This book is a spectacular introduction to the modern mathematical discipline known as the Theory of Games. Harold Kuhn first presented these lectures at Princeton University in 1952. They succinctly convey the essence of the theory, in part through the prism of the most exciting developments at its frontiers half a century ago. Kuhn devotes considerable space to topics that, while not strictly the subject matter of game theory, are firmly bound to it. These are taken mainly from the geometry of convex sets and the theory of probability distributions. The book opens by addressing "matrix games," a name first introduced in these lectures as an abbreviation for two-person, zero-sum games in normal form with a finite number of pure strategies. It continues with a treatment of games in extensive form, using a model introduced by the author in 1950 that quickly supplanted von Neumann and Morgenstern's cumbersome approach. A final section deals with games that have an infinite number of pure strategies for the two players. Throughout, the theory is generously illustrated with examples, and exercises test the reader's understanding. A historical note caps off each chapter.
For readers familiar with the calculus and with elementary matrix theory or vector analysis, this book offers an indispensable store of vital insights on a subject whose importance has only grown with the years.
Harold W. Kuhn is Professor Emeritus of Mathematics at Princeton University. Joint winner of the 1980 von Neumann Prize in Theory, he is internationally known for co-authoring a paper that initiated the theory of "nonlinear programming." Kuhn is the editor or coeditor of several books (all Princeton), including "The Essential John Nash", "Classics in Game Theory, Linear Inequalities and Related Systems", and "Contributions to the Theory of Games, I and II".
Author's Note vii Preface ix Chapter 1. What Is the Theory of Games? 1 Notes 3 Chapter 2. Matrix Games 5 2.1 Two Examples 5 2.2 The Definition of a Matrix Game 9 2.3 The Fundamental Theorem for 2 x 2 Matrix Games 10 2.4 The Geometry of Convex Sets 12 2.5 Fundamental Theorem for All Matrix Games 21 2.6 A Graphical Method of Solution 24 2.7 An Algorithm for Solving All Matrix Games 28 2.8 Simplified Poker 36 Notes 45 Appendix 48 Chapter 3. Extensive Games 59 3.1 Some Preliminary Restrictions 59 3.2 The Axiom System 59 3.3 Pure and Mixed Strategies 64 3.4 Games with Perfect Information 66 3.5 A Reduction of the Game Matrix 67 3.6 An Instructive Example 70 3.7 Behavior Strategies and Perfect Recall 72 3.8 Simplified Poker Reconsidered 77 Notes 78 Chapter 4. Infinite Games 81 4.1 Some Preliminary Restrictions 81 4.2 An Illuminating Example 81 4.3 Mixed Strategies and Expectation 83 4.4 The Battle of the Maxmin versus Supinf 88 4.5 The Fundamental Theorem 92 4.6 The Solution of Games on the Unit Square 94 Notes 103 Index 105