A Course in Game Theory

Page iii A Course in Game Theory Martin J. Osborne Ariel Rubinstein The MIT Press Cambridge, Massachusetts London, England Page iv Copyright © 1994 Massachusetts Institute of Technology All rights reserved. No part of this book may be reproduced in any form by any electronic. or mechanical means (including photocopying, recording, or information storage and retrieval) without permission in writing from the publisher. This book was typeset by the authors, who are greatly indebted to Donald Knuth (the creator of T E X), Leslie Lamport (the creator of LAT E X), and Eberhard Mattes (the creator of emT E X) for generously putting superlative software in the public domain. Camera-ready copy was produced by Type 2000, Mill Valley, California, and the book was printed and bound by The Maple-Vail Book Manufacturing Group, Binghamton, New York. Osborne, Martin J. A course in game theory I Martin J. Osborne, Ariel Rubinstein. p. cm. Includes bibliographical references and index. ISBN 0-262-15041-7.—ISBN 0-262-65040-1 (pbk.) 1. Game theory. I. Rubinstein, Ariel. II. Title. HB 144.0733 1994 658.4'0353-dc20 94-8308 CIP Fifth printing, 1998 Page v CONTENTS Preface xi 1 Introduction 1 1.1 Game Theory 1 1.2 Games and Solutions 2 1.3 Game Theory and the Theory of Competitive Equilibrium 3 1.4 Rational Behavior 4 1.5 The Steady State and Deductive Interpretations 5 1.6 Bounded Rationality 6 1.7 Terminology and Notation 6 Notes 8 I Strategic Games 9 2 Nash Equilibrium 11 2.1 Strategic Games 11 2.2 Nash Equilibrium 14 2.3 Examples 15 2.4 Existence of a Nash Equilibrium 19 2.5 Strictly Competitive Games 21 2.6 Bayesian Games: Strategic Games with Imperfect Information 24 Notes 29 Page vi 3 Mixed, Correlated, and Evolutionary Equilibrium 31 3.1 Mixed Strategy Nash Equilibrium 31 3.2 Interpretations of Mixed Strategy Nash Equilibrium 37 3.3 Correlated Equilibrium 44 3.4 Evolutionary Equilibrium 48 Notes 51 4 Rationalizability and Iterated Elimination of Dominated Actions 53 4.1 Rationalizability 53 4.2 Iterated Elimination of Strictly Dominated Actions 58 4.3 Iterated Elimination of Weakly Dominated Actions 62 Notes 64 5 Knowledge and Equilibrium 67 5.1 A Model of Knowledge 67 5.2 Common Knowledge 73 5.3 Can People Agree to Disagree? 75 5.4 Knowledge and Solution Concepts 76 5.5 The Electronic Mail Game 81 Notes 84 II Extensive Games with Perfect Information 87 6 Extensive Games with Perfect Information 89 6.1 Extensive Games with Perfect Information 89 6.2 Subgame Perfect Equilibrium 97 6.3 Two Extensions of the Definition of a Game 101 6.4 The Interpretation of a Strategy 103 6.5 Two Notable Finite Horizon Games 105 6.6 Iterated Elimination of Weakly Dominated Strategies 108 Notes 114 7 Bargaining Games 117 7.1 Bargaining and Game Theory 117 7.2 A Bargaining Game of Alternating Offers 118 7.3 Subgame Perfect Equilibrium 121 7.4 Variations and Extensions 127 Notes 131 Page vii 8 Repeated Games 133 8.1 The Basic Idea 133 8.2 Infinitely Repeated Games vs. Finitely Repeated Games 134 8.3 Infinitely Repeated Games: Definitions 136 8.4 Strategies as Machines 140 8.5 Trigger Strategies: Nash Folk Theorems 143 8.6 Punishing for a Limited Length of Time: A Perfect Folk Theorem for the Limit of Means Criterion 146 8.7 Punishing the Punisher: A Perfect Folk Theorem for the Overtaking Criterion 149 8.8 Rewarding Players Who Punish: A Perfect Folk Theorem for the Discounting Criterion 150 8.9 The Structure of Subgame Perfect Equilibria Under the Discounting Criterion 153 8.10 Finitely Repeated Games 155 Notes 160 9 Complexity Considerations in Repeated Games 163 9.1 Introduction 163 9.2 Complexity and the Machine Game 164 9.3 The Structure of the Equilibria of a Machine Game 168 9.4 The Case of Lexicographic Preferences 172 Notes 175 10 Implementation Theory 177 10.1 Introduction 177 10.2 The Implementation Problem 178 10.3 Implementation in Dominant Strategies 180 10.4 Nash Implementation 185 10.5 Subgame Perfect Equilibrium Implementation 191 Notes 195 Page viii III Extensive Games with Imperfect Information 197 11 Extensive Games with Imperfect Information 199 11.1 Extensive Games with Imperfect Information 199 11.2 Principles for the Equivalence of Extensive Games 204 11.3 Framing Effects and the Equivalence of Extensive Games 209 11.4 Mixed and Behavioral Strategies 212 11.5 Nash Equilibrium 216 Notes 217 12 Sequential Equilibrium 219 12.1 Strategies and Beliefs 219 12.2 Sequential Equilibrium 222 12.3 Games with Observable Actions: Perfect Bayesian Equilibrium 231 12.4 Refinements of Sequential Equilibrium 243 12.5 Trembling Hand Perfect Equilibrium 246 Notes 254 IV Coalitional Games 255 13 The Core 257 13.1 Coalitional Games with Transferable Payoff 257 13.2 The Core 258 13.3 Nonemptiness of the Core 262 13.4 Markets with Transferable Payoff 263 13.5 Coalitional Games without Transferable Payoff 268 13.6 Exchange Economies 269 Notes 274 14 Stable Sets, the Bargaining Set, and the Shapley Value 277 14.1 Two Approaches 277 14.2 The Stable Sets of yon Neumann and Morgenstern 278 14.3 The Bargaining Set, Kernel, and Nucleolus 281 14.4 The Shapley Value 289 Notes 297 [...]