PD as our 
primary model. Instead, our initial base appropriation model involved n-players in a 
one-shot game where the return that any one player obtained from harvesting resource 
units depended on the level of harvesting undertaken by others. Since most natural 
resource systems are characterized by nonlinear relationships, we used a quadratic 
production function that meant that the initial units appropriated are more valuable 
than units appropriated later. If appropriators withdrew a sufficiently large number 
of resource units, however, the outcome was counterproductive. The maximum 
outcome occurred when appropriators harvested some, but not all, of the resource 




389 



units available. A rational player in this game, without communication and external 
enforcement of agreements, would be led at equilibrium to harvest more resource 
units than was optimal. The aggregate payoff to the players at any equilibrium for this 
game was substantially less than the maximum aggregate payoff, on the order of 
40%. If this game adequately reflected the game that appropriators in field settings 
were playing, how did they obtain outcomes better than those available at this 
equilibrium? 



3 Comparing the Social Desirability of 
Institutional Arrangements 

In field settings, as opposed to the above noncooperative game model, players tend 
to play an appropriation game more than once, thus creating the possibility of 
repeated game effects. However, as long as the one-shot game is finitely repeated, its 
subgame perfect equilibria (Selten, 1975) will continue to be inefficient (the 
equilibrium efficiency is only 40% in the example above). (Of course, if players 
delude themselves into thinking they are playing an infinitely repeated game, better 
subgame perfect equilibria are available.) It would not appear that mere repetition of 
a game will lead to an easy escape from a dilemma outcome. Repetition of the 
dilemma outcome is what subgame perfect equilibrium predicts. 

Again, players in field settings have the opportunity to communicate with one 
another. The importance of individuals communicating with one another in settings 
where mutual promises are not enforced by an external authority has been heavily 
discounted in game-theoretic analysis (see Harsanyi and Selten, 1988). Talk is cheap. 
Without external enforcement, individuals do not have an incentive to keep promises 
that are out of equilibrium. Thus, even with communication, subgame perfect 
equilibrium outcomes of the repeated game would still be inefficient (see Ostrom, 
Walker, and Gardner 1992 for details). 

Field studies, however, show that many self-organized groups have developed and 
agreed upon their own rules, monitor each other's performance, and engage in 
graduated sanctions for those instances where individuals do not keep their promises 
(see literature reviewed in Ostrom, 1994 and Ostrom, Gardner, and Walker, 1994). 
Thus, the institutions of communication, monitoring, and sanctioning appear to 
improve observed outcomes in such groups. These institutions transform the game in 
a way that the model does not capture. 

Exactly how these institutions accomplish this transformation, however, is 
impossible to determine from field studies alone. There are several reasons why. 
First, in contrast to a solved game model, scholars can rarely obtain quantitative data 
about the potential benefits that could be achieved if participants cooperate at an 
optimal level or about the level of inefficiency yielded when they act independently. 
Second, it is difficult to determine how much improvement has been achieved as 
opposed to the same setting without particular institutions in place. Third, without 
using expensive time series designs, studies only include those institutional 
arrangements that survive and the proportion of similar cases that did not survive 
remains unknown. Fourth, many variables differ from one case to the next. A large 
number of cases is required to gain statistical control of the relative importance of 




390 



diverse variables. Some of our own students have been able to conduct studies of a 
relatively large number of field cases so as to engage in multivariate statistical 
analysis (see Tang, 1992; Schlager, 1990; Lam, 1994). To complement such studies, 
the techniques of laboratory investigation play an important role. 



4 Pursuit of Model-Based Empirical Research 

The designer of an experiment knows exactly what the achievable optimum is for 
each experimental set of conditions, and how far from optimum any equilibrium is. 
In an experiment, one can measure the effect of communication, for instance, by how 
close an experimental group that agrees to some joint strategy sticks to that agreement 
and achieves a good outcome. Groups that fail to achieve an agreement and continue 
to make independent and inefficient decisions continue to be a part of the data 
produced by experiments. Thus, the proportion of groups achieving a Joint agreement 
under diverse conditions and the level of conformance to agreements are known. 
Finally, a major advantage of the experimental method is better control of relevant 
variables. Laboratory settings are sparse in contrast to natural settings, but that is part 
of their advantage. 

Experimental studies have generated data that is highly consistent with the cases 
describing smaller and more homogeneous field settings (see Ostrom and Walker, 
forthcoming). Individuals (typically, students in groups of 8) using CPRs and acting 
independently tend to overharvest, congest, and/or destroy these resources. When 
given an opportunity to communicate on a face-to-face basis about the situation they 
are in, however, these same individuals tend to agree upon joint strategies and carry 
these out. In repeated settings, individuals use communication as a mode of 
sanctioning others (Ostrom, Walker, and Gardner, 1992).^ Moreover, when given an 
option to impose sanctions on themselves in a repeated setting, individuals who agree 
upon a joint strategy and on sanctions achieve close to 100% efficiency. Often, they 
do not even need to use the sanctioning capability they have devised for themselves 
(ibid.). 

