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A Gentle Introduction to Game Theory
Saul Stahl, University of Kansas, Lawrence, KS
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Mathematical World
1999; 176 pp; softcover
Volume: 13
Reprint/Revision History:
reprinted 2000
ISBN-10: 0-8218-1339-0
ISBN-13: 978-0-8218-1339-3
List Price: US$29
Member Price: US$23.20
Order Code: MAWRLD/13
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See also:

Models of Conflict and Cooperation - Rick Gillman and David Housman

The mathematical theory of games was first developed as a model for situations of conflict, whether actual or recreational. It gained widespread recognition when it was applied to the theoretical study of economics by von Neumann and Morgenstern in Theory of Games and Economic Behavior in the 1940s. The later bestowal in 1994 of the Nobel Prize in economics on Nash underscores the important role this theory has played in the intellectual life of the twentieth century.

This volume is based on courses given by the author at the University of Kansas. The exposition is "gentle" because it requires only some knowledge of coordinate geometry; linear programming is not used. It is "mathematical" because it is more concerned with the mathematical solution of games than with their applications.

Existing textbooks on the topic tend to focus either on the applications or on the mathematics at a level that makes the works inaccessible to most non-mathematicians. This book nicely fits in between these two alternatives. It discusses examples and completely solves them with tools that require no more than high school algebra.

In this text, proofs are provided for both von Neumann's Minimax Theorem and the existence of the Nash Equilibrium in the \(2 \times 2\) case. Readers will gain both a sense of the range of applications and a better understanding of the theoretical framework of these two deep mathematical concepts.

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Readership

Undergraduates in any area, interested in game theory.

Reviews

"This book is an excellent introduction to the mathematical aspects of game theory for beginners without a background in calculus."

-- Journal of Mathematical Psychology

"Game theory, in the sense of von Neumann and Morgenstern, studies models of competition in situations of uncertainty. It provides a means for both deriving desirable strategies and explaining naturally occurring behavior; it finds applications ranging from economics and politics to evolutionary biology. All this and its intrinsic human interest (read here how it elucidates the outcome of the Cuban Missile Crisis) make it a favorite undergraduate topic, particularly for students majoring outside mathematics. There is not a faster read in the realm of higher mathematics. Recommended for college libraries. Undergraduates and up."

-- CHOICE

Table of Contents

  • Introduction
  • The formal definitions
  • Optimal responses to specific strategies
  • The maximin strategy
  • The minimax strategy
  • Solutions of zero-sum games
  • \(2 \times n\) and \(m \times 2\) games
  • Dominance
  • Symmetric games
  • Poker-like games
  • Pure maximin and minimax strategies
  • Pure nonzero-sum games
  • Mixed strategies for nonzero-sum games
  • Finding mixed Nash equilibria for \(2 \times 2\) nonzero-sum games
  • Bibliography
  • Solutions to selected exercises
  • Index
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