What is included with this book?
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.
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 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