Game Theory: An Introduction

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While most books on game theory are either too applied or too abstract, this book finds a nice balance between the two, and in addition, it does not focus on linear programming. This is the first book to utilize computer software (MapleTM and Mathematica) to do the types of linear programming involved in game theory since Maple can solve linear and nonlinear programs very quickly and easily. This allows students and readers to solve many more advanced and interesting games without spending time on the theory of linear programming. The focus of the book is not on proofs, but some proofs are provided for important results. Algorithms for solution of the games are presented in detail, and interesting applications are used to illustrate the theory. The sequence of topics is (1) 2 person zero sum matrix games; (2) Nonzero sum games and the reduction to nonlinear programming; (3) Cooperative games covering both the Nucleolus concept and the Shapley value; and (4) Bargaining, including threat strategies.

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Emmanuel N. Barron PhD, is Professor of Mathematical Sciences in the Department of Mathematics and Statistics at Loyola University Chicago. He is the author of over fifty journal articles, and his teaching experience includes optimal control, stochastic processes, differential games, analysis, operations research, game theory, and financial mathematics, to name a few. Dr. Barron received his PhD in Mathematics from Northwestern University in 1974. Table of Contents Preface. Acknowledgments. Introduction. 1. Matrix 2 person games. 1.1 The Basics. Problems. 1.2 The von Neumann Minimax Theorem. Problems. 1.3 Mixed strategies. 1.3.1 Dominated Strategies. 1.4 Solving 2 x 2 games graphically. Problems. 1.5 Graphical solution of 2 x m and n x 2 games. Problems. 1.6 Best Response Strategies. Problems. 2. Solution Methods for Matrix Games. 2.1 Solution of some special games. 2.1.1 2 x 2 games again. Problems. 2.2 Invertible matrix games. Problems. 2.3 Symmetric games. Problems. 2.4 Matrix games and linear programming. 2.4.1 A direct formulation without transforming. Problems. 2.5 Linear Programming and the Simplex Method (Optional). 2.5.1 The Simplex Method Step by Step. Problems. 2.6 A Game Theory Model of Economic Growth (Optional). Problems. 3. Two Person Nonzero Sum Games. 3.1 The Basics. Problems. 3.2 2 x 2 Bimatrix Games. Problems. 3.3 Interior Mixed Nash Points by Calculus. Problems. 3.3.1 Proof that there is a Nash Equilibrium for Bimatrix Games (Optional). 3.4 Nonlinear Programming Method for Nonzero Sum 2 person Games. Problems. 3.5 Choosing among several Nash Equilibria (Optional). Problems. 4. N Person Nonzero Sum Games with a Continuum of Strategies. 4.1 The Basics. 4.2 Economics applications of Nash equilibria. Problems. 4.2.1 Duels. Problems. 4.3 Auctions (Optional). 4.3.1 Complete Information 208. Problems. 4.3.2 Incomplete Information. 4.3.3 Symmetric Independent Private Value Auctions. Problems. 4.3.4 Symmetric Individual private value auctions again. Problems. 5. Cooperative games. 5.1 Coalitions and Characteristic Functions. Problems. 5.1.1 Finding the least core. Problems. 5.2 The Nucleolus. Problems. 5.3 The Shapley Value. Problems. 5.4 Bargaining. 5.4.1 The Nash model with security point. 5.4.2 Threats. Problems. 6. Evolutionary Stable Strategies and Population games. 6.1 Evolution. Problems. 6.2 Population games. Problems. Appendix A: The essentials of matrix analysis. Appendix B: The essentials of probability. B.0.1 Order Statistics. Appendix C: The Essentials of Maple. Appendix D: The Mathematica commands. Appendix E: Biographies.Appendix F: Solutions to selected Problems. Problem Solutions. References. Index.