STRATEGY

AN INTRODUCTION TO GAME THEORY

JOEL WATSON

University of California, San Diego

Third Edition

W. W. NORTON & COMPANY NEW YORK • LONDON

CONTENTS*

PREFACE xiii

1 INTRODUCTION 1

Noncooperative Game Theory 3 Contract and Cooperative Game Theory 4 The Meaning of "Game" 5

PART I: REPRESENTATIONS AND BASIC ASSUMPTIONS 7

2 THE EXTENSIVE FORM 9

Other Examples and Conventions 15 Guided Exercise 17 Exercises 18

3 STRATEGIES AND THE NORMAL FORM 22

Terminology and Notation for Strategies 23 The Normal Form 26 Classic Normal-Form Games 28 Interpretation of the Normal Form 30 Guided Exercise 31 Exercises 33

4 BELIEFS, MIXED STRATEGIES, AND EXPECTED PAYOFFS 37

Guided Exercise 39 Exercises 40

* Chapters in italics are applications chapters.

vii

viii Contents

5 GENERAL ASSUMPTIONS AND METHODOLOGY 42

Rationality 43 Common Knowledge 44 Overview of Solution Concepts 45 The Issue of Realism 46

PART II: ANALYZING BEHAVIOR IN STATIC SETTINGS 47

6 DOMINANCE AND BEST RESPONSE 49 Dominance 49 The First Strategic Tension and the Prisoners' Dilemma 52 The Concept of Efficiency 53 Best Response 54 Dominance and Best Response Compared 57 Weak Dominance 60 Guided Exercise 61 Exercises 63

7 RATIONALIZABILITY AND ITERATED DOMINANCE 67 The Second Strategic Tension 71 Guided Exercise 73 Exercises 74

8 LOCATION, PARTNERSHIP, AND SOCIAL UNREST 78 A Location Game 7g A Partnership Game: Strategic Complementarities 81 Social Unrest gy Guided Exercise gQ Exercises g -j

9 NASH EQUILIBRIUM g5

Nash Equilibrium gy Equilibrium of the Partnership Game 1 QQ Coordination and Social Welfare n.

Contents ix

The Third Strategic Tension 104 Congruous Sets 104 Aside: Experimental Game Theory 106 Guided Exercise 107 Exercises 108

10 OLIGOPOLY, TARIFFS, CRIME, AND VOTING 113

Cournot Duopoly Model 113 Bertrand Duopoly Model 115 Tariff Setting by Two Countries 117 A Model of Crime and Police 118 The Median Voter Theorem 118 Strategic Voting 120 Guided Exercise 123 Exercises 124

11 MIXED-STRATEGY NASH EQUILIBRIUM 132

Randomization in Sports 135 Technical Notes 136 Example: Duopoly with Capacity Constraints 137 Guided Exercise 140 Exercises 140

12 STRICTLY COMPETITIVE GAMES AND SECURITY STRATEGIES 148

Guided Exercise 151 Exercises 152

13 CONTRACT, LAW, AND ENFORCEMENT IN STATIC SETTINGS 154

Complete Contracting in Discretionary Environments 158 Contracting with Court-Imposed Breach Remedies 161 Guided Exercise 166 Exercises 167

X Contents

PART III: ANALYZING BEHAVIOR IN DYNAMIC SETTINGS 173

14 DETAILS OF THE EXTENSIVE FORM 175

Recalling the Strategy Definition 179 Guided Exercise 180 Exercises 181

15 SEQUENTIAL RATIONALITY AND SUBGAME PERFECTION 183

Incredible Threats in the Stackelberg Duopoly Game 184 Sequential Rationality 185 Backward Induction 186 Subgame Perfection 188 The SPE of the Stackelberg Duopoly Game 191 Technical Notes 192 Conditional Dominance and Forward Induction 195 Guided Exercise 198 Exercises 199

16 TOPICS IN INDUSTRIAL ORGANIZATION 204

Advertising and Competition 204 A Model of Limit Capacity 206 Dynamic Monopoly 209 Price Guarantees as a Commitment to High Prices 21 3 Dynamic Price Competition with Capacity Constraints 21 5 Guided Exercise 217 Exercises 21 9

17 PARLOR GAMES

Guided Exercise

18 BARGAINING PROBLEMS

Bargaining: Value Creation and Division An Abstract Representation of Bargaining Problems An Example The Standard Bargaining Solution A Note About Legal Default Rules Guided Exercise Exercises

224

226 226

230

230 232 234 236 238 240 241

Contents xi

19 ANALYSIS OF SIMPLE BARGAINING GAMES 244

Ultimatum Games: Power to the Proposer 244 Two-Period, Alternating-Offer Games: Power to the Patient 246 Infinite-Period, Alternating-Offer Game 250 Multilateral B argaining 2 51 Guided Exercise 254 Exercises 255

20 GAMES WITH JOINT DECISIONS; NEGOTIATION EQUILIBRIUM 259

Joint Decisions 260 Negotiation Equilibrium 262 Example: Contracting for High-Powered Incentives 263 Guided Exercise 265 Exercises 267

21 UNVERIFIABLE INVESTMENT, HOLD UP, OPTIONS, AND OWNERSHIP 274

Hold-Up Example 275 Up-Front Contracting and Option Contracts 278 Asset Ownership 281 Guided Exercise 283 Exercises 285

22 REPEATED GAMES AND REPUTATION 291

A Two-Period Repeated Game 292 An Infinitely Repeated Game 297 The Equilibrium Payoff Set with Low Discounting 302 Guided Exercise 306 Exercises 307

23 COLLUSION, TRADE AGREEMENTS, AND GOODWILL 311

Dynamic Oligopoly and Collusion 311 Enforcing International Trade Agreements 313 Goodwill and Trading a Reputation 314 Guided Exercise 317 Exercises 318

xii Contents

PART IV: INFORMATION 325

24 RANDOM EVENTS AND INCOMPLETE INFORMATION 327

Guided Exercise 331 Exercises 332

25 RISK AND INCENTIVES IN CONTRACTING 336

Risk Aversion 336 A Principal-Agent Game 340 Guided Exercise 345 Exercises 346

26 BAYESIAN NASH EQUILIBRIUM AND RATIONALIZABILITY 350

Guided Exercise 353 Exercises 354

27 LEMONS, AUCTIONS, AND INFORMATION AGGREGATION 360

Markets and Lemons 360 Auctions 363 Information Aggregation 368 Guided Exercise 373 Exercises 374

28 PERFECT BAYESIAN EQUILIBRIUM 378

Conditional Beliefs about Types 379 Sequential Rationality 380 Consistency of Beliefs 381 Equilibrium Definition 382 Guided Exercise 3g4 Exercises 3gg

29 JOB-MARKET SIGNALING AND REPUTATION 392 Jobs and School 392 Reputation and Incomplete Information 395 Guided Exercise 3gg Exercises 4QQ

APPENDICES 407

INDEX 4?7