Findings from experiments where no communication is allowed are consistent 
with the outcome predicted by subgame perfect equilibrium of the finitely repeated 
game. Findings from experiments where communication is allowed are not consistent 
with that same theory. The finding that "mere talk" can produce an agreement that is 
enforced by the participants themselves needs to be explained theoretically and not 
simply with ad hoc adjustments. 

The problem of explaining behavior observed in the laboratory has long inspired 
Reinhard Selten's theory of bounded rationality (see Selten, 1978 and 1991, for 
example). A variant of a suggestion by Selten and his coworkers has proved crucial 
to our own work (see Selten, Mitzkewitz, and Uhlich, 1988). Suppose that in a game 



^ In an interesting variation of our communication experiments, Rocco and Warglien (1995) 
replaced face-to-face communication with communication via email. Subjects tended to come 
closer to optimum in their agreements, but had greater difficulty in conforming to agreements 
they had made. 





391 



with communication an agreement has been reached. Consider a player in that game 
who has just observed a deviation from the agreement. We call the response of such 
a player measured if, when the deviation observed is small, the player's own 
subsequent deviation (if any) is small. In this theory, a deviation is small if it lies 
between the agreement and the equilibrium for that player. A measured response is 
a heuristic for dealing with deviation from equilibrium, and as such is fundamentally 
different from the threats and punishments involved in subgame perfect equilibrium. 

We have developed a data-based theory of measured response to explain how 
individuals who adapt this heuristic can agree upon reasonably efficient joint 
strategies and sustain those agreements. What is crucial here is not just the 
opportunity to communicate but also the extent to which deviations remain small. As 
long as no more than 10% of the deviations are large, then an agreement is sustained 
and efficiencies of at least 80% (up from 40%) are observed. On the other hand, if 
more than 10% of deviations are large, then agreements break down and play reverts 
to that predicted by subgame perfection. Using measured responses, players can 
achieve high levels of efficiency without external enforcers (see Ostrom, Gardner, 
and Walker, 1994: ch. 9, for details). 



5 Where the Dialogue Goes from Here 

The study of CPRs being used by a small number of relatively homogeneous 
participants has produced a progressive research program over the past decade. While 
theory produced a set of predictions that cooperation was unlikely in CPR situations, 
field research produced some contrary evidence. Carefully designed experiments 
produced evidence that was consistent with received theory under one set of 
institutional conditions, and with field studies, under another set. 

In addition, these advances have spurred the building of new theories besides that 
of measured response. Hester and Guth (1994) have recently demonstrated that 
individual preferences including a concern for the payoffs received by others (called 
"altruism" by Hester and Gtith) are evolutionarily stable under specific conditions. In 
another paper, Gtith and Kliemt (1994) have shown that having internal costs 
associated with breaking promises is also evolutionarily stable under similar 
conditions (see also Crawford and Ostrom, 1995). In both cases, the identity and 
"type" of all players are known (or can be known at low cost) to other players before 
playing a round of a game. This kind of low cost information about the 
trustworthiness and other-regardingness of participants is more likely to be available 
in small, homogeneous communities that have been studied in the field and in the lab. 

Typically in science, research starts on the simpler systems and only later moves 
to more complex systems. A simple CPR setting exists when the incentives are highly 
salient and clear to the participants, who are themselves relatively homogeneous and 
few in number. Relaxing any of these features results in complexity. The most 
complex CPRs are global in nature: the open seas, the atmosphere, outer space. 
Nevertheless, we are confident that the dialogue that we have started will continue 
to prove useful in the face of additional complexity. 




392 



6 A Continuation of the Dialogue into the Future 

One way of moving toward explanations of behavior in larger and more complex 
CPRs is to take incremental steps in our theory, field, and laboratory studies in that 
direction. In the field, we are now studying local institutions as they affect the 
behavior of appropriators harvesting from forests in Bolivia, India, Nepal, and 
Uganda (see Gibson, McKean, and Ostrom, forthcoming). This is a logical next step 
for field studies, since appropriation fi*om forests produces concentrated local impacts 
as well as affecting larger systems at a landscape and global level. In the laboratory, 
we are conducting two series of experiments that take small incremental steps. In one 
of these, subjects face asymmetric payoff matrices. In the other, subjects participate 
in a repeated base CPR game and also vote (using simple majority rule, majority rule 
with symmetric proposals, and unanimity) on allocation rules for the later rounds of 
the game they are playing (Walker et al., 1995). These new empirical thrusts 
challenge us to extend our theoretical understanding of the issues surrounding 
multiple, simultaneous scales, asymmetries among players, and the choice of game 
structure by those involved. 



References 

Bester, Helmut, and Werner Guth. 1994. "Is Altruism Evolutionarily Stable?" Tilburg, The 
Netherlands: Tilburg University, Center for Economic Research. 

