Game Theory

E. N. BARRON, PhD, is Professor of Mathematics and Statistics in the Department of Mathematics and Statistics at Loyola University Chicago and the author of more than sixty journal articles on optimal control, differential games, nonlinear partial differential equations, and mathematical finance.

Preface for the Second Edition xi Preface for the First Edition xv

Acknowledgments xvii

Introduction 1

1 Matrix Two-Person Games 5

1.1 The Basics 5

Problems 16

1.2 The von Neumann Minimax Theorem 18

1.2.1 Proof of von Neumann’s Minimax Theorem (Optional) 21

Problems 24

1.3 Mixed Strategies 25

1.3.1 Properties of Optimal Strategies 35

1.3.2 Dominated Strategies 38

1.4 Solving 2 × 2 Games Graphically 41

Problems 43

1.5 Graphical Solution of 2 × m and n × 2 Games 44

Problems 50

1.6 Best Response Strategies 53

Problems 57

1.6.1 MapleTM/Mathematica R 58

Bibliographic Notes 59

2 Solution Methods for Matrix Games 60

2.1 Solution of Some Special Games 60

2.1.1 2 × 2 Games Revisited 60

Problems 64

2.2 Invertible Matrix Games 65

2.2.1 Completely Mixed Games 68

Problems 74

2.3 Symmetric Games 76

Problems 81

2.4 Matrix Games and Linear Programming 82

2.4.1 Setting Up the Linear Program: Method 1 83

2.4.2 A Direct Formulation Without Transforming: Method 2 89

Problems 94

2.5 Appendix: Linear Programming and the Simplex Method 98

2.5.1 The Simplex Method Step by Step 101

Problems 108

2.6 Review Problems 108

2.7 Maple/Mathematica 109

2.7.1 Invertible Matrices 109

2.7.2 Linear Programming: Method 1 110

2.7.3 Linear Programming: Method 2 111

Bibliographic Notes 113

3 Two-Person Nonzero Sum Games 115

3.1 The Basics 115

Problems 123

3.2 2 × 2 Bimatrix Games, Best Response, Equality of Payoffs 125

3.2.1 Calculation of the Rational Reaction Sets for 2 × 2 Games 125

Problems 132

3.3 Interior Mixed Nash Points by Calculus 135

3.3.1 Calculus Method for Interior Nash 135

Problems 143

3.3.2 Proof that There is a Nash Equilibrium for Bimatrix Games (Optional) 146

3.4 Nonlinear Programming Method for Nonzero Sum Two-Person Games 148

3.4.1 Summary of Methods for Finding Mixed Nash Equilibria 156

Problems 158

3.5 Correlated Equilibria 159

3.5.1 LP Problem for a Correlated Equilibrium 165

Problems 166

3.6 Choosing Among Several Nash Equilibria (Optional) 167

Problems 172

3.6.1 Maple/Mathematica 173

3.6.2 Mathematica for Lemke–Howson Algorithm 173

Bibliographic Notes 175

4 Games in Extensive Form: Sequential Decision Making 176

4.1 Introduction to Game Trees—Gambit 176

Problems 189

4.2 Backward Induction and Subgame Perfect Equilibrium 190

Problems 193

4.2.1 Subgame Perfect Equilibrium 194

4.2.2 Examples of Extensive Games Using Gambit 200

Problems 209

Bibliographic Notes 212

5 n-Person Nonzero Sum Games and Games with a Continuum of Strategies 213

5.1 The Basics 213

Problems 235

5.2 Economics Applications of Nash Equilibria 242

5.2.1 Cournot Duopoly 243

5.2.2 A Slight Generalization of Cournot 245

5.2.3 Cournot Model with Uncertain Costs 247

5.2.4 The Bertrand Model 250

5.2.5 The Stackelberg Model 252

5.2.6 Entry Deterrence 254

Problems 256

5.3 Duels (Optional) 259

5.3.1 Silent Duel on [0,1] (Optional) 262

Problem 266

5.4 Auctions (Optional) 266

5.4.1 Complete Information 271

Problems 272

5.4.2 Incomplete Information 272

5.4.3 Symmetric Independent Private Value Auctions 275

Problem 286

Bibliographic Notes 287

6 Cooperative Games 288

6.1 Coalitions and Characteristic Functions 288

Problems 307

6.1.1 More on the Core and Least Core 310

Problems 317

6.2 The Nucleolus 319

6.2.1 An Exact Nucleolus for Three-Player Games 327

Problems 333

6.3 The Shapley Value 335

Problems 347

6.4 Bargaining 352

6.4.1 The Nash Model with Security Point 358

6.4.2 Threats 365

6.4.3 The Kalai–Smorodinsky Bargaining Solution 377

6.4.4 Sequential Bargaining 379

Problems 384

Review Problems 386

6.5 Maple/Mathematica 386

6.5.1 Finding the Nucleolus One Step at a Time 386

6.5.2 Mathematica Code for Three-Person Nucleolus 391

6.5.3 The Shapley Value with Maple 393

6.5.4 Maple and Bargaining 393

Bibliographic Notes 394

7 Evolutionary Stable Strategies and Population Games 395

7.1 Evolution 395

7.1.1 Properties of an ESS 402

Problems 408

7.2 Population Games 409

Problems 428

Bibliographic Notes 430

Appendix A: The Essentials of Matrix Analysis 432

Appendix B: The Essentials of Probability 436

Appendix C: The Essentials of Maple 442

Appendix D: The Mathematica Commands 448

Appendix E: Biographies 463

Problem Solutions 465

References 549

Index 551