Classics in Game Theory 9781400829156

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Classics in Game Theory

F R O N T I E R S

O F

E C O N O M I C

R E S E A R C H

Series Editors

David M. Kreps

Thomas J. Sargent

Classics in Game Theory Edited by Harold W. Kuhn

P R I N C E T O N

U N I V E R S I T Y

P R I N C E T O N ,

N E W

PRESS

J E R S E Y

Copyright © 1997 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, Chichester, West Sussex All Rights Reserved Library of Congress Cataloging-in-Publication Data

Classics in game theory / edited by Harold W. Kuhn p. cm. — Frontiers of Economic Research Includes bibliographical references and index. ISBN 0-691-01193-1 (cloth: alk. paper)—ISBN 0-691-01192-3 (pbk.: alk. paper) 1. Game theory. 2. Mathematical economics. I. Kuhn, Harold W. (Harold William), 1925- . II. Series. HB144.C53 1997 519.3—dc20 96-20693 This book has been composed in 10/13 Times Roman Princeton University Press books are printed on acid-free paper and meet the guidelines for permanence and durability of the Committee on Production Guidelines for Book Longevity of the Council on Library Resources Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 10 9 8 7 6 5 4 3 2 1 (Pbk.)

CONTENTS

Permissions

vii

H. W. KUHN

Foreword

ix

DAVID KREPS AND ARIEL RUBINSTEIN

An Appreciation

xi

1. JOHN F. NASH, JR.

Equilibrium Points in n-Person Games. PNAS 36 (1950) 48-49.

3

2. JOHN F. NASH, JR.

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

5

3. JOHN NASH

Non-Cooperative Games. Annals of Mathematics 54 (1951) 286-295.

14

4. JULIA ROBINSON

An Iterative Method of Solving a Game. Annals of Mathematics 54 (1951) 296-301.

27

5. F. B. THOMPSON

Equivalence of Games in Extensive Form. RAND Memo RM-759 (1952).

36

6. H. W. KUHN

Extensive Games and the Problem of Information. Contributions to the Theory of Games II (1953) 193-216.

46

7. L. S. SHAPLEY

A Value for n-Person Games. Contributions to the Theory of Games II (1953) 307-317.

69

8. L. S. SHAPLEY

Stochastic Games. PNAS 39 (1953) 1095-1100.

80

VI

CONTENTS

9. H. EVERETT

Recursive Games. Contributions to the Theory of Games III (1957) 47-78.

87

10. R. J. AUMANN AND B. PELEG

Von Neumann-Morgenstern Solutions to Cooperative Games without Side Payments. Bulletin AMS 66 (1960) 173-179. 119 11. GERARD DEBREU AND HERBERT SCARF

A Limit Theorem on the Core of an Economy. International Economic Review 4 (1963) 235-246.

127

12. ROBERT J. AUMANN AND MICHAEL MASCHLER

The Bargaining Set for Cooperative Games. Advances in Game Theory (1964) 443-477.

140

13. ROBERT J. AUMANN

Existence of Competitive Equilibria in Markets with a Continuum of Traders. Econometrica 34 (1966) 1-17.

170

14. HERBERT E. SCARF

The Core of an ^-Person Game. Econometrica 35 (1967) 50-69.

192

15. JOHN C. HARSANYI

Games with Incomplete Information Played by "Bayesian" Players. Part I: The Basic Model. Man. Sci. 14 (1967) 159-182 Part II: Bayesian Equilibrium Points. Man. Sci. 14 (1968) 320-334 Part III: The Basic Probability Distribution of the Game. Man. Sci. 14 (1968) 486-502.

216 247 268

16. DAVID BLACKWELL AND T. S. FERGUSON

The Big Match. Ann. Math. Stat. 39 (1968) 159-163.

289

17. LLOYD S. SHAPLEY AND MARTIN SHUBIK

On Market Games. IE. T. 1 (1969) 9-25.

296

18. R. SELTEN

Reexamination of the Perfectness Concept for Equilibrium Points in Extensive Games. Int. J. Game Th. 4 (1975) 25-55.

