Game Theory - E. N. Barron

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Résumé :

Authoritative and quantitative approach to modern game theory with applications from areas including economics, political science, computer science, and engineering

Game Theory acknowledges the role of mathematics in making logical and advantageous decisions in adversarial situations and provides a balanced treatment of the subject that is both conceptual and applied. This newly updated and revised Third Edition streamlines the text to introduce readers to the basic theories behind games in a less technical but still mathematically rigorous way, with many new real-world examples from various fields of study, including economics, political science, military science, finance, biological science, and general game playing.

The text introduces topics like repeated games, Bayesian equilibria, signaling games, bargaining games, evolutionary stable strategies, extensive games, and network and congestion games, which will be of interest across a wide range of disciplines. Separate sections in each chapter illustrate the use of Mathematica and Gambit software to create, analyze, and implement effective decision-making models.

A companion website contains the related Mathematica and Gambit data sets and code. Solutions, hints, and methods used to solve most problems to enable self-learning are in an Appendix.

Game Theory includes detailed information on:

• The von Neumann Minimax Theorem and methods for solving any 2-person zero sum matrix game.
• Two-person nonzero sum games solved for a Nash Equilibrium using nonlinear programming software or a calculus method. Nash Equilibria and Correlated Equilibria. Repeated games and punishment strategies to enforce cooperation
• Games in Extensive Form for solving Bayesian and perfect information games using Gambit.
• N-Person nonzero sum games, games with a continuum of strategies and many models in economics applications, duels, auctions, of Nash Equilibria, and the Stable Matching problem
• Coalitions and characteristic functions of cooperative games, an exact nucleolus for three-player games, bargaining
• Game theory in evolutionary processes and population games

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

Biographie:

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

Prob...

Sommaire:

E. N. Barron, PhD, is Professor of Mathematics and Statistics in the Department of Mathematics and Statistics at Loyola University Chicago....