Introducing Game Theory and its Applications

Elliott Mendelson is the late professor emeritus at Queens College in Flushing, New York, USA. Dr. Mendelson obtained his bachelor's at Columbia University and his master's and doctoral degrees at Cornell University and was elected afterward to the Harvard Society of Fellows. In addition to his other writings, he is the author of another CRC Press book, Introduction to Mathematical Logic, Sixth Edition. Dan Zwillinger has more than 35 years of proven technical expertise in numerous areas of engineering and the physical sciences. He earned a PhD in applied mathematics from the California Institute of Technology. He is the editor of CRC Standard Mathematical Tables and Formulas, 33rd Edition and also Table of Integrals, Series, and Products, by Gradshteyn and Ryzhik. He serves as the series editor of the CRC Series of Advances in Applied Mathematics.

Preface

Introduction

1 Combinatorial games

1.1 Definition of combinatorial games

1.2 Fundamental theorem of combinatorial games

1.3 Nim

1.4 Hex and other games

1.5 Tree games

1.6 Grundy functions

1.7 Bogus Nim-sums

1.8 Chapter summary

2 Two-person zero-sum games

2.1 Games in normal form

2.2 Saddle points and equilibrium pairs

2.3 Maximin and minimax

2.4 Mixed strategies

2.5 2-by-2 matrix games

2.6 2-by-n, m-by-2 and 3-by-3 matrix games

2.7 Linear programming

2.8 Chapter summary

3 Solving two-person zero-sum games using LP

3.1 Perfect canonical linear programming problems

3.2 The simplex method

3.3 Pivoting

3.4 The perfect phase of the simplex method

3.5 The Big M method

3.6 Bland's rules to prevent cycling

3.7 Duality and the simplex method

3.8 Solution of game matrices

3.9 Chapter summary

4 Non-zero-sum games and k-person games

4.1 The general setting

4.2 Nash equilibria

4.3 Graphical method for 2 × 2 matrix games

4.4 Inadequacies of Nash equilibria & cooperative games

4.5 The Nash arbitration procedure

4.6 Games with two or more players

4.7 Coalitions

4.8 Games in coalition form

4.9 The Shapley value

4.10 The Banzhaf power index

4.11 Imputations

4.12 Strategic equivalence

4.13 Stable sets

4.14 Chapter summary

5 Imperfect Information Games

5.1 The general setting

5.2 Complete information games in extensive form

5.3 Imperfect information games in extensive form

5.4 Games with random effects

5.5 Chapter summary

6 Computer solutions to games

6.1 Zero-sum games - invertible matrices

6.2 Zero sum games - linear program problem (LP)

6.3 Special Linear Programming Capabilities

6.4 Non-zero sum games - linear complementarity problem (LCP)

6.5 Special game packages

6.6 Chapter summary

Appendices

Appendix A Utility theory

Appendix B Nash's theorem

Appendix C Finite probability theory

Appendix D Calculus & Differentiation

Appendix E Linear Algebra

Appendix F Linear Programming

Appendix G Named Games and Game Data

Answers to selected exercises

Bibliography

Index