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Game Theory
An Introduction
E. N. Barron (Author)
9781394169115, Wiley
Hardback, published 30 April 2024
576 pages
25.7 x 18.5 x 3.1 cm, 0.93 kg
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: 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.
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
Subject Areas: Business & management [KJ]