Crawford, Sue E.S., and Elinor Ostrom. 1995. "A Grammar of Institutions." American 
Political Science Review 89(3) (Sept.): 582-600. 

Dawes, Robyn M. 1973. "The Commons Dilemma Game: An N-person Mixed-Motive Game 
with a Dominating Strategy for Defection." Oregon Research Institute Research Bulletin 
13:1-12. 

Gardner, Roy. 1995. Games for Business and Economics. New York: John Wiley. 

Gardner, Roy, and Elinor Ostrom. 1991. "Rules and Games." Public Choice 70(2) (May): 
121-49. 

Gardner, Roy, Elinor Ostrom, and James Walker. 1990. "The Nature of Common-Pool 
Resource Problems." Rationality and Society 2(3) (July): 335-58. 

Gibson, Clark, Margaret McKean, and Elinor Ostrom, eds. Forthcoming. Forest Resources 
and Institutions. Forests, Trees and People Programme, Phase II, Working Paper no. 3. 
Rome: Food and Agriculture Organization of the United Nations. 

Guth, Werner, and Hartmut Kliemt. 1994. "Competition or Co-operation. On the Evolutionary 
Economics of Trust, Exploitation and Moral Attitudes." Metroeconomica 45(1394): 155- 
87. 

Hardin, Garrett. 1968. "The Tragedy of the Commons." Science 162:1243-48. 

Harsanyi, John, and Reinhard Selten. 1988. A General Theory of Equilibrium Selection in 
Games. Cambridge: MIT Press. 

Lam, Wai Fung. 1994. "Institutions, Engineering Infrastructure, and Performance in the 
Governance and Management of Irrigation Systems: The Case of Nepal." Ph.D. diss., 
Indiana University, Bloomington. 

National Research Council. 1986. Proceedings of the Conference on Common Property 
Resource Management. Washington, D.C.: National Academy Press. 

Ostrom, Elinor. 1990. Governing the Commons: The Evolution of Institutions for Collective 
Action. New York: Cambridge University Press. 

Ostrom, Elinor. 1994. "Constituting Social Capital and Collective Action." Journal of 




393 



Theoretical Politics 6(4) (Oct.): 527-62. 

Ostrom, Elinor, Roy Gardner, and James Walker. 1994. Rules, Games, and Common-Pool 
Resources. Ann Arbor: University of Michigan Press. 

Ostrom, Elinor and James Walker. Forthcoming. "Neither Markets Nor States: Linking 
Transformation Processes in Collective Action Arenas." In Perspectives on Public Choice, 
ed. Dennis Mueller. New York: Cambridge University Press. 

Ostrom, Elinor, James Walker, and Roy Gardner. 1992. "Covenants With and Without a 
Sword: Self-Governance Is Possible." American Political Science Review 86(2) (.June): 
404-17. 

Rocco, Elena, and Massimo Warglien. 1995. "Computer Mediated Communication and the 
Emergence of 'Electronic Opportunism'." Venice, Italy: University of Venice, Department 
of Economics, Laboratory of Experimental Economics. 

Schlager, Edella. 1990. "Model Specification and Policy Analysis: The Governance of Coastal 
Fisheries." Ph.D. diss., Indiana University, Bloomington. 

Selten, Reinhard. 1975. "Reexamination of the Perfectness Concept for Equilibrium Points in 
Extensive Games." International Journal of Game Theory 4:25-55. 

Selten, Reinhard. 1978. "The Chain Store Paradox." Theory and Decision 9:127-59. 

Selten, Reinhard. 1986. "Institutional Utilitarianism." In Guidance, Control, and Evaluation 
in the Public Sector, ed. Franz-Xaver Kaufmann, Giandomenico Majone, and Vincent 
Ostrom, 251-64. Berlin: de Gruyter. 

Selten, Reinhard. 1991. "Evolution, Learning, and Economic Behavior." Games and 
Economic Behavior 3:3-24. 

Selten, Reinhard, Michael Mitzkewitz, and Gerald Uhlich. 1988. "Duopoly Strategies 
Programmed by Experienced Players." Discussion Paper no. B-106. Bonn, Germany: 
University of Bonn. 

Stevenson, Glenn G. 1991. Common Property Economics: A General Theory and Land Use 
Applications. New York: Cambridge University Press. 

Tang, Shui Yan. 1992. Institutions and Collective Action: Self-Governance in Irrigation. San 
Francisco, Calif: ICS Press. 

Walker, James, Roy Gardner, Elinor Ostrom, and Andrew Herr. 1995. "Voting on Allocation 
Rules in a Commons without Face-to-Face Communication: Theoretical Issues and 
Experimental Results." Paper to be presented at the conference on "Game Theory and the 
Behavioral Sciences," Tucson, Arizona, October 10-12, 1995. 

Weissing, Franz J., and Elinor Ostrom. 1991. "Irrigation Institutions and the Games Irrigators 
Play: Rule Enforcement without Guards." In Game Equilibrium Models II: Methods, 
Morals, and Markets, ed. Reinhard Selten, 188-262. Berlin: Springer- Verlag. 

