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 TEX), Leslie Lamport (the creator of LATEX), and Eberhard Mattes (the creator of emTEX) 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 1 Introduction 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 9 Strategic Games 2 11 Nash Equilibrium 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 31 Mixed, Correlated, and Evolutionary Equilibrium 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 53 Rationalizability and Iterated Elimination of Dominated Actions 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 67 Knowledge and Equilibrium 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 87 Extensive Games with Perfect Information 6 89 Extensive Games with Perfect Information 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 7 117 Bargaining Games 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 8 133 Repeated Games 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 146 the Limit of Means Criterion 8.7 Punishing the Punisher: A Perfect Folk Theorem for the Overtaking 149 Criterion 8.8 Rewarding Players Who Punish: A Perfect Folk Theorem for the 150 Discounting Criterion 8.9 The Structure of Subgame Perfect Equilibria Under the Discounting 153 Criterion 8.10 Finitely Repeated Games 155 Notes 160 9 163 Complexity Considerations in Repeated Games 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 10 177 Implementation Theory 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 197 Extensive Games with Imperfect Information 11 199 Extensive Games with Imperfect Information 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 12 219 Sequential Equilibrium 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 255 Coalitional Games 13 257 The Core 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 14 277 Stable Sets, the Bargaining Set, and the Shapley Value 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 Page ix 15 299 The Nash Solution 15.1 Bargaining Problems 299 15.2 The Nash Solution: Definition and Characterization 301 15.3 An Axiomatic Definition 305 15.4 The Nash Solution and the Bargaining Game of Alternating 310 Offers 15.5 An Exact Implementation of the Nash Solution 311 Notes 312 List of Results 313 References 321 Index 341 Page xi PREFACE This book presents some of the main ideas of game theory. It is designed to serve as a textbook for a one-semester graduate course consisting of about 28 meetings each of 90 minutes. The topics that we cover are those that we personally would include in such a one-semester course. We do not pretend to provide a complete reference book on game theory and do not necessarily regard the topics that we exclude as unimportant. Our selection inevitably reflects our own preferences and interests. (Were we to start writing the book now we would probably add two chapters, one on experimental game theory and one on learning and evolution.) We emphasize the foundations of the theory and the interpretation of f the main concepts. Our style is to give precise definitions and full proofs of results, sacrificing generality and limiting the scope of the material when necessary to most easily achieve these goals. We have made a serious effort to give credit for an the concepts, results, examples, and exercises (see the "Notes" at the end of each chapter). We regret any errors and encourage you to draw our attention to them. Structure of the Book The book consists of four parts; in each part we study a group of related models. The chart on the next page summarizes the interactions among the chapters. A basic course could consist of Chapters 2, 3, 6, 11, 12, and 13. Page xii The main interactions between the chapters. The areas of the boxes in which the names of the chapters appear are proportional to the lengths of the chapters. A solid arrow connecting two boxes indicates that one chapter depends on the other; a dotted arrow indicates that only the main ideas of one chapter are used in the other. A basic course could consist of the six chapters in heavy boxes. Page xiii Exercises Many of the exercises are challenging; we often use exercises to state subsidiary results. Instructors will probably want to assign additional straightforward problems and adjust (by giving hints) the level of our exercises to make it appropriate for their students. Solutions are available to instructors on the web site for the book (see page xv). Disagreements Between the Authors We see no reason why a jointly authored book should reflect a uniform view. At several points, as in the following note, we briefly discuss issues about which we disagree. A Note on Personal Pronouns We disagree about how to handle English third-person singular pronouns. AR argues that we should use a ''neutral" pronoun and agrees to the use of "he", with the understanding that this refers to both men and women. Continuous reminders of the he/she issue simply divert the reader's attention from the main issues. Language is extremely important in shaping our thinking, but in academic material it is not useful to wave it as a flag, as is common in some circles. MJO argues that no language is "neutral". In particular, there is a wealth of evidence, both from experiments and from analyses of language use, that "he" is not generally perceived to encompass both females and males. To quote the American Heritage Dictionary (third edition, page 831), "Thus he is not really a gender-neutral pronoun; rather it refers to a male who is to be taken as the representative member of the group referred to by its antecedent. The traditional usage, then, is not simply a grammatical convention; it also suggests a particular pattern of thought." Further, the use of "he" to refer to an individual of unspecified sex did not even arise naturally, but was imposed as a rule by (male) prescriptive grammarians in the eighteenth and nineteenth centuries who were upset by the widespread use of "they" as a singular pronoun and decided that, since in their opinion men were more important than women, "he" should be used. The use of "he" to refer to a generic individual thus both has its origins in sexist attitudes and promotes such attitudes.