317

List of Contributors

355

Index

357

PERMISSIONS

1. JOHN F. NASH, JR.

Equilibrium Points in n-Person Games. Reprinted from PNAS 36 (1950) 48-49, by permission of the author. 2. JOHN F. NASH, JR.

The Bargaining Problem. Reprinted from Econometrica 18 (1950) 155-162, by permission of The Econometric Society. 3. JOHN NASH

Non-Cooperative Games. Reprinted from Annals of Mathematics Journal 54 (1951) 286-295, by permission of the Annals of Mathematics Journal. 4. JULIA ROBINSON

An Iterative Method of Solving a Game. Reprinted from Annals of Mathematics Journal 54 (1951) 296-301, by permission of the Annals of Mathematics Journal. 5. F. B. THOMPSON

Equivalence of Games in Extensive Form. January 1952, RM-759, 12pp. Used by Permission of RAND. 6. H. W. KUHN

Extensive Games and the Problem of Information. Reprinted from Contributions to the Theory of Games II (1953) 193-216, by permission of Princeton University Press. 7. L. S. SHAPLEY

A Value for n-Person Games. Reprinted from Contributions to the Theory of Games II (1953) 307-317, by permission of Princeton University Press. 8. L. S. SHAPLEY

Stochastic Games. PNAS 39 (1953) 1095-1100, by permission of PNAS and L. S. Shapley. 9. H. EVERETT

Recursive Games. Reprinted from Contributions to the Theory of Games III (1957) 47-78, by permission of Princeton University Press. 10. R. J. AUMANN AND B. PELEG Von Neumann-Morgenstern Solutions to Cooperative Games with-

viii

PERMISSIONS

out Side Payments. Reprinted from Bulletin of the American Mathematical Society 66 (1960) 173-179, by permission of the American Mathematical Society. 11. GERARD DEBREU AND HERBERT E. SCARF

A Limit Theorem on the Core of an Economy. Reprinted from The International Economic Review 4 (1963) 235-246, by permission of the International Economic Review. 12. ROBERT J. AUMANN AND MICHAEL MASCHLER

The Bargaining Set for Cooperative Games. Reprinted from Advances in Game Theory (1964) 443-477, by permission of Princeton University Press. 13. ROBERT J. AUMANN

Existence of Competitive Equilibria in Markets with a Continuum of Traders. Reprinted from Econometrica 34 (1966) 1-17, by permission of The Econometric Society. 14. HERBERT E. SCARF

The Core of an n-Person Game. Reprinted from Econometrica 35 (1967) 50-69, by permission of The Econometric Society. 15. Reprinted by permission, JOHN C. HARSANYI, Games with Incomplete Information Played by "Bayesian" Players. Part I: The Basic Model, Management Science 14 (1967) 159-182; Part II: Bayesian Equilibrium Points, Management Science 14 (1968) 320-334; Part III: The Basic Probability Distribution of the Game, Management Science 14 (1968) 486-502, The Institute of Management Sciences (currently INFORMS), 2 Charles Street, Suite 300, Providence, RI. 16. DAVID BLACKWELL AND T. S. FERGUSON

The Big Match. Reprinted from Annals Mathematical Statistics 39 (1968) 159-163, by permission of the Institute for Mathematical Statistics. 17. LLOYD S. SHAPLEY AND MARTIN SHUBIK

On Market Games. Reprinted from the Journal of Economic Theory 1 (1969) 9-25, by permission of Academic Press, Inc., P.O. Box 860630, Orlando, FL 32886-0630 18. R. SELTEN

Reexamination of the Perfectness Concept for Equilibrium Points in Extensive Games. International Journal of Game Theory 4 (1975) 25-55. Reprinted by permission of Physica-Verlag, GmbH & Co. KG, TiergartenstraBe 17, D-69121 Heidelberg, Germany.

