A Gentle Introduction to Game Theory (Mathematical World, Vol. 13) (Mathematical World)

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 2×2 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.ReadershipUndergraduates in any area, interested in game theory. 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 X 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.--book Jacket. The Formal Definitions -- Optimal Responses To Specific Strategies -- The Maximin Strategy -- The Minimax Strategy -- Solutions Of Zero-sum Games -- 2 X N And M X 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 X 2 Nonzero-sum Games. Saul Stahl. Includes Bibliographical References And Index. 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. Cover 1 Title 6 Copyright 7 Table of Contents 10 Preface 12 Chapter 1. Introduction 14 Chapter 2. The formal definitions 26 Chapter 3. Optimal responses to specific strategies 38 Chapter 4. The maximin strategy 48 Chapter 5. The minimax strategy 58 Chapter 6. Solutions of zero-sum games 66 Chapter 7. 2 x n and m x 2 games 84 Chapter 8. Dominance 96 Chapter 9. Symmetric games 102 Chapter 10. Poker-like games 110 Chapter 11. Pure maximin and minimax strategies 128 Chapter 12. Pure nonzero-sum games 134 Chapter 13. Mixed strategies for nonzero-sum games 148 Chapter 14. Finding mixed Nash equilibria for 2 x 2 nonzero-sum games 168 Bibliography 182 Solutions to selected exercises 184 Index 188 Back Cover 190