# Mathematical Methods and Theory in Games, Programming, and Economics

### Matrix Games, Programming, and Mathematical Economics

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Matrix Games, Programming, and Mathematical Economics deals with game theory, programming theory, and techniques of mathematical economics in a single systematic theory. The principles of game theory and programming are applied to simplified problems related to economic models, business decisions, and military tactics. The book explains the theory of matrix games and some of the tools used in the analysis of matrix games. The text describes optimal strategies for matrix games which have two basic properties, as well as the construction of optimal strategies. The book investigates the structure of sets of solutions of discrete matrix games, with emphasis on the class of games whose solutions are unique. The examples show the use of dominance concepts, symmetries, and probabilistic arguments that emphasize the principles of game theory. One example involves two opposing political parties in an election campaign, particularly, how they should distribute their advertising efforts for wider exposure. The text also investigates how to determine an optimal program from several choices that results with the maximum or minimum objective. The book also explores the analogs of the duality theorem, the equivalence of game problems to linear programming problems, and also the inter-industry nonlinear activity analysis model requiring special mathematical methods. The text will prove helpful for students in advanced mathematics and calculus. It can be appreciated by mathematicians, engineers, economists, military strategists, or statisticians who formulate decisions using mathematical analysis and linear programming.

## Table of Contents

Introduction. the Nature of the Mathematical Theory of Decision Processes

1 The Background

2 The Classification of the Mathematics of Decision Problems

3 The Main Disciplines

Part I. The Theory of Matrix Games

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 Matrix Games

1.5 General Min-Max Theorem

1.6 Problems

Notes and References

Chapter 2. The Nature of Optimal Strategies for Matrix Games

2.1 Properties of Optimal Strategies

2.2 Types of Strict Dominance

2.3 Construction of Optimal Strategies

2.4 Characterization of Extreme-Point Optimal Strategies

2.5 Completely Mixed Matrix Games

2.6 Symmetric Games

2.7 Problems

Notes and References

Chapter 3. Dimension Relations for Sets of Optimal Strategies

3.1 The Principal Theorems

3.2 Proof of Theorem 3.1.1

3.3 Proof of Theorem 3.1.2

3.4 The Converse of Theorem 3.1.2

3.5 Uniqueness of Optimal Strategies

3.6 Problems

Notes and References

Chapter 4. Solutions of Some Discrete Games

4.1 Colonel Blotto Game

4.2 Identification of Friend and Foe (I.F.F. Game)

4.3 Poker Game

4.4 An Advertising Example

4.5 a Bargaining Example

4.6 Problems

Notes and References

Solutions to Problems of Chapters 1-4

Part II. Linear and Nonlinear Programming and Mathematical Economics

Chapter 5. Linear Programming

5.1 Formulation of the Linear Programming Problem

5.2 The Linear Programming Problem and its Dual

5.3 The Principal Theorems of Linear Programming (Preliminary Results)

5.4 The Principal Theorems of Linear Programming (Continued)

5.5 Connections Between Linear Programming Problems and Game Theory

5.6 Extensions of the Duality Theorem

5.7 Warehouse Problem

5.8 Optimal Assignment Problem

5.9 Transportation and Flow Problem

5.10 Maximal-Flow, Minimal-Cut Theorem

5.11 The Caterer's Problem

5.12 Price Speculation Model

5.13 Problems

Notes and References

Chapter 6. Computational Methods for Linear Programming and Game Theory

6.1 The Simplex Method

6.2 Auxiliary Simplex Methods

6.3 An Illustration of the Use of the Simplex Method

6.4 Computation of Network Flow

6.5 A Method of Approximating the Value of a Game

6.6 Proof of the Convergence

6.7 A Differential-Equations Method for Determining the Value of a Game

6.8 Problems

Notes and References

Chapter 7. Nonlinear Programming

7.1 Concave Programming

7.2 Examples of Concave Programming

7.3 The Arrow-Hurwicz Gradient Method

7.4 The Vector Maximum Problem

7.5 Conjugate Functions

7.6 Composition of Conjugate Functions

7.7 Conjugate Concave Functions

7.8 A Duality Theorem of Nonlinear Programming

7.9 Applications of the Theory of Conjugate Functions to Convex Sets

7.10 Problems

Notes and References

Chapter 8. Mathematical Methods in the Study of Economic Models

8.1 Open and Closed Linear Leontief Models

8.2 The Theory of Positive Matrices

8.3 Applications of the Theory of Positive Matrices to the Study of Linear Models of Equilibrium and Exchange

8.4 The Theory of Production

8.5 Efficient Points of a Leontief-Type Model

8.6 The Theory of Consumer Choice

8.7 Nonlinear Models of Equilibrium

8.8 The Arrow-Debreu Equilibrium Model of a Competitive Economy

8.9 Problems

Notes and References

Chapter 9. Mathematical Methods in the Study of Economic Models (Continued)

9.1 Welfare Economics

9.2 The Stability of a Competitive Equilibrium

9.3 Local Stability

9.4 Global Stability of Price Adjustment Processes

9.5 Global Stability (Continued)

9.6 A Difference-Equations Formulation of Global Stability

9.7 Stability and Expectations (Model I)

9.8 Stability and Expectations (Model II)

9.9 The Von Neumann Model of an Expanding Economy

9.10 A General Model of Balanced Growth

9.11 Problems

Notes and References

Solutions to Problems of Chapters 5-9

Appendix A. Vector Spaces and Matrices

A.1 Euclidean and Unitary Spaces

A.2 Subspaces, Linear Independent, 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

Bibliography

Index

## Product details

- No. of pages: 444
- Language: English
- Copyright: © Pergamon 1959
- Published: January 1, 1959
- Imprint: Pergamon
- eBook ISBN: 9781483222981

## About the Author

### Samuel Karlin

#### Affiliations and Expertise

Stanford University and The Weizmann Institute of Science