FOREWORD

In 1988 several colleagues in the Economics Department proposed that I select a group of papers in game theory that would be published as a set of readings to supplement the small group of textbooks in the subject for the burgeoning courses in both mathematics and economics departments. The original proposal was that the papers to be included would be drawn from the Annals of Mathematics Studies devoted to game theory (Contributions to the Theory of Games, Vols. I-IV, and Advances in Game Theory). However, the more thought that I gave to the project, the more I believed that this would be unduly restrictive, and that papers should be sought from other sources. Accordingly, I asked the advice of a number of friends and colleagues whose judgment I trust, enclosing a tentative list of papers to be included. There were some differences of opinion, but nevertheless a great deal of agreement about what should be considered the "classics in game theory." Although no one of them bears responsibility for the final list, I am happy to thank the following for their advice and comments: Ken Arrow, Paolo Caravani, V. P. Crawford, Gerard Debreu, Avinash Dixit, Sergiu Hart, Ehud Kalai, Roger Meyerson, Herve Moulin, Guillermo Owen, John Roberts, Herbert Scarf, David Schmeidler, Martin Shubik, William Thompson, Robert Willig, Robert Wilson, and Peyton Young. The title "Classics in Game Theory" will suggest different things to different people, but the heart of this volume is the basic building blocks on which the current edifice of game theory is built. With a confirmed procrastinator (myself) in charge of the project, the final list of papers was chosen by 1990, but the most urgent efforts of Jack Repcheck, the economics editor at Princeton University Press, could not shake it loose from me. At one time I intended to prepare an introductory essay that would incorporate some "prehistoric" excerpts (say, by Montmort, Zermelo, and von Neumann). The essay would have also given me the opportunity to give the volume some historical perspective and, incidentally, to explain some of the criteria used in the selection of the papers. However, that was not to be.

FOREWORD

In 1988 several colleagues in the Economics Department proposed that I select a group of papers in game theory that would be published as a set of readings to supplement the small group of textbooks in the subject for the burgeoning courses in both mathematics and economics departments. The original proposal was that the papers to be included would be drawn from the Annals of Mathematics Studies devoted to game theory (Contributions to the Theory of Games, Vols. I-IV, and Advances in Game Theory). However, the more thought that I gave to the project, the more I believed that this would be unduly restrictive, and that papers should be sought from other sources. Accordingly, I asked the advice of a number of friends and colleagues whose judgment I trust, enclosing a tentative list of papers to be included. There were some differences of opinion, but nevertheless a great deal of agreement about what should be considered the "classics in game theory." Although no one of them bears responsibility for the final list, I am happy to thank the following for their advice and comments: Ken Arrow, Paolo Caravani, V. P. Crawford, Gerard Debreu, Avinash Dixit, Sergiu Hart, Ehud Kalai, Roger Meyerson, Herve Moulin, Guillermo Owen, John Roberts, Herbert Scarf, David Schmeidler, Martin Shubik, William Thompson, Robert Willig, Robert Wilson, and Peyton Young. The title "Classics in Game Theory" will suggest different things to different people, but the heart of this volume is the basic building blocks on which the current edifice of game theory is built. With a confirmed procrastinator (myself) in charge of the project, the final list of papers was chosen by 1990, but the most urgent efforts of Jack Repcheck, the economics editor at Princeton University Press, could not shake it loose from me. At one time I intended to prepare an introductory essay that would incorporate some "prehistoric" excerpts (say, by Montmort, Zermelo, and von Neumann). The essay would have also given me the opportunity to give the volume some historical perspective and, incidentally, to explain some of the criteria used in the selection of the papers. However, that was not to be.

X

FOREWORD

The project assumed a new urgency upon the announcement of the 1994 Nobel Memorial Prize in Economic Science, which recognized the central importance of the theory of games for economic theory by honoring John Nash, John Harsanyi, and Reinhard Selten. I am pleased to say that the key works for which they were honored were five of the final list of eighteen papers that was ready but not acted upon in 1990. At this point Peter Dougherty, publisher in social science and public affairs at Princeton University Press, entered the fray. After discussions with David Kreps and Ariel Rubinstein, who were active supporters of the collection, they kindly agreed to relieve me of the responsibility of writing the introductory essay. They have kindly written the "appreciation" that follows this foreword. To all of the people who have kept my procrastination from preventing this collection's reaching the light of day, I hereby express my sincerest thanks. I can only hope that their patience has been worth the effort. Harold W. Kuhn Princeton, New Jersey

