This page intentionally left blank
Games, Strategies, and Decision Making
Joseph E. Harrington, Jr. Johns Hopkins University
Worth Publishers
Senior Publisher: Craig Bleyer
Acquisitions Editor: Sarah Dorger
Senior Marketing Manager: Scott Guile
Market Research and Development: Steven Rigolosi
Developmental Editor: Carol Pritchard-Martinez
Consulting Editor: Paul Shensa
Associate Editor: Matthew Driskill
Photo Editor: Bianca Moscatelli
Art Director: Babs Reingold
Photo Researcher: Julie Tesser
Senior Designer, Cover Designer: Kevin Kall
Interior Designer: Lissi Sigillo
Associate Managing Editor: Tracey Kuehn
Project Editors: Dana Kasowitz
Dennis Free, Aptara®, Inc.
Production Manager: Barbara Anne Seixas
Composition: Aptara®, Inc.
Printing and Binding: RR Donnelley
Library of Congress Control Number: 2008925817
ISBN-13: 978-0-7167-6630-8
ISBN-10: 0-7167-6630-2
© 2009 by Worth Publishers
All rights reserved
Printed in the United States of America
First printing
Worth Publishers
41 Madison Avenue
New York, NY 10010
www.worthpublishers.com
Games, Strategies, and Decision Making
To Colleen and Grace, who as children taught me love,
and who as teenagers taught me strategy.
This page intentionally left blank
vi
Joseph E. Harrington, Jr., is Professor of Economics at Johns Hopkins University. He has served on various editorial boards, including those of the RAND Journal of Economics, Foundations and Trends in Microeconomics, and the Southern Economic Journal. His research has appeared in top journals in a variety of disciplines, including economics (e.g., the American Economic Review, Journal of Political Economy, and Games and Economic Behavior), po- litical science (Economics and Politics, Public Choice), sociology (American Journal of Sociology), management science (Management Science), and psy- chology (Journal of Mathematical Psychology). He is a coauthor of Economics of Regulation and Antitrust, which is in its fourth edition.
Brief Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
C H A P T E R 1
Introduction to Strategic Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
C H A P T E R 2
Building a Model of a Strategic Situation . . . . . . . . . . . . . . . . . . . 17
C H A P T E R 3
Eliminating the Impossible: Solving a Game when Rationality Is Common Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
C H A P T E R 4
Stable Play: Nash Equilibria in Discrete Games with Two or Three Players . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
C H A P T E R 5
Stable Play: Nash Equilibria in Discrete n-Player Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
C H A P T E R 6
Stable Play: Nash Equilibria in Continuous Games . . . 147
C H A P T E R 7
Keep ’Em Guessing: Randomized Strategies . . . . . . . . . . . . . . 181
C H A P T E R 8
Taking Turns: Sequential Games with Perfect Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
C H A P T E R 9
Taking Turns in the Dark: Sequential Games with Imperfect Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
vii
BRIEF CONTENTS viii
C H A P T E R 1 0
I Know Something You Don’t Know: Games with Private Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
C H A P T E R 1 1
What You Do Tells Me Who You Are: Signaling Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
C H A P T E R 1 2
Lies and the Lying Liars That Tell Them: Cheap Talk Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
C H A P T E R 1 3
Playing Forever: Repeated Interaction with Infinitely Lived Players . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391
C H A P T E R 1 4
Cooperation and Reputation: Applications of Repeated Interaction with Infinitely Lived Players . . . . . . . . . . . . . . . . . . . . 423
C H A P T E R 1 5
Interaction in Infinitely Lived Institutions . . . . . . . . . . . . . . . . . 451
C H A P T E R 1 6
Evolutionary Game Theory and Biology: Evolutionarily Stable Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479
C H A P T E R 1 7
Evolutionary Game Theory and Biology: Replicator Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507
Answers to “Check Your Understanding” Questions . . . . . . . . . . . . . . . . . . . . . . S-1
Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G-1
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-1
ix
Competition for Elected Office . . . . . . . . . . . . . . . . . . . . . 38 The Science 84 Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.6 Moving from the Extensive Form and Strategic Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Baseball, II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Galileo Galilei and the Inquisition, II . . . . . . . . . . . . . . . . 40 Haggling at an Auto Dealership, II . . . . . . . . . . . . . . . . . 41
2.7 Going from the Strategic Form to the Extensive Form . . . . . . . . . . . . . . . . . . . . . . . . 42 2.8 Common Knowledge . . . . . . . . . . . . . . . . . . . . . 43 2.9 A Few More Issues in Modeling Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