Weissing, Franz J., and Elinor Ostrom. 1993. "Irrigation Institutions and the Games Irrigators 
Play: Rule Enforcement on Government- and Farmer-Managed Systems." In Games in 
Hierarchies and Networks: Analytical and Empirical Approaches to the Study of 
Governance Institutions, ed. Fritz W. Scharpf, 387-428. Frankfurt am Main: Campus 
Verlag; Boulder, Colo.: Westview Press. 




Naive Strategies in Competitive Games^ 



Ariel Rubinstein*, Amos Tversky^, and Dana Heller^ 

’ The School of Economics, Tel-Aviv University, Tel-Aviv, Israel, 
e-mail: rariel@ccsg.tau.ac.il 

^ Department of Psychology, Stanford University, Stanford, CA 94305-2130, 
e-mail: amos@psych.stanford.edu 

^ Graduate School of Business, Stanford University, Stanford, CA 94305-5015, 
e-mail: heller@ecu.stanford.edu 



Abstract. We investigate the behavior of players in a two-person competitive 
game. One player "hides" a treasure in one of four locations, and the other player 
"seeks" the treasure in one of these locations. The seeker wins if her choice 
matches the hider’s choice; the hider wins if it does not. According to the classical 
game-theoretic analysis, both players should choose each item with probability of 
.25 . In contrast, we found that both hiders and seekers tended to avoid the 
endpoints. This bias produces a positive correlation between the players' choices, 
giving the seeker a considerable advantage over the hider. 

J.E.L Classification numbers: C7,C9. 



1. Introduction 

In this article we investigate the behavior of players in several two-person 
competitive (i.e., zero-sum) games. In each game, one player "hides" a treasure in 
one of four locations, and the other player "seeks" the treasure in one of these 
locations. The seeker wins if her choice matches the hider's choice; the hider wins 
if it does not. Because the game has only two possible outcomes, and each player 
would rather win than lose, a rational player should maximize the probability of 
winning. 

This game has a unique Nash equilibrium with mixed strategies: Each player 
selects the four alternatives with equal probability (i.e. .25). The proof of this 



^ This paper replaces working paper “Naive Strategies in Zero-Sum Games”, The 
Sackler Institute of Economic Studies, Tel-Aviv University, No. 17-93 (September, 1993). 

We acknowledge support from United States-Israel Binational Science Foundation, 
Grant no. 1011-341, to the first author, and from the National Science Foundation, Grant 
no. SBR-9408684, to the second author. 

We are grateful to Anat Alexandron, Tony Bastardi and Bradley Ruffle for their help. 




395 



proposition reveals the strategic considerations induced by this game. In 
equilibrium, player 2 (the seeker) does not choose any alternative unless she 
believes that it is (one of) the most likely alternatives to be chosen by player 1. 
Similarly, in equilibrium, player 1 (the hider) does not choose any alternative 
unless he believes that it is among the alternatives that are least likely to be chosen 
by player 2. Thus in equilibrium the probability of any alternative that may be 
chosen by player 2 cannot exceed .25 . Consequently, player 2 (and analogously 
player 1) selects each of the alternatives with equal probability, and player 2 wins 
with probability .25 . 

The following study compares the solution prescribed by game theory to the 
choices made by subjects. In particular, we explore the effects of three factors : 
the spatial position of the alternatives (endpoint versus middle point), the relative 
distinctiveness of the alternatives (focal versus non-focal items), and the role of 
the player (hider versus seeker). Although these factors are not considered in the 
formal game theoretical analysis, there are reasons to believe that they might 
influence the players’ choice. 



2. Method 

The subjects in the present study were undergraduate students at Stanford 
University who had taken an introductory psychology course, and were required 
to participate in a few experiments. All subjects were presented with the same 
form, consisting of six sets with four items each, as shown in Figure 1. 

The six sets of items, each of which corresponds to a different game, were 
constructed as follows. Three of the sets consist of pictorial items (sets 1,3 and 5), 
and the other three of verbal items (letters or words). Each set includes one 
distinctive (focal) item. We constructed three types of focal items: neutral (sets 1 
and 4); positive, on the background of negative items (sets 5 and 6); and negative, 
on the background of positive items (sets 2 and 3). Each type consists of one 
verbal and one pictorial set. We also distinguish between the items in the two 
ends, called endpoints, and the two items in the middle. 

Each participant in the competitive games was assigned to play either the role 
of the hider or the role of the seeker in all six games. The players were informed 
that they will be randomly paired with a person playing the other role. They were 
told that the hider's task is "to hide the treasure behind one of the four items", 
whereas the seeker's role is "to guess the location of the treasure". If the seeker 
guesses correctly - she wins $10 and the hider receives nothing; if the seeker does 
not guess correctly - she receives nothing and the hider receives $10. The 
participants were told that "both players are informed of the rules of the game". 
They were promised that five pairs of players will be selected at random and will 
be paid according to the outcome of the game. 




