Game Theory

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

• Preface for the Third Edition xiPreface for the Second Edition xiiiPreface for the First Edition xviAcknowledgments xixIntroduction xxi1 Matrix Two-Person Games 11.1 What Is Game Theory? 11.2 Motivating Examples 21.2.1 Three Card Poker 31.2.2 Simplified Baseball 61.2.3 2 × 2 NIM 91.3 Mathematical Setup 111.3.1 Definition of a Matrix Game 111.3.2 Saddle Points: What It Means to be Optimal 14Problems 151.4 Mixed Strategies 171.4.1 Definition of Mixed Strategies 171.4.2 Optimal Mixed Strategies 181.4.3 Best Response Strategies 231.4.4 Dominated Strategies 27Problems 301.5 The Indifference Principle and Completely Mixed Games 321.5.1 2 × 2 Games 351.5.2 Completely Mixed Games and Invertible Matrix Games 371.5.3 An Application: Optimal Target Choice and Defense 40Problems 451.6 Finding Saddle Points in General 491.6.1 Graphical Methods 491.6.2 The n × m Case and Linear Programming 521.6.3 Using Calculus 581.6.4 Symmetric Games 59Problems 621.7 Existence of Saddle Points: The Von Neumann Minimax Theorem 671.7.1 Statement of the Minimax Theorem 671.7.2 Von Neumann’s Theorem Guarantees Matrix Games Have Saddle Points 69Problems 691.8 Review Problems 75Problems 751.9 Appendix: A Proof of the von Neumann Minimax Theorem 762 Two-Person Nonzero Sum Games 812.1 The Basics 812.1.1 Prisoner’s Dilemma 83Problems 882.2 2 × 2 Bimatrix Games, Best Response, Equality of Payoffs 90Problems 962.3 Interior Mixed Nash Points by Calculus 982.3.1 Calculus Method for Interior Nash 98Problems 1052.3.2 Existence of a Nash Equilibrium for Bimatrix Games 1072.4 Nonlinear Programming Method for Nonzero Sum Two-Person Games 108Summary of Methods for Finding Mixed Nash Equilibria 111Problems 1122.5 Correlated Equilibria 1142.5.1 Motivating Example 1142.5.2 Definition of Correlated Equilibrium and Social Welfare 115Problems 1222.6 Choosing Among Several Nash Equilibria (Optional) 123Problems 128Bibliographic Notes 1283 Games in Extensive Form: Sequential Decision Making 1293.1 Introduction to Game Trees/Extensive form of Games 1293.1.1 Gambit 129Problems 1403.2 Backward Induction and Subgame Perfect Equilibrium 143Problems 1473.2.1 Subgame Perfect Equilibrium 1493.2.2 Examples of Extensive Games Using Gambit 1543.3 Behavior Strategies in Extensive Games 157Problems 1593.4 Extensive Games with Imperfect Information 1653.4.1 Bayesian Games and Bayesian Equilibria 1703.4.1.1 Separating and Pooling PBEs 182Problems 189Bibliographic Notes 1984 N-Person Nonzero Sum Games and Games with a Continuum of Strategies 1994.1 Motivating Examples 1994.2 The Basics 2024.2.1 Do We Have Mixed Strategies in Continuous Games 2064.2.2 Existence of Pure NE 214Problems 2274.3 Economics Applications of Nash Equilibria 234Problems 2484.4 Duels 252Problems 2594.5 Auctions 2604.5.1 Complete Information 264Problems 2654.5.2 Symmetric Independent Private Value Auctions 265Problems 2724.6 Stable Matching, Marriage, and Residencies 2724.6.1 Finding a Stable Marriage Using Mathematica 277Problems 2784.7 Selected Chapter Problems 280Problems 280Bibliographic Notes 2835 Repeated Games 2855.1 Games Repeated Until 2885.2 Grim-Trigger in General 2955.2.1 A Better Estimate for the Discount Factor 2995.2.2 Folk Theorems 300Problems 301Bibliographic Notes 3056 Cooperative Games 3076.1 What Is a Cooperative Game? 3076.2 Coalitions and Characteristic Functions 308Problems 3246.2.1 More on the Core and Least Core 327Problems 3336.3 The Nucleolus 3346.3.1 An Exact Nucleolus for Three Player Games 341Problems 3466.4 The Shapley Value 348Problems 359Bibliographic Notes 3647 Bargaining 3677.1 Introduction 3677.2 The Nash Model with Security Point 3737.3 Threats 3797.3.1 Finding the Threat Strategies 3817.3.1.1 Summary Approach for Bargaining with Threat Strategies 3837.3.1.2 Another Way to Derive the Threat Strategies Procedure 3847.4 The Kalai–Smorodinsky Bargaining Solution 3897.5 Sequential Bargaining 391Problems 396Bibliographic Notes 3998 Evolutionary Stable Strategies and Population Games 4018.1 Evolution 4018.1.1 Properties of an ESS 407Problems 4128.2 Population Games 4138.3 The Von Neumann Minimax Theorem from Replicator Dynamics 429Problems 431Bibliographic Notes 437Appendix A The Essentials of Matrix Analysis 439Appendix B The Essentials of Probability 443Appendix C The Mathematica Commands 447C.1 The Upper and Lower Values of a Game 447C.2 The Value of an Invertible Matrix Game with Mixed Strategies 448C.3 Solving Matrix Games 448C.4 Interior Nash Points 449C.5 Lemke–Howson Algorithm for Nash Equilibrium 450C.6 Is the Core Empty? 450C.7 Find and Plot the Least Core 451C.8 Nucleolus Procedure and Shapley Value 453C.9 Mathematica Code for Three-Person Nucleolus 454C.10 Plotting the Payoff Pairs 456C.11 Bargaining Solutions 457C.12 Mathematica for Replicator Dynamics 459Appendix D Biographies 461D.1 John Von Neumann 461D.2 John Forbes Nash 462Selected Problem Solutions 463References 545Index 547