... concepts in game theory In this chapter we describe it in the context of a strategic game and in the related context of a Bayesian game 2.1 Strategic Games 2.1.1 Definition A strategic game is a model of interactive decision-making in which each decision-maker chooses his plan of action once and for all, and these choices are made simultaneously The model consists of a finite set N of players and, for each... interaction (The model of a repeated game discussed in Chapter 8 deals with series of strategic interactions in which such intertemporal links do exist.) When referring to the actions of the players in a strategic game as "simultaneous" we do not necessarily mean that these actions are taken at the same point in time One situation that can be modeled as a strategic game is the following The players are... nomenclature we call the game "BoS" BoS models a situation in which players wish to coordinate their behavior, but have conflicting interests The game has two Nash equilibria: (Bach, Bach) and (Stravinsky, Stravinsky) That is, there are two steady states: one in which both players always choose Bach and one in which they always choose Stravinsky Page 16 Figure 16.1 Bach or Stravinsky? (BoS) (Example... Figure 16.2 A coordination game (Example 16.1) • Example 16.1 (A coordination game) As in BoS, two people wish to go out together, but in this case they agree on the more desirable concert A game that captures this situation is given in Figure 16.2 Like BoS, the game has two Noah equilibria: (Mozart, Mozart) and (Mahler, Mahler) In contrast to BoS, the players have a mutual interest in reaching one of... convex since is quasi-concave on Ai; B has a closed graph since each is continuous Thus by Kakutani's theorem B has a fixed point; as we have noted any fixed point is a Nash equilibrium of the game Note that this result asserts that a strategic game satisfying certain conditions has at least one Nash equilibrium; as we have seen, a game can have more than one equilibrium (Results that we do not discuss... which a player can base his expectation of the other players' behavior Another interpretation, which we adopt through most of this book, is that a player can form his expectation of the other players' behavior on Page 14 the basis of information about the way that the game or a similar game was played in the past (see Section 1.5) A sequence of plays of the game can be modeled by a strategic game only... the Department of Economics at the University of Canterbury, New Zealand; I am grateful to the members of the department for their generous hospitality I am grateful also to the Social Science and Humanities Research Council of Canada and the Natural Sciences and Engineering Research Council of Canada for financially supporting my research in game theory over the last six years AR I have used parts of... formal: we state definitions and results precisely, interspersing them with motivations and interpretations of the concepts The use of mathematical models creates independent mathematical interest In this book, however, we treat game theory not as a branch of mathematics but as a social science whose aim is to understand the behavior of interacting decision-makers; we do not elaborate on points of mathematical... strategic game and in Parts II and III the concept of an extensive game A strategic game is a model of a situation in which each player chooses his plan of action once and for all, and all players' decisions are made simultaneously (that is, when choosing a plan of action each player is not informed of the plan of action chosen by any other player) By contrast, the model of an extensive game specifies... symmetric: A1 = A2 and if and only if for all and Use Kakutani's Page 21 theorem to prove that there is an action such that is a Nash equilibrium of the game (Such an equilibrium is called a symmetric equilibrium.) Give an example of a finite symmetric game that has only asymmetric equilibria 2.5 Strictly Competitive Games We can say little about the set of Nash equilibria of an arbitrary strategic game; . 7 Bargaining Games 117 7.1 Bargaining and Game Theory 117 7.2 A Bargaining Game of Alternating Offers 118 7.3 Subgame Perfect Equilibrium 121 7.4 Variations. Cooperative Games In all game theoretic models the basic entity is a player. A player may be interpreted as an individual or as a group of individuals making