396 



1 . 







2 . 



polite rude honest friendly 



■ © © © (X) 



ABA 



A 




hate 



detest 



love dislike 



Figure 1 . The six sets of items used in all games. The focal item is marked in each set. 



We also investigated two non-competitive two-person games based on the 
same sets of items. In the coordination game, both players receive $10 if they 
select the same item, and nothing if they select different items. In the 
discoordination game, both players receive $10 if they select different items, and 
nothing if they select the same item. Two different groups of subjects played the 
coordination and discoordination games. As before, subjects were told that they 
will be randomly paired with another player, and that five pairs of players will be 
selected at random and will be paid according to the outcome of the game. 




397 



3. Results 

We first describe the results of the non-competitive games, and then turn to the 
analysis of the competitive games, which is the focus of this paper. 

Non-Competitive Games 

Tables la and lb present the percentage of subjects who chose each of the four 
items in each of the coordination and the discoordination games, respectively. 



la - Coordination Games (N=50) 



Set 


1 


2 


3 


4 


P 


1 


86 


0 


10 


4 


0.001 


2 


6 


54 


12 


28 


0.001 


3 


6 


6 


14 


74 


0.001 


4 


8 


76 


16 


0 


0.001 


5 


6 


88 


6 


0 


0.001 


6 


2 


6 


88 


4 


0.001 



lb - Discoordination Games (N=49) 



Set 


1 


2 


3 


4 


P 


1 


39 


14 


18 


29 


NS 


2 


28 


20 


32 


20 


NS 


3 


17 


27 


23 


33 


NS 


4 


6 


20 


55 


18 


0.001 


5 


17 


40 


29 


15 


0.05 


6 


16 


29 


26 


29 


NS 



Table 1. Percentage of subjects who chose each of the four items in each of the six sets, in 
the coordination games (la), and the discoordination games (lb). The focal item in each set 
appears in bold face. The significance level associated with the Chi-square test of each set 
is denoted by p. 



The data in table la show, as expected, that in all six coordination games, the 
majority of subjects (overall mean=78%) chose the focal item. The hypothesis that 
each item is selected with equal probability (i.e., .25) is rejected by a Chi-square 
test as indicated by the p-values in the right hand column. The pattern of 




398 



responses in the discoordination games , displayed in table lb, is quite different. 
Here, there is no tendency to select the focal item, and the distribution of choices 
in most games, does not depart significantly from random selection. 

To investigate individuals' responses, we counted for each player the number of 
games (from 0 to 6) in which the player selected the focal item. Figure 2 presents, 
for both the coordination and discoordination games, the distributions of these 
values, along with the binomial distribution (with p=.25), that is expected under 
the hypothesis that all items are selected with equal probability. 




Number of focal points chosen by players 



Figure 2. Distribution of patterns of focal point choices for the coordination and 
discoordination games compared to the predicted binomial distribution. 



Figure 2 shows that in the coordination game the distribution of choices was 
markedly different than expected by chance. In particular, more than 50% of the 
players selected the focal item in all six games. In the discoordination game, in 
contrast, the distribution of choices of the focal item is very similar to that 
expected under random selection. These data indicate that in the presence of a 
single distinctive item, subjects were able to achieve a reasonable level of 
coordination (about 65%). In the absence of an effective method of achieving 
discoordination, players chose more or less in random. 

Compe ti t i ve (jTOQS 

Tables 2a and 2b present the results of the competitive games for the seekers 
and the hiders respectively. The data show that, contrary to the game theoretical 
analysis, the players did not choose the four items in each game with equal 
probability. A Chi-square test indicates that with only one exception (set 3 for the 
hider) all the distributions of choices depart significantly from random selection. 




399 



We next discuss, in turn, the effects of endpoints, focal items, and the players’ 
role. 



2a - Seeker (N=62) 



Set 


1 


2 


3 


4 


P 


1 


29 


24 


42 


5 


0.001 


2 


8 


40 


40 


11 


0.001 


3 


7 


25 


34 


34 


0.01 


4 


13 


31 


45 


11 


0.001 


5 


16 


55 


21 


8 


0.001 


6 


20 


21 


55 


14 


0.001 



2b - Hider (N=53) 



Set 


1 


2 


3 


4 


p 


1 


23 


23 


43 


11 


0.01 


2 


15 


26 


51 


8 


0.001 


3 


21 


26 


34 


19 


NS 


4 


9 


36 


40 


15 


0.01 


5 


15 


40 


34 


11 


0.01 


6 


11 


23 


38 


28 


0.05 



Table 2. Percentage of subjects who chose each of the four items in each of the six sets, in 
the Competitive games ( 2a : Seeker, 2b : Hider ). The focal item in each set appears in 
bold face. The significance level associated with the Chi-square test of each set is denoted 
by p. 



