Game Theory (Hardcover, 3rd)

· 제목 : Game Theory (Hardcover, 3rd) (An Introduction)
· 분류 : 외국도서 > 과학/수학/생태 > 수학 > 게임 이론
· ISBN : 9781394169115
· 쪽수 : 576쪽
· 출판일 : 2024-04-09

Preface for the Third Edition xi

Preface for the Second Edition xiii

Preface for the First Edition xvi

Acknowledgments xix

Introduction xxi

1 Matrix Two-Person Games 1

1.1 What Is Game Theory? 1

1.2 Motivating Examples 2

1.2.1 Three Card Poker 3

1.2.2 Simplified Baseball 6

1.2.3 2 × 2 NIM 9

1.3 Mathematical Setup 11

1.3.1 Definition of a Matrix Game 11

1.3.2 Saddle Points: What It Means to be Optimal 14

Problems 15

1.4 Mixed Strategies 17

1.4.1 Definition of Mixed Strategies 17

1.4.2 Optimal Mixed Strategies 18

1.4.3 Best Response Strategies 23

1.4.4 Dominated Strategies 27

Problems 30

1.5 The Indifference Principle and Completely Mixed Games 32

1.5.1 2 × 2 Games 35

1.5.2 Completely Mixed Games and Invertible Matrix Games 37

1.5.3 An Application: Optimal Target Choice and Defense 40

Problems 45

1.6 Finding Saddle Points in General 49

1.6.1 Graphical Methods 49

1.6.2 The n × m Case and Linear Programming 52

1.6.3 Using Calculus 58

1.6.4 Symmetric Games 59

Problems 62

1.7 Existence of Saddle Points: The Von Neumann Minimax Theorem 67

1.7.1 Statement of the Minimax Theorem 67

1.7.2 Von Neumann’s Theorem Guarantees Matrix Games Have Saddle Points 69

Problems 69

1.8 Review Problems 75

Problems 75

1.9 Appendix: A Proof of the von Neumann Minimax Theorem 76

2 Two-Person Nonzero Sum Games 81

2.1 The Basics 81

2.1.1 Prisoner’s Dilemma 83

Problems 88

2.2 2 × 2 Bimatrix Games, Best Response, Equality of Payoffs 90

Problems 96

2.3 Interior Mixed Nash Points by Calculus 98

2.3.1 Calculus Method for Interior Nash 98

Problems 105

2.3.2 Existence of a Nash Equilibrium for Bimatrix Games 107

2.4 Nonlinear Programming Method for Nonzero Sum Two-Person Games 108

Summary of Methods for Finding Mixed Nash Equilibria 111

Problems 112

2.5 Correlated Equilibria 114

2.5.1 Motivating Example 114

2.5.2 Definition of Correlated Equilibrium and Social Welfare 115

Problems 122

2.6 Choosing Among Several Nash Equilibria (Optional) 123

Problems 128

Bibliographic Notes 128

3 Games in Extensive Form: Sequential Decision Making 129

3.1 Introduction to Game Trees/Extensive form of Games 129

3.1.1 Gambit 129

Problems 140

3.2 Backward Induction and Subgame Perfect Equilibrium 143

Problems 147

3.2.1 Subgame Perfect Equilibrium 149

3.2.2 Examples of Extensive Games Using Gambit 154

3.3 Behavior Strategies in Extensive Games 157

Problems 159

3.4 Extensive Games with Imperfect Information 165

3.4.1 Bayesian Games and Bayesian Equilibria 170

3.4.1.1 Separating and Pooling PBEs 182

Problems 189

Bibliographic Notes 198

4 N-Person Nonzero Sum Games and Games with a Continuum of Strategies 199

4.1 Motivating Examples 199

4.2 The Basics 202

4.2.1 Do We Have Mixed Strategies in Continuous Games 206

4.2.2 Existence of Pure NE 214

Problems 227

4.3 Economics Applications of Nash Equilibria 234

Problems 248

4.4 Duels 252

Problems 259

4.5 Auctions 260

4.5.1 Complete Information 264

Problems 265

4.5.2 Symmetric Independent Private Value Auctions 265

Problems 272

4.6 Stable Matching, Marriage, and Residencies 272

4.6.1 Finding a Stable Marriage Using Mathematica 277

Problems 278

4.7 Selected Chapter Problems 280

Problems 280

Bibliographic Notes 283

5 Repeated Games 285

5.1 Games Repeated Until 288

5.2 Grim-Trigger in General 295

5.2.1 A Better Estimate for the Discount Factor 299

5.2.2 Folk Theorems 300

Problems 301

Bibliographic Notes 305

6 Cooperative Games 307

6.1 What Is a Cooperative Game? 307

6.2 Coalitions and Characteristic Functions 308

Problems 324

6.2.1 More on the Core and Least Core 327

Problems 333

6.3 The Nucleolus 334

6.3.1 An Exact Nucleolus for Three Player Games 341

Problems 346

6.4 The Shapley Value 348

Problems 359

Bibliographic Notes 364

7 Bargaining 367

7.1 Introduction 367

7.2 The Nash Model with Security Point 373

7.3 Threats 379

7.3.1 Finding the Threat Strategies 381

7.3.1.1 Summary Approach for Bargaining with Threat Strategies 383

7.3.1.2 Another Way to Derive the Threat Strategies Procedure 384

7.4 The Kalai?Smorodinsky Bargaining Solution 389

7.5 Sequential Bargaining 391

Problems 396

Bibliographic Notes 399

8 Evolutionary Stable Strategies and Population Games 401

8.1 Evolution 401

8.1.1 Properties of an ESS 407

Problems 412

8.2 Population Games 413

8.3 The Von Neumann Minimax Theorem from Replicator Dynamics 429

Problems 431

Bibliographic Notes 437

Appendix A The Essentials of Matrix Analysis 439

Appendix B The Essentials of Probability 443

Appendix C The Mathematica Commands 447

C.1 The Upper and Lower Values of a Game 447

C.2 The Value of an Invertible Matrix Game with Mixed Strategies 448

C.3 Solving Matrix Games 448

C.4 Interior Nash Points 449

C.5 Lemke?Howson Algorithm for Nash Equilibrium 450

C.6 Is the Core Empty? 450

C.7 Find and Plot the Least Core 451

C.8 Nucleolus Procedure and Shapley Value 453

C.9 Mathematica Code for Three-Person Nucleolus 454

C.10 Plotting the Payoff Pairs 456

C.11 Bargaining Solutions 457

C.12 Mathematica for Replicator Dynamics 459

Appendix D Biographies 461

D.1 John Von Neumann 461

D.2 John Forbes Nash 462

Selected Problem Solutions 463

References 545

Index 547