AN APPRECIATION

The Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel, 1994, was awarded jointly to John C. Harsanyi, John F. Nash, and Reinhard Selten, "for their pioneering analysis of equilibria in the theory of noncooperative games." In so doing, the Royal Swedish Academy of Sciences took note of a revolutionary change in the language and style of analysis of economics and economists, in which game-theoretic ideas have become commonplace. Game theory has provided economists with a flexible language for discussing many issues central to economic inquiry, from two-person bargaining, to highly personal, repeated, and long-run exchange, to the theoretical foundations of economic models of monopoly and perfect competition. Most fields in economics and certainly economic theory have been dramatically affected by these ideas. But game theory is not only a subfield of economics. It analyzes abstractly conflicts of interest. Thus game theory has expanded far beyond economics. We find a growing tendency to use game-theoretic concepts and models in a variety of fields: Political scientists use game theory to examine political institutions. Philosophers find game theory a tool for reexamination of norms and social institutions. Biologists find game theory a framework to analyze the conflicting interests between creatures in nature. The history of game theory in economics may be told as follows. Basic game-theoretic concepts of equilibrium, in the context of competition with a small number of participants, were developed all but formally by Cournot, von Stackelberg, and Bertrand. Mathematical models of games of strategy had been broached by the French mathematician Borel, the Polish mathematician Steinhaus, and the German mathematician Zermelo (for the special case of chess). The first attempt at a general theory with a solution concept for zero-sum two-person games and the associated minimax theorem was published by von Neumann in 1928. However, formal game theory reached a larger audience with the publication of von Neumann and Morgenstern's treatise Theory of

AN APPRECIATION

The Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel, 1994, was awarded jointly to John C. Harsanyi, John F. Nash, and Reinhard Selten, "for their pioneering analysis of equilibria in the theory of noncooperative games." In so doing, the Royal Swedish Academy of Sciences took note of a revolutionary change in the language and style of analysis of economics and economists, in which game-theoretic ideas have become commonplace. Game theory has provided economists with a flexible language for discussing many issues central to economic inquiry, from two-person bargaining, to highly personal, repeated, and long-run exchange, to the theoretical foundations of economic models of monopoly and perfect competition. Most fields in economics and certainly economic theory have been dramatically affected by these ideas. But game theory is not only a subfield of economics. It analyzes abstractly conflicts of interest. Thus game theory has expanded far beyond economics. We find a growing tendency to use game-theoretic concepts and models in a variety of fields: Political scientists use game theory to examine political institutions. Philosophers find game theory a tool for reexamination of norms and social institutions. Biologists find game theory a framework to analyze the conflicting interests between creatures in nature. The history of game theory in economics may be told as follows. Basic game-theoretic concepts of equilibrium, in the context of competition with a small number of participants, were developed all but formally by Cournot, von Stackelberg, and Bertrand. Mathematical models of games of strategy had been broached by the French mathematician Borel, the Polish mathematician Steinhaus, and the German mathematician Zermelo (for the special case of chess). The first attempt at a general theory with a solution concept for zero-sum two-person games and the associated minimax theorem was published by von Neumann in 1928. However, formal game theory reached a larger audience with the publication of von Neumann and Morgenstern's treatise Theory of