Perhaps the most striking feature of these data is players’ tendency to avoid the 
endpoints. Overall, the two endpoints were selected by the hiders and the seekers 
on 31% and 28% of the games, respectively, significantly less than the 50% 
expected under random choice (p<0.001). The tendency to avoid the endpoints is 
even stronger in sets 2,4,5,6 in which the endpoint is not a focal item. 

To investigate individuals’ choices, we counted for each player the number of 
games (from 0 to 6) in which the endpoint was selected. Figure 3 presents, for the 
hiders and seekers separately, the distributions of these values, along with a 
binomial distribution (p=0.5) expected under random choice. It is evident from the 
figure that both hiders and seekers were reluctant to select the endpoints: 
Avoiding the endpoints in all six games was the single most common pattern, 
selected by 28% of the hiders and 37% of the seekers. The probabilities of 
obtaining such results by chance are vanishingly small. 




400 




Number af endpoints chosen by players 



Figure 3. Distribution of patterns of endpoint choices for seekers and hiders compared to 
the predicted binomial distribution 



We have initially hypothesized that hiders will tend to avoid the focal item, in 
order not to be "predictable". Thus, we expected the hiders to choose the focal 
item less frequently than the seekers and less frequently than expected by chance 
(i.e., 25%). The results of the present study support the former hypothesis but not 
the latter. Overall, the focal items were selected by the seekers 41% of the time 
and by the hiders only 30% of the time (p<0.5), but both groups selected the items 
significantly more than expected by chance. 

Two factors might have contributed to this result. First, the observed tendency 
to avoid the endpoints is bound to increase the percentage of choice of other 
alternatives, including the focal items. Second, the popularity of. the focal items 
was produced primarily by the sets that included a positive focal item. In the 
verbal sets (2,4 and 6), where the focal items were not the endpoints, the positive 
focal item (love) was selected 47% of the times (across hiders and seekers) 
whereas the neutral (B) and the negative (rude) focal items were selected 33% of 
the times (p<.05). The results for the pictorial sets were similar but here the 
comparison was problematic because the neutral and negative focal points (sets 1 
and 3) were endpoints whereas the positive (set 5) was not. 

To test whether the endpoint effect depends on the presence of a focal item, we 
conducted a follow-up study using identical procedure, except that none of the six 
sets included a focal item. As before three of the sets were verbal and three 
pictorial. In three of the sets all four items were identical, except for spatial 
position. In the remaining three sets all four items were distinct, but without a 
focal item (e.g. south, east, north, west). There were 103 subjects who were 
divided about equally between the two roles. 





401 



In this study too, players tended to avoid the endpoints, although this tendency 
was less pronounced than before. Seekers chose the endpoints 36% of the time 
and hiders chose it 43% of the time. Both values are significantly smaller than the 
value of 50% expected by chance (p<.05). To summarize, although the picture is 
not entirely clear, it appears that seekers tend to avoid the endpoints more than 
hiders, while hiders tend to avoid the focal point more than seekers. 



4. Discussion 

The main finding of the present study is that both hiders and seekers tended to 
avoid the endpoints, thereby departing from the classical game theoretical 
solution. This tendency was not observed in the coordination game, where players 
tended to choose the focal items, or in the discoordination game, where players 
chose more or less at random. The tendency to avoid the endpoints has also been 
observed in other contexts. When people are faced with the choice among 3,4 or 5 
identical items, people tend to avoid the endpoints and select the middle item. 
This bias has been observed in picking products from a supermarket shelf, 
selecting a bathroom stall, or picking an arbitrary symbol (Christenfeld, 1995). 

In the context of a competitive game, the reluctance to select the endpoints has 
obvious strategic implications. In the present study it induces a high correlation 
between the choices of the hider and the seeker, which gives the seeker a 
considerable advantage despite the symmetry of the game. Note that if the 
distribution of the seeker's choices is (Pi,P 2 ,P 3 ,P 4 ) and the distribution of the 
hider's choices is (qbq 2 j 93594 ), then the probability of a match is 
i^=Piqi+P 2 q 2 +P 3 q 3 +P 4 q 4 » and the expected payoff of the seeker is $10m. If either 
player chooses items with equal probabilities than m=.25. In contrast, the 
observed values of m for the six games in the main study were: .31, .33, .26, .32, 
.32, .31, with an average of m=.31. To interpret these values note that if the two 
players employ the mixed strategy (.4,.2,.2,.2), where one item is chosen twice as 
often than any other, then m is .28; if both players employ the strategy (.4,.4,.l,.l) 
then m=.34. Thus, the observed m-values reflect substantial departure from the 
equal-probability strategies. 

A possible explanation for the seeker's advantage involves the psychological 
asymmetry between the hider and the seeker. Although the games are 
simultaneous, that is, players make their choices without knowledge of the other 
player's choice, it is natural to regard the seeker as "responding" to the hider's 
choice. Because it seems easier to contemplate the first than the second move in a 
game, the seeker may have a better insight into the behavior of the hider than vice 
versa. Thus, the seeker's tendency to avoid the endpoints could reflect her valid 
belief that the hider is not likely to select these items. 

