The mathematical study of games is an intriguing endeavor with implications and applications that reach far beyond tic-tac-toe, chess, and poker to economics, business, and even biology and politics. Most texts on the subject, however, are written at the graduate level for those with strong mathematics, economics, or business backgrounds.
In a clear and refreshing departure from this trend, Introducing Game Theory and its Applications presents an easy-to-read introduction to the basic ideas and techniques of game theory. After a brief introduction, the author begins with a chapter devoted to combinatorial games--a topic neglected or treated minimally in most other texts. The focus then shifts to two-person zero-sum games and their solution. Here the author presents the simplex method, based on linear programming, for solving these games and develops within his presentation the required background in linear programming. The final chapter presents some of the fundamental ideas and tools of non-zero-sum games and games with more than two players, including an introduction to cooperative game theory.
This book will not only satisfy the curiosity of those whose interest in the subject was piqued by the 1994 Nobel Prize awarded to Harsanyi, Nash, and Selten. It also prepares its readers for more advanced study of game theory's applications in economics, business, and the physical, biological, and social sciences.
Table of Contents
Definition of Combinatorial Game
The Fundamental Theorem for Combinatorial Games
Hex and Other Games
TWO-PERSON ZERO-SUM GAMES
Games in Normal Form
Saddle Points and Equilibrium Pairs
Maximin and Minimax
2 x 2 Matrix Games
2 x n, m x 2, and 3 x 3 Matrix Games
Linear Programming. Von Neumann's Theorem
THE SIMPLEX METHOD. THE FUNDAMENTAL THEOREM OF DUALITY. SOLUTION OF TWO-PERSON ZERO-SUM GAMES.
Slack Variables. Perfect Canonical Linear Programming Problems
The Simplex Method
The Perfect Phase of the Simplex Method
The Big M Method
Bland's Rules to Prevent Cycling
Duality and the Simplex Method
Solution of Game Matrices
Proofs of Facts 1-4
NON-ZERO-SUM GAMES AND k-PERSON GAMES
The General Setting
Graphical Method for Finding Nash Equilibria for 2 ´ 2 Matrices
Inadequacies of Nash Equilibria in Non-Zero-Sum Games. Cooperative Games
The Nash Arbitration Procedure
Games with Two or More Players
Games in Coalition Form
The Shapley Value
Finite Probability Theory
Answers to Selected Exercises
"This is an important contribution - one that will make demand for this book high. The value of the book can be seen in three distinct characteristics. First, the book does not skimp on providing full details of the theorems (and accompanying proofs) that underpin the strategic study of games. …At a minimum, this book is a wonderful source of games for classroom exercises. …A better understanding of the theory of those games is fundamental to any empirical enterprise, and Introducing Game Theory and Its Applications provides a fine starting point for interested students."
-Interfaces, Andrew B. Whitford, University of Georgia
"In this introductory textbook aimed at anyone trying to understand the implications and applications of game theory, Mendelson provides basic methods for various games that relate to mathematics, economics and business."
-Columbia College Today
"It can be recommended to readers with a limited mathematical knowledge who are interested in game theory and its applications in economics, political science and biology."
-EMS Newsletter, March 2005