C H A P T E R 3
Eliminating the Impossible: Solving a Game when
Rationality Is Common Knowledge 55
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.2 Solving a Game when Players Are Rational . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.2.1 Strict Dominance . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
White Flight and Racial Segregation in Housing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Banning Cigarette Advertising on Television . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.2.2 Weak Dominance . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Bidding at an Auction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 The Proxy Bid Paradox at eBay . . . . . . . . . . . . . . . . . . . . 66
3.3 Solving a Game when Players Are Rational and Players Know that Players Are Rational . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
Team-Project Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
C H A P T E R 1
Introduction to Strategic Reasoning 1
1.1 Who Wants to Be a Game Theorist? . . . 1 1.2 A Sampling of Strategic Situations . . . . . 3 1.3 Whetting Your Appetite: The Game of Concentration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Psychological Profile of a Player . . . . . . . 8 1.4.1 Preferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4.2 Beliefs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4.3 How Do Players Differ? . . . . . . . . . . . . . . . . . . . 12
1.5 Playing the Gender Pronoun Game . . . 13 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
C H A P T E R 2
Building a Model of a Strategic Situation 17
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 Extensive Form Games: Perfect Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Baseball, I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Galileo Galilei and the Inquisition, I . . . . . . . . . . . . . . . . 22 Haggling at an Auto Dealership, I . . . . . . . . . . . . . . . . . 24
2.3 Extensive Form Games: Imperfect Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Mugging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 U.S. Court of Appeals for the Federal Circuit . . . . . . . 30 The Iraq War and Weapons of Mass Destruction . . . 32
2.4 What Is a Strategy? . . . . . . . . . . . . . . . . . . . . . . . 34 2.5 Strategic Form Games . . . . . . . . . . . . . . . . . . . 36
Tosca . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Contents
Existence-of-God Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 Boxed-Pigs Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.4 Solving a Game when Rationality Is Common Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.4.1 The Doping Game: Is It Rational for Athletes to Use Steroids? . . . . . . . . . . . . . . . . . . . . . . . . 73
3.4.2 Iterative Deletion of Strictly Dominated Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.5 Appendix: Strict and Weak Dominance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3.6 Appendix: Rationalizability (Advanced) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
C H A P T E R 4
Stable Play: Nash Equilibria in Discrete Games with Two
or Three Players 89
4.1 Defining Nash Equilibrium . . . . . . . . . . . . . . 89 4.2 Classic Two-Player Games . . . . . . . . . . . . . . 92
Prisoners’ Dilemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 A Coordination Game—Driving Conventions . . . . . . . 95 A Game of Coordination and Conflict—Telephone . . 95 An Outguessing Game—Rock–Paper–Scissors . . . . . 97 Conflict and Mutual Interest in Games . . . . . . . . . . . . . 99
4.3 The Best-Reply Method . . . . . . . . . . . . . . . . . 99 4.4 Three-Player Games . . . . . . . . . . . . . . . . . . . . 101
American Idol Fandom . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Voting, Sincere or Devious? . . . . . . . . . . . . . . . . . . . . . . 102 Promotion and Sabotage . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.5 Foundations of Nash Equilibrium . . . 109 4.5.1 Relationship to Rationality Is Common Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.5.2 The Definition of a Strategy, Revisited . 110
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.6 Appendix: Formal Definition of Nash Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . 116 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
C H A P T E R 5
Stable Play: Nash Equilibria in Discrete n-Player
Games 117
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.2 Symmetric Games . . . . . . . . . . . . . . . . . . . . . . . 118
The Sneetches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Airline Security . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 Operating Systems: Mac or Windows? . . . . . . . . . . . . 125 Applying for an Internship . . . . . . . . . . . . . . . . . . . . . . . . 128