xii

AN APPRECIATION

Games and Economic Behavior, first published in 1944. Following the Second World War (and in some instances evolving out of work done during the war), the subject underwent explosive development in the 1950s and 1960s. Concepts which today form the heart of game theory as it is applied to economics and other disciplines—Nash equilibrium, the theory of extensive form games, axiomatic bargaining theory, the Shapley value, the core and its connections to competitive equilibrium, most other cooperative game-theory solution concepts, the folk theorem, games of incomplete information, and basic notions of perfection —all were created during this heroic period. A few fundamental developments came a bit later, most notably Selten's further development of the idea of perfection and foundational issues of common knowledge. But by the end of the 1960s, the tools were essentially in place. The revolution in economics occurred quite shortly thereafter. Beginning in the 1970s, and especially in the context of Industrial Organization, the language and techniques of game theory moved from being an esoteric and limited tool of microeconomic theorists to become part of the mainstream language of the discipline. In this volume, Prof. Harold Kuhn has collected eighteen papers from the heroic era of game theory. Included here you will find the following: Nash equilibrium. If any concept has achieved primacy in game theory, it is Nash's equilibrium concept regarding the model of a game in strategic form. Chapters 1 and 3 contain the first formal statements of this concept in the literature. Evolution of equilibrium. In the late 1940s, the RAND Corporation mathematician George Brown proposed an adaptive behavior algorithm (called "fictitious play") for solving zero-sum two-person games. It was not known whether the method converged (a money prize was offered by the RAND Corporation to settle this question) until the publication of Julia Robinson's paper (Chapter 3). With convergence established by her elegant arguments, the model of fictitious play has become a cornerstone of recent work on evolutionary and adaptive learning models. Extensive form games and perfect recall. Whereas the strategic form of a game lacks a dynamic structure, the extensive form allows the analysis of dynamic considerations. Thompson's paper (Chapter 5) proposes a set of transformations that connects strategically equivalent extensive form games together and to their equivalent strategic form counterpart.

AN APPRECIATION

xiii

His paper uses Kuhn's model for extensive form games (Chapter 6) which has become the standard model. Kuhn analyzes games with and without perfect information, and introduces the concepts of subgame and perfect recall. This paper contains fundamental insights on the role of perfect recall in answering the question of when games can be solved with "behavioral" modes of play as effectively as by mixed strategies. Games with incomplete information. How can game-theoretic models be used to analyze competitive situations in which some parties have information that others lack, about their own capabilities or tastes, or about the underlying state of nature? In Chapter 15, John Harsanyi provides the standard answer to this question. After Nash equilibrium, Harsanyi's definition of games with incomplete information is perhaps the single most important innovation from the point of view of modern economic applications. Perfect equilibrium. In 1965, Selten first formulated the notion of subgame perfection, a criterion for studying whether a Nash equilibrium of an extensive form game is based on credible threats/strategies. Subgame perfection had a somewhat limited reach, and in his 1975 paper, reprinted here as Chapter 18, Selten reformulated subgame perfection, giving a notion of perfection that applies to all extensive form games with perfect recall. The concept itself is a crucial tool for studying dynamic competitive interactions but, as importantly or more so, it is a cornerstone of Selten's seminal program for studying competitive dynamics as something more than simple static and simultaneous choice of strategies. Repeated games. Several works in this volume are the pioneering work regarding the analysis of the model of dynamic zero-sum games: Shapley (Chapter 8), Everett (Chapter 9), and Blackwell-Ferguson (Chapter 11) give seminal analyses of different varieties of these dynamic games, suggesting some of the most innovative mathematical techniques used in game theory. Games and markets. Game theory has been used to study the foundations of economic (price-mediated) equilibrium in markets, giving insights into the sources of market power and the implications of "competitive" agents for the existence and character of market equilibrium. Two of the seminal works in this line of thought are provided here: Aumann's paper on the existence of competitive equilibrium with (truly)

