Mathematical Methods and Theory in Games, Programming, and Economics

Mathematical Methods and Theory in Games, Programming, and Economics

Volume 2: The Theory of Infinite Games

1st Edition - January 1, 1959

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  • Author: Samuel Karlin
  • eBook ISBN: 9781483224008

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Mathematical Methods and Theory in Games, Programming, and Economics, Volume II provides information pertinent to the mathematical theory of games of strategy. This book presents the mathematical tools for manipulating and analyzing large sets of strategies. Organized into nine chapters, this volume begins with an overview of the fundamental concepts in game theory, namely, strategy and pay-off. This text then examines the identification of strategies with points in Euclidean n-space, which is a convenience that simplifies the mathematical analysis. Other chapters provide a discussion of the theory of finite convex games. This book discusses as well the extension of the theory of convex continuous games to generalized convex games, which leads to the characterization that such games possess optimal strategies of finite type. The final chapter deals with the components of a simple two-person poker game. This book is a valuable resource for mathematicians, statisticians, economists, social scientists, and research workers.

Table of Contents

  • Chapter 1. The Definition of a Game and the Min-Max Theorem

    1.1 Introduction. Games in Normal Form

    1.2 Examples

    1.3 Choice of Strategies

    1.4 The Min-Max Theorem for Finite Matrix Games

    1.5 General Min-Max Theorem

    1.6 Problems

    Notes and References

    Chapter 2. The Nature and Structure of Infinite Games

    2.1 Introduction

    2.2 Games on the Unit Square

    2.3 Classes of Games on the Unit Square

    2.4 Infinite Games Whose Strategy Spaces are Known Function Spaces

    2.5 How to Solve Infinite Games

    2.6 Problems

    Notes and References

    Chapter 3. Separable and Polynomial Games

    3.1 General Finite Convex Games

    3.2 The Fixed-Point Method for Finite Convex Games

    3.3 Dimension Relations for Solutions of Finite Convex Games

    3.4 The Method of Dual Cones

    3.5 Structure of Solution Sets of Separable Games

    3.6 General Remarks on Convex Sets in En

    3.7 The Reduced Moment Spaces

    3.8 Polynomial Games

    3.9 Problems

    Notes and References

    Chapter 4. Games with Convex Kernels and Generalized Convex Kernels

    4.1 Introduction

    4.2 Convex Continuous Games

    4.3 Generalized Convex Games

    4.4 Games with Convex Pay-Off in En

    4.5 A Theorem on Convex Functions

    4.6 Problems

    Notes and References

    Chapter 5. Games of Timing of One Action for Each Player

    5.1 Examples of Games of Timing

    5.2 The Integral Equations of Games of Timing and Their Solutions

    5.3 Integral Equations with Positive Kernels

    5.4 Existence Proofs

    5.5 The Silent Duel with General Accuracy Functions

    5.6 Problems

    Notes and References

    Chapter 6. Games of Timing (Continued)

    6.1 Games of Timing of Class I

    6.2 Examples

    6.3 Proof of Theorem 6.1.1

    6.4 Games of Timing Involving Several Actions

    6.5 Butterfly-Shaped Kernels

    6.6 Problems

    Notes and References

    Chapter 7. Miscellaneous Games

    7.1 Games with Analytic Kernels

    7.2 Bell-Shaped Kernels

    7.3 Bell-Shaped Games

    7.4 Other Types of Continuous Games

    7.5 Invariant Games

    7.6 Problems

    Notes and References

    Chapter 8. Infinite Classical Games Not Played Over the Unit Square

    8.1 Preliminary Results (The Neyman-Pearson Lemma)

    8.2 Application of the Neyman-Pearson Lemma to a Variational Problem

    8.3 The Fighter-Bomber Duel

    8.4 Solution of the Fighter-Bomber Duel

    8.5 The Two-Machine-Gun Duel

    8.6 Problems

    Notes and References

    Chapter 9. Poker and General Parlor Games

    9.1 A Simplified Blackjack Game

    9.2 A Poker Model with One Round of Betting and One Size of Bet

    9.3 A Poker Model with Several Sizes of Bet

    9.4 Poker Model with Two Rounds of Betting

    9.5 Poker Model with K Raises

    9.6 Poker with Simultaneous Moves

    9.7 The Le Her Game

    9.8 "High Hand Wins"

    9.9 Problems

    Notes and References

    Solutions to Problems

    Appendix A. Vector Spaces and Matrices

    A.1 Euclidean and Unitary Spaces

    A.2 Subspaces, Linear Independence, Basis, Direct Sums, Orthogonal Complements

    A.3 Linear Transformations, Matrices, and Linear Equations

    A.4 Eigenvalues, Eigenvectors, and the Jordan Canonical Form

    A.5 Transposed, Normal, and Hermitian Matrices; Orthogonal Complement

    A.6 Quadratic Form

    A.7 Matrix-Valued Functions

    A.8 Determinants; Minors, Cofactors

    A.9 Some Identities

    A.10 Compound Matrices

    Appendix B. Convex Sets and Convex Functions

    B.1 Convex Sets in En

    B.2 Convex Hulls of Sets and Extreme Points of Convex Sets

    B.3 Convex Cones

    B.4 Convex and Concave Functions

    Appendix C. Miscellaneous Topics

    C.1 Semicontinuous and Equicontinuous Functions

    C.2 Fixed-Point Theorems

    C.3 Set Functions and Probability Distributions



Product details

  • No. of pages: 398
  • Language: English
  • Copyright: © Pergamon 1959
  • Published: January 1, 1959
  • Imprint: Pergamon
  • eBook ISBN: 9781483224008

About the Author

Samuel Karlin

Affiliations and Expertise

Stanford University and The Weizmann Institute of Science

About the Editor

Z. W. Birnbaum

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