5.3 Asymmetric Games . . . . . . . . . . . . . . . . . . . . . 130 Entry into a Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 Civil Unrest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
5.4 Selecting among Nash Equilibria . . . 137 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
C H A P T E R 6
Stable Play: Nash Equilibria in Continuous Games 147
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 6.2 Solving for Nash Equilibria without Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
Price Competition with Identical Products . . . . . . . . . 149 Neutralizing Price Competition with
Price-Matching Guarantees . . . . . . . . . . . . . . . . . . . . . 152 Competing for Elected Office . . . . . . . . . . . . . . . . . . . . . 154
6.3 Solving for Nash Equilibria with Calculus (Optional) . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
Price Competition with Differentiated Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .160
Tragedy of the Commons— The Extinction of the Woolly Mammoth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
Charitable Giving and the Power of Matching Grants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
x CONTENTS
C H A P T E R 7
Keep ’Em Guessing: Randomized Strategies 181
7.1 Police Patrols and the Drug Trade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 7.2 Making Decisions under Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 7.2.1 Probability and Expectation . . . . . . . . . 182
7.2.2 Preferences over Uncertain Options . . . 185
7.2.3 Ordinal vs. Cardinal Payoffs . . . . . . . . . 186
7.3 Mixed Strategies and Nash Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 7.3.1 Back on the Beat . . . . . . . . . . . . . . . . . 187
7.3.2 Some General Properties of a Nash Equilibrium in Mixed Strategies . . . . . 191
7.4 Examples . . . . . . . . . . . . . . . . . . . .192 Avranches Gap in World War II . . . . . . . . . . . . . . . 193 Entry into a Market . . . . . . . . . . . . . . . . . . . . . . . 197
7.5 Advanced Examples . . . . . . . . . . . . 198 Penalty Kick in Soccer . . . . . . . . . . . . . . . . . . . . . 198 Slash ’em Up: Friday the 13th . . . . . . . . . . . . . . . 201 Bystander Effect . . . . . . . . . . . . . . . . . . . . . . . . . . 204
7.6 Games of Pure Conflict and Cautious Behavior . . . . . . . . . . . . . 207 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
7.7 Appendix: Formal Definition of Nash Equilibrium in Mixed Strategies . . . . . . . . . . . . . . . . . . . . . . .215 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
C H A P T E R 8
Taking Turns: Sequential Games with Perfect
Information 219
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 8.2 Backward Induction and Subgame Perfect Nash Equilibrium . . . . . . . . . . . . . . . . . . . 221
8.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 Cuban Missile Crisis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 Enron and Prosecutorial
Prerogative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 Racial Discrimination and Sports . . . . . . . . . . . . . . . . . 229
8.4 Waiting Games: Preemption and Attrition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 8.4.1 Preemption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
8.4.2 War of Attrition . . . . . . . . . . . . . . . . . . . . . . . . . . 238
8.5 Do People Reason Using Backward Induction? . . . . . . . . . . . . . . . . . . . . . . . . 239 8.5.1 Experimental Evidence and Backward Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
8.5.2 A Logical Paradox with Backward Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .242
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
C H A P T E R 9
Taking Turns in the Dark: Sequential
Games with Imperfect Information 255
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 9.2 Subgame Perfect Nash Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
British Intelligence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
9.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 OS/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 Agenda Control in the Senate . . . . . . . . . . . . . . . . . . . . 268
9.4 Commitment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 9.4.1 Deterrence of Entry . . . . . . . . . . . . . . . . . . . . . . 270
9.4.2 Managerial Contracts and Competition: East India Trade in the Seventeenth Century . . . . . . . . . . . . . . . . . . . . . . . . 277
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
CONTENTS xi
C H A P T E R 1 0
I Know Something You Don’t Know: Games with Private
Information 291
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 10.2 A Game of Incomplete Information: The Munich Agreement . . . . 291 10.3 Bayesian Games and Bayes–Nash Equilibrium . . . . . . . . . . . . . . . . . . . . 296
Gunfight in the Wild West . . . . . . . . . . . . . . . . . . . . . . . . 298
10.4 When All Players Have Private Information: Auctions .