xiv

AN APPRECIATION

competitive agents (Chapter 13), and Shapley and Shubik's model and analysis of market games (Chapter 17). Cooperative game theory. The Nobel Prize for 1994 was granted for achievements in noncooperative game theory. Cooperative game theory is the other "half of the subject. A cooperative game specifies the details about what a coalition (and not just a single player) can achieve without the agreement of the other players. This part of game theory may have made less of an impact on the broad community of economists so far, but we believe that its impact will be felt in the future. In Chapter 10, Aumann and Peleg analyze the von Neumann-Morgenstern Solution for games without side payments; in Chapter 12, Aumann and Maschler define the Bargaining Set. The core. The central solution concept in cooperative game theory, and especially in cooperative game theory as it has been applied to economics, is the core. The core is especially important for its connection to competitive equilibria in "large" economies. In Chapter 14, Scarf discusses fundamental issues connected with the core; in Chapter 11, Debreu and Scarf formalize the intuition of Edgeworth, that in the competitive equilibrium of a large economy, every agent gets just what they "contribute" to the society. Nash bargaining solution. In "small" economies, where bargaining can take place, game theory has developed axioms for saying what is a fair and/or reasonable outcome. In the context of two-person bargaining, Nash (Chapter 2) is the seminal reference. This paper sets up the general problem and then solves it axiomatically, paving the way for the huge literature of axiomatic bargaining theory that followed. It is remarkable in that it was written originally as a paper for an undergraduate course before Nash was familiar with the work of von Neumann and Morgenstern. Shapley value. Along with the Nash Bargaining Solution, the Shapley Value, denned and axiomatized in Chapter 7, is the premier axiomatic standard of "equity" in game theory, the subject of intensive reaxiomatization and (more importantly) of enormous use in applications from cost allocation to recent work in corporate finance. These eighteen papers include virtually all the foundation stones of game theory. They have also established the formal style of game

AN APPRECIATION

XV

theory. The emphasis on clear definition and well-knit proofs sets the tone for the further development of game theory. Over the past five years or so, a number of excellent advanced textbooks in game theory have appeared. With the benefit of hindsight, they cover many of these foundation stones without some of the awkwardness that can accompany original efforts. But to our mind, this collection of papers stands out for their clarity and vision. To see these ideas as they were first expressed is a pleasure for us, the beneficiaries of the heroic era of game theory, and we are pleased to sit back with you and enjoy again these classics. David Kreps and Ariel Rubinstein

Classics in Game Theory

EQUILIBRIUM POINTS IN n-PERSON GAMES JOHN F. NASH, JR.* PRINCETON UNIVERSITY

Communicated by S. Lefschetz, November 16,1949

ONE MAY define a concept of an n-person game in which each player has a finite set of pure strategies and in which a definite set of payments to the n players corresponds to each n-tuple of pure strategies, one strategy being taken for each player. For mixed strategies, which are probability distributions over the pure strategies, the pay-off functions are the expectations of the players, thus becoming polylinear forms in the probabilities with which the various players play their various pure strategies. Any n -tuple of strategies, one for each player, may be regarded as a point in the product space obtained by multiplying the n strategy spaces of the players. One such n-tuple counters another if the strategy of each player in the countering n-tuple yields the highest obtainable expectation for its player against the n — 1 strategies of the other players in the countered n-tuple. A self-countering n-tuple is called an equilibrium point. The correspondence of each n-tuple with its set of countering ntuples gives a one-to-many mapping of the product space into itself. From the definition of countering we see that the set of countering points of a point is convex. By using the continuity of the pay-off functions we see that the graph of the mapping is closed. The closedness is equivalent to saying: if P1,P2,... and QvQ2,...,Qn,... are sequences of points in the product space where Qn -» Q, Pn -> P and Qn counters Pn then Q counters P. Since the graph is closed and since the image of each point under the mapping is convex, we infer from Kakutani's theorem 1 that the mapping has a fixed point (i.e., point contained in its image). Hence there is an equilibrium point. *The author is indebted to Dr. David Gale for suggesting the use of Kakutani's theorem to simplify the proof and to the A. E. C. for financial support. 'Kakutani, S., Duke Math. /., 8, 457-459 (1941).

NASH

In the two-person zero-sum case the "main theorem"2 and the existence of an equilibrium point are equivalent. In this case any two equilibrium points lead to the same expectations for the players, but this need not occur in general.

Von Neumann, J., and Morgenstern, O., The Theory of Games and Economic Behaviour, Chap. 3, Princeton University Press, Princeton, 1947.

THE BARGAINING PROBLEM1 JOHN F. NASH, JR.

A new treatment is presented of a classical economic problem, one which occurs in many forms, as bargaining, bilateral monopoly, etc. It may also be regarded as a nonzero-sum two-person game. In this treatment a few general assumptions are made concerning the behavior of a single individual and a group of two individuals in certain economic environments. From these, the solution (in the sense of this paper) of the classical problem may be obtained. In the terms of game theory, values are found for the game.

