Strategy, Decision Making ( Harrington) Problem Chapter 11 Exercise 8

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Games, Strategies, and Decision Making

Joseph E. Harrington, Jr. Johns Hopkins University

Worth Publishers

 

 

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Games, Strategies, and Decision Making

 

 

To Colleen and Grace, who as children taught me love,

and who as teenagers taught me strategy.

 

 

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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 . . . . . . . . . . . . . . . . . . . . . . . . 301

Independent Private Values and Shading Your Bid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302

Common Value and the Winner’s Curse . . . . . . . . . . . 304

10.5 Voting on Committees and Juries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

10.5.1 Strategic Abstention . . . . . . . . . . . . . . . . . . . . 307

10.5.2 Sequential Voting in the Jury Room . . . 309