Game theory : an introduction

Game theory arises in almost every fact of human and inhuman interaction since oftentimes during these communications objectives are opposed or cooperation is viewed as an option. From economics and finance to biology and computer science, researchers and practitioners are often put in complex decision-making scenarios, whether they are interacting with each other or working with evolving technology and artificial intelligence. Acknowledging the role of mathematics in making logical and advantageous decisions, Game Theory: An Introduction uses modern software applications to create, analyze, and implement effective decision-making models.

While most books on modern game theory are either too abstract or too applied, this book provides a balanced treatment of the subject that is both conceptual and hands-on. Game Theory introduces readers to the basic theories behind games and presents real-world examples from various fields of study such as economics, political science, military science, finance, biological science as well as general game playing. A unique feature of this book is the use of Maple to find the values and strategies of games, and in addition, it aids in the implementation of algorithms for the solution or visualization of game concepts. Maple is also utilized to facilitate a visual learning environment of game theory and acts as the primary tool for the calculation of complex non-cooperative and cooperative games.

Important game theory topics are presented within the following five main areas of coverage:

Two-person zero sum matrix games

Nonzero sum games and the reduction to nonlinear programming

Cooperative games, including discussion of both the Nucleolus concept and the Shapley value

Bargaining, including threat strategies

Evolutionary stable strategies and population games

Although some mathematical competence is assumed, appendices are provided to act as a refresher of the basic concepts of linear algebra, probability, and statistics. Exercises are included at the end of each section along with algorithms for the solution of the games to help readers master the presented information. Also, explicit Maple and Mathematica® commands are included in the book and are available as worksheets via the book's related Web site. The use of this software allows readers to solve many more advanced and interesting games without spending time on the theory of linear and nonlinear programming or performing other complex calculations.

With extensive examples illustrating game theory's wide range of relevance, this classroom-tested book is ideal for game theory courses in mathematics, engineering, operations research, computer science, and economics at the upper-undergraduate level. It is also an ideal companion for anyone who is interested in the applications of game theory.

Author Notes

E. 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 research journal articles, and his teaching experience includes optimal control, stochastic processes, differential games, analysis, operations research, game theory, and financial mathematics, among others.

Preface	p. xi
Acknowledgments	p. xv
Introduction	p. xvii
1 Matrix Two-Person Games	p. 1
1.1 The Basics	p. 1
Problems	p. 14
1.2 The von Neumann Minimax Theorem	p. 15
Problems	p. 21
1.3 Mixed Strategies	p. 22
1.3.1 Dominated Strategies	p. 34
1.4 Solving 2 x 2 Games Graphically	p. 37
Problems	p. 38
1.5 Graphical Solution of 2 x m and n x 2 Games	p. 39
Problems	p. 46
1.6 Best Response Strategies	p. 47
Problems	p. 53
2 Solution Methods for Matrix Games	p. 55
2.1 Solution of Some Special Games	p. 55
2.1.1 2 x 2 Games Revisited	p. 55
Problems	p. 59
2.2 Invertible Matrix Games	p. 60
Problems	p. 65
2.3 Symmetric Games	p. 68
Problems	p. 72
2.4 Matrix Games and Linear Programming	p. 73
2.4.1 A Direct Formulation Without Transforming: Method 2	p. 81
Problems	p. 87
2.5 Linear Programming and the Simplex Method (Optional)	p. 90
2.5.1 The Simplex Method Step by Step	p. 93
Problems	p. 100
2.6 A Game Theory Model of Economic Growth (Optional)	p. 101
Problems	p. 107
3 Two-Person Nonzero Sum Games	p. 109
3.1 The Basics	p. 109
Problems	p. 117
3.2 2 x 2 Bimatrix Games	p. 118
Problems	p. 128
3.3 Interior Mixed Nash Points by Calculus	p. 129
Problems	p. 139
3.3.1 Proof that There Is a Nash Equilibrium for Bimatrix Games (Optional)	p. 141
3.4 Nonlinear Programming Method for Nonzero Sum Two-Person Games	p. 143
Problems	p. 153
3.5 Choosing Among Several Nash Equilibria (Optional)	p. 155
Problems	p. 160
4 N-Person Nonzero Sum Games with a Continuum of Strategies	p. 163
4.1 The Basics	p. 163
4.2 Economics Applications of Nash equilibria	p. 169
Problems	p. 184
4.3 Duels (optional)	p. 186
Problems	p. 193
4.4 Auctions (Optional)	p. 195
4.4.1 Complete Information	p. 200
Problems	p. 201
4.4.2 Incomplete Information	p. 202
4.4.3 Symmetric Independent Private Value Auctions	p. 205
Problems	p. 217
5 Cooperative Games	p. 219
5.1 Coalitions and Characteristic Functions	p. 219
Problems	p. 240
5.1.1 Finding the Least Core	p. 241
Problems	p. 248
5.2 The Nucleolus	p. 249
Problems	p. 263
5.3 The Shapley Value	p. 264
Problems	p. 276
5.4 Bargaining	p. 277
5.4.1 The Nash Model with Security Point	p. 285
5.4.2 Threats	p. 292
Problems	p. 305
6 Evolutionary Stable Strategies and Population Games	p. 309
6.1 Evolution	p. 310
Problems	p. 319
6.2 Population Games	p. 320
Problems	p. 340
Appendix A The Essentials of Matrix Analysis	p. 345
Appendix B The Essentials of Probability	p. 351
Appendix C The Essentials of Maple	p. 357
Appendix D The Mathematica Commands	p. 363
Appendix E Biographies	p. 379
Problem Solutions	p. 383
References	p. 409
Index	p. 411

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