INTRODUCTION

A two-person bargaining situation involves two individuals who have the opportunity to collaborate for mutual benefit in more than one way. In the simpler case, which is the one considered in this paper, no action taken by one of the individuals without the consent of the other can affect the well-being of the other one. The economic situations of monopoly versus monopsony, of state trading between two nations, and of negotiation between employer and labor union may be regarded as bargaining problems. It is the purpose of this paper to give a theoretical discussion of this problem and to obtain a definite "solution"—making, of course, certain idealizations in order to do so. A "solution" here means a determination of the amount of satisfaction each individual should expect to get from the situation, or, rather, a determination of how much it should be worth to each of these individuals to have this opportunity to bargain. This is the classical problem of exchange and, more specifically, of bilateral monopoly as treated by Cournot, Bowley, Tintner, Fellner, and others. A different approach is suggested by von Neumann and Morgenstern in Theory of Games and Economic Behavior2 which permits the lr The author wishes to acknowledge the assistance of Professors von Neumann and Morgenstern who read the original form of the paper and gave helpful advice as to the presentation. 2 John von Neumann and Oskar Morgenstern, Theory of Games and Economic Behavior, Princeton: Princeton University Press, 1944 (Second Edition, 1947), pp. 15-31.

6

NASH

identification of this typical exchange situation with a nonzero-sum two-person game. In general terms, we idealize the bargaining problem by assuming that the two individuals are highly rational, that each can accurately compare his desires for various things, that they are equal in bargaining skill, and that each has full knowledge of the tastes and preferences of the other. In order to give a theoretical treatment of bargaining situations we abstract from the situation to form a mathematical model in terms of which to develop the theory. In making our treatment of bargaining we employ a numerical utility, of the type developed in Theory of Games, to express the preferences, or tastes, of each individual engaged in bargaining. By this means we bring into the mathematical model the desire of each individual to maximize his gain in bargaining. We shall briefly review this theory in the terminology used in this paper. UTILITY THEORY OF THE INDIVIDUAL

The concept of an "anticipation" is important in this theory. This concept will be explained partly by illustration. Suppose Mr. Smith knows he will be given a new Buick tomorrow. We may say that he has a Buick anticipation. Similarly, he might have a Cadillac anticipation. If he knew that tomorrow a coin would be tossed to decide whether he would get a Buick or a Cadillac, we should say that he had a \ Buick, \ Cadillac anticipation. Thus an anticipation of an individual is a state of expectation which may involve the certainty of some contingencies and various probabilities of other contingencies. As another example, Mr. Smith might know that he will get a Buick tomorrow and think that he has half a chance of getting a Cadillac too. The \ Buick, \ Cadillac anticipation mentioned above illustrates the following important property of anticipations: if 0 < p < 1 and A and B represent two anticipations, there is an anticipation, which we represent by pA + (1 - p)B, which is a probability combination of the two anticipations where there is a probability p of A and 1 - p of B. By making the following assumptions we are enabled to develop the utility theory of a single individual: 1. An individual offered two possible anticipations can decide which is preferable or that they are equally desirable.

THE BARGAINING PROBLEM

7

2. The ordering thus produced is transitive; if A is better than B and B is better than C then A is better than C. 3. Any probability combination of equally desirable states is just as desirable as either. 4. If A, B, and C are as in assumption (2), then there is a probability combination of A and C which is just as desirable as C. This amounts to an assumption of continuity. 5. If 0 < p < 1 and A and 5 are equally desirable, then pA + (1 - p)C and /?£ + (1 — p)C are equally desirable. Also, if A and -B are equally desirable, A may be substituted for B in any desirability ordering relationship satisfied by B. These assumptions suffice to show the existence of a satisfactory utility function, assigning a real number to each anticipation of an individual. This utility function is not unique, that is, if u is such a function then so also is au + b, provided a > 0. Letting capital letters represent anticipations and small ones real numbers, such a utility function will satisfy the following properties: (a) u(A) > u(B) is equivalent to A is more desirable than B, etc. (b) If 0