Mathematical Methods and Theory in Games, Programming, and Economics - 1st Edition - ISBN: 9781483198972, 9781483224008

Mathematical Methods and Theory in Games, Programming, and Economics

1st Edition

Volume 2: The Theory of Infinite Games

Authors: Samuel Karlin
Editors: Z. W. Birnbaum
eBook ISBN: 9781483224008
Imprint: Pergamon
Published Date: 1st January 1959
Page Count: 398
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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




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© Pergamon 1959
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About the Author

Samuel Karlin

Affiliations and Expertise

Stanford University and The Weizmann Institute of Science

About the Editor

Z. W. Birnbaum

Ratings and Reviews