**An exciting new edition of the popular introduction to ****game theory and its applications**

The thoroughly expanded *Second Edition *presents a unique, hands-on approach to game theory. While most books on the subject are too abstract or too basic for mathematicians, *Game Theory: An Introduction, Second Edition *offers a blend of theory and applications, allowing readers to use theory and software to create and analyze real-world decision-making models.

With a rigorous, yet accessible, treatment of mathematics, the book focuses on results that can be used to determine optimal game strategies. *Game Theory: An Introduction, Second Edition *demonstrates how to use modern software, such as Maple™, Mathematica®, and Gambit, to create, analyze, and implement effective decision-making models. Coverage includes the main aspects of game theory including the fundamentals of two-person zero-sum games, cooperative games, and population games as well as a large number of examples from various fields, such as economics, transportation, warfare, asset distribution, political science, and biology. The *Second Edition *features:

• A new chapter on extensive games, which greatly expands the implementation of available models

• New sections on correlated equilibria and exact formulas for three-player cooperative games

• Many updated topics including threats in bargaining games and evolutionary stable strategies

• Solutions and methods used to solve all odd-numbered problems

• A companion website containing the related Maple and Mathematica data sets and code

A trusted and proven guide for students of mathematics and economics, *Game Theory: An Introduction, Second Edition *is also an excellent resource for researchers and practitioners in economics, finance, engineering, operations research, statistics, and computer science.

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