However, the results of a study we conducted earlier do not support this 
hypothesis. In that study subjects played a competitive game (using the A B A A 
set) in which the hider places a mine instead of a treasure. In this game the 




402 



seeker's goal is to avoid the item chosen by the hider, whereas the hider’s role is to 
match the choice of the seeker. If the seeker has a psychological advantage, she 
should be able to avoid the mine more often than expected by chance. The data, 
however, yielded a positive correlation (m=.28) between the choices of the two 
players, which favored the hider over the seeker. This result suggests that the 
advantage of the seeker in the treasure game and of the hider in the mine game 
reflects a common response tendency rather than a psychological advantage of 
one role over the other. 

In summary, our players were attentive to the payoff function, and employed 
different strategies in the three types of games. As expected, they selected the 
focal item in the coordination games, and chose more or less at random in the 
discoordination games. In the competitive games, however, the players employed 
a naive strategy (avoiding the endpoints), that is not guided by valid strategic 
reasoning. In particular, the hiders in this experiment either did not expect that the 
seekers too, will tend to avoid the endpoints, or else did not appreciate the 
strategic consequences of this expectation (Shafir and Tversky, 1992). 



References 

N. Christenfeld, Choices from identical options. Psychological Science, vol. 6, 
no. 1, 1995 (50-55). 

E. Shafir, and A. Tversky, "Thinking Through Uncertainty: Noncon- 
sequentialism in Reasoning and Choice". Cognitive Psychology, 24:449-474, 
1992. 




Induction vs. Deterrence in the Chain Store 
Game: How Many Potential Entrants are Needed 
to Deter Entry? 



James A. Sundali^ and Amnon Rapoport^ 

* Department of Administrative Sciences, Graduate School of Management, Kent State 
University, Kent, OH 44240, USA 

^ Department of Management and Policy, College of Business and Public Administration, 
University of Arizona, Tucson, AZ 85721 



Abstract. We report the results of two experiments designed to test competitively 
the induction against the deterrence theory in Selten's (1978) chain store game with 
complete and perfect information. Our major purpose is to determine the number of 
entrants (m) needed for rendering the deterrence argument effective. Our results show 
that if we increase m from 10 to 15, support for the deterrence theory increases 
significantly. But even with m=15, the aggressive behavior of the incumbent does not 
deter most entrants. We then compare these results with previous studies of the chain 
store game allowing for incomplete information, several choices by the same entrant, 
and multiple iterations of the game. 

Keywords, game theory, experimental economics, chain store paradox, induction, 
deterrence, sequential equilibrium, reputation 



1 Introduction 

There is no agreement in the economic literature whether preemptive actions taken 
by an incumbent firm in order to maintain its monopoly in the face of threatened 
entry can be maintained as a viable threat. There are those who argue that predatory 
pricing is an irrational strategy for attempting to maintain a monopoly position and 
that it is, therefore, unlikely to be adopted in practice (McGee, 1958, 1980; Carlton 
& Perloff, 1990), and there are others who maintain the opposite position. As noted 
by Trockel (1986), the crucial point in this debate is the credibility of the threat. It is, 
therefore, important to determine analytically whether effective threatening of 
potential entrants by a monopolist is rational. 

The first and most influential analysis of this problem is Selten’s formal model 
known as the Chain Store game (1978). The model is presented as a noncooperative 
(m+ 1 )-person game with a single monopolist (player M) and m potential entrants 
(players 1,2,..., m). The game is played under complete information over a sequence 
of m consecutive periods (stage games) 1, ... , m. At the beginning of period k, player 
k (k=l, 2, .. , m) must decide between entering the kth market (IN) or staying out 
(OUT). Once she has made her decision, all other players are fully informed of her 




404 



choice. No further decisions are called for in period k, if player k's decision is OUT. 
If her decision is IN, then player M has to choose between two alternative courses of 
action called "cooperative" (CO) and "aggressive" (AG). (Here, "cooperative" is 
better described as peaceful reaction or acquiescence, rather than proper cooperation, 
and "aggressive" is described as an economic fight). Once player M has made his 
choice, all other players are immediately informed of his decision, too. Then, for k=l, 
2, ..., m-1, the (k+l)th period starts and is played according to the same rules. The 
game ends after m periods. 

It is assumed that all m+1 players are profit maximizers. For the monopolist this 
means maximization of his profit across all m markets, and for each of the entrants 
maximization of her profit in the stage game. Players cannot commit themselves to 
threats or promises, and side payments are not allowed. The payoffs for the stage 
game are common knowledge; they are displayed in Fig. 1 in the order (M, k). 
Although the specific choice of numbers in Fig. 1 is consistent with what are believed 
to be the relative sizes of monopoly and duopoly profits, in fact for each player only 
the ordinal ranking of the outcomes, independent of the opponent's outcomes, is 
important. 



