Mathematical methods and theory in games, programming, and economics 数学方法及博弈论、程序和经济

The mathematics involved in decision making plays a crucial role in the analy-sis of management problems, in economic studies, military tactics and opera-tions research. Moreover, applications in these areas have resulted in the development of new types of mathematical structures, including game theory,linear and nonlinear programming and mathematical economics. In this book the author attempts to synthesize the concepts of game theory, programming theory and the concepts and techniques of mathematical economics into a sin-gle systematic theory.
The book is divided into two parts: (1) theory of matrix games and (2) linear and nonlinear programming and mathematical economics. Part One coverage includes: definition of a game and the min-max theorem; nature of optimal strategies of matrix games; dimension relations for sets of optimal strategies;solutions of some discrete games. Topics in Part Two include linear program-ming; computational methods for linear programming and game theory; non-linear programming; and mathematical methods in the study of economic mod-els. In both parts, the principles of game theory and programming are appliedto a variety of simplified problems based on economic models, business deci-sions and military tactics in an attempt to clarify the key mathematical con-cepts involved and suggest their applicability to other similar problems.Throughout, the presentation of each subject is self-contained and completely rigorous, but every effort has been made to bring out the conceptual depth and formal elegance of the overall theory.
Each chapter contains problems, with solutions, of various degrees of difficul-ty, many of them leading to extensions of the theory. Solutions to most of theproblems and hints for solving others are given at the end of each part.

INTRODUCTION. THE NATURE OF THE MATHEMATICAL THEORY OF DECI-SION 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.l 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 mchod 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
CHAPTER 8. MATHEMATICAL METHODS IN THE STUDY OF ECONOMIC MODELS
CHAPTER 9. MATHEMATICAL METHODS IN THE STUDY OF ECONOMIC MODELS(Continued)
SOLUTIONS TO PROBLEMS OF CHAPTERS 5-9
APPENDIX A. VECTOR SPACES AND MATRICES
APPENDIX B. CONVEX SETS AND CONVEX FUNCTIONS
APPENDIX C. MISCELLANEOUS TOPICS
BIBLIOGRAPHY
INDEX