Figure 1. The Chain Store game in extensive and normal form. 



(2,2) (0,0) (1,5) 




Player k 
IN OUT 



2,2 


5, 1 


0,0 


5, 1 



The resulting game is, therefore, an m-fold repetition of a two-person game in 
extensive form with complete and perfect information. Selten proposed two theories 
for what he calls an "adequate behavior in the game" (1978, p. 130). The first, called 




405 



the induction theory, invokes the standard backward induction argument to show that 
each of the entrants 1, 2, , m should always choose IN and the monopolist should 

always respond with CO. This solution is supported as perfect equilibrium (Selten, 
1975) and sequential equilibrium (Kreps & Wilson, 1982a) as well as by repeated 
elimination of dominated strategies. In particular, the induction theory excludes 
reputation building by player M and keeps the m markets independent. Player M 
cannot convince any potential entrant that he is a predator, because they know that 
he is not. Critical to this theory is the assumption of common knowledge of 
rationality (Aumann, 1976). 

The alternative theory proposed by Selten, called the deterrence theory, reasons 
that when the stage game is repeated in time, player M has a monetary incentive to 
respond aggressively to entry by early entrants. He may establish this reputation for 
toughness by aggressively responding to entry in early rounds, thereby deterring later 
players k from entering. Given the payoffs for the stage game in Fig. 1 and assuming 
m=20, player M needs only deter 50 percent of the potential entrants to realize a 
higher payoff than the always CO choice proposed by the induction theory (assuming 
cooperative behavior in the last three periods). Specifically, Selten suggests that 
player M should use his intuition for how many of the last periods of the game he 
wishes to adhere to the induction argument. Supposing that m=20, and he wishes to 
accept this argument for periods 18-20 but not 1-17. Then in periods 1-17 he should 
respond AG to IN, and in periods 18-20 he should respond CO. Anticipating this kind 
of behavior, players 1, 2, .. , m should behave accordingly. If up to period k-1 not 
very many of the players 1, 2, ... , k-1 chose IN and player M's response was always 
AG, then player k should play OUT unless she believes that period k is sufficiently 
near the end of the game to make it probable that player M will accept the induction 
argument for periods k, k+1, .. , m. Thus, the deterrence theory gives rise to 
equilibrium behavior, only it is not sustained as subgame perfect. 

Selten concedes that "the deterrence theory does not yield precise rules of 
behavior, since some details are left to the intuition of the players" (1978, p. 132). 
This imprecision, argues Selten, should not detract from its plausibility and 
practicality; rather, he finds the deterrence theory considerably more convincing. "My 
experience suggests" he writes, "that mathematically trained persons recognize the 
logical validity of the induction argument, but they refuse to accept it as a guide to 
practical behavior" (1978, p. 133). The failure of backward induction in two-person 
ultimatum bargaining games with a relatively short horizon (see Roth, in press, for 
an extensive review) is in strong support with Selten's conjecture. 

There have been several attempts to account for reputation building and 
predatory pricing by modifying some aspects of the chain store game (Davis, 1985; 
Kreps & Wilson, 1982b; Milgrom & Roberts, 1982; Trockel, 1986). The starting 
point for the former three papers is the insight that relaxation of the complete 
information assumption destroys the subgame structure, thereby leading to a 
breakdown of the backward induction argument. Kreps and Wilson (1982b) and 
Milgrom and Roberts (1982) altered the game by assuming a small amount of doubt 
on the side of the entrants about whether the perceived model of the game is correct. 
They showed that there exist sequential equilibria for these slightly altered versions 
of the game prescribing predation in the early stages for player M and, consequently, 
staying out of the market for most entrants. Trockel (1986) provided yet another 




406 



resolution of the paradox, which renders possible reputation building and deterrence, 
by maintaining the assumption of complete but not perfect information. For 
experimental evidence supporting predatory pricing see, e.g., Camerer and Weigelt 
(1988) and Jung, Kagel, and Levin (1994). 

Our study departs from these previous experiments by having a closer match 
between theory and experimental design. In particular, we do not introduce various 
types of monopolists, as do Camerer and Weigelt and Jung et al., nor do we allow 
entrants to respond more than once, as do Jung et al. in their sessions without 
experimenter-induced strong monopolist. Rather, our major focus in this paper is on 
establishing the necessary conditions for entry deterrence in chain store like games 
with complete and perfect information. Selten stated that "it will be useful to assume 
that there are m potential competitors, where m may be any positive integer. 
Nevertheless, it is convenient to focus attention on m=20, since the game changes its 
character if m becomes too small" (1978, p.l28). We contend that this is not a matter 
of convenience but of substance. While willing to accept Selten's claim for the 
deterrence argument when m is relatively high (say, m>20), our major purpose in this 
study is to investigate the gradual weakening of the force of this argument as m gets 
smaller. In essence, we ask: how many potential entrants are needed for the 
deterrence argument to become effective? This question is of some importance if, as 
Selten claims, the dete