Introduction. the Nature of the Mathematical Theory of Decision Processes
1
1 The Background
2
2 The Classification of the Mathematics of Decision Problems
3
3 The Main Disciplines
Chapter 1. The Definition of a Game and the Min-Max Theorem
1 1.1 Introduction. Games in Normal Form
1
1.2 Examples
1
1.3 Choice of Strategies
1
1.4 The Min-Max Theorem for Matrix Games
1
1.5 General Min-Max Theorem
1.6 Problems
Notes and References
Chapter 2. The Nature of Optimal Strategies for Matrix Games
2 2.1 Properties of Optimal Strategies
2
2.2 Types of Strict Dominance
2
2.3 Construction of Optimal Strategies
2
2.4 Characterization of Extreme-Point Optimal Strategies
2
2.5 Completely Mixed Matrix Games
2
2.6 Symmetric Games
2.7 Problems
Notes and References
Chapter 3. Dimension Relations for Sets of Optimal Strategies
3
3.1 The Principal Theorems
3
3.2 Proof of Theorem 3.1.1
3
3.3 Proof of Theorem 3.1.2
3
3.4 The Converse of Theorem 3.1.2
3
3.5 Uniqueness of Optimal Strategies
3.6 Problems
Notes and References
Chapter 4. Solutions of Some Discrete Games
4
4.1 Colonel Blotto Game
4
4.2 Identification of Friend and Foe (I.F.F. Game)
4
4.3 Poker Game
4
4.4 An Advertising Example
4
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
5.1 Formulation of the Linear Programming Problem
5
5.2 The Linear Programming Problem and its Dual
5
5.3 The Principal Theorems of Linear Programming (Preliminary Results)
5
5.4 The Principal Theorems of Linear Programming (Continued)
5
5.5 Connections Between Linear Programming Problems and Game Theory
5
5.6 Extensions of the Duality Theorem
5
5.7 Warehouse Problem
5
5.8 Optimal Assignment Problem
5
5.9 Transportation and Flow Problem
5
5.10 Maximal-Flow, Minimal-Cut Theorem
5
5.11 The Caterer's Problem
5
5.12 Price Speculation Model
5.13 Problems
Notes and References
Chapter 6. Computational Methods for Linear Programming and Game Theory
6
6.1 The Simplex Method
6
6.2 Auxiliary Simplex Methods
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6.3 An Illustration of the Use of the Simplex Method
6
6.4 Computation of Network Flow
6
6.5 A Method of Approximating the Value of a Game
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6.6 Proof of the Convergence
6
6.7 A Differential-Equations Method for Determining the Value of a Game
6.8 Problems
Notes and References
Chapter 7. Nonlinear Programming
7
7.1 Concave Programming
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7.2 Examples of Concave Programming
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7.3 The Arrow-Hurwicz Gradient Method
7
7.4 The Vector Maximum Problem
7
7.5 Conjugate Functions
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7.6 Composition of Conjugate Functions
7
7.7 Conjugate Concave Functions
7
7.8 A Duality Theorem of Nonlinear Programming
7
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
8.1 Open and Closed Linear Leontief Models
8
8.2 The Theory of Positive Matrices
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8.3 Applications of the Theory of Positive Matrices to the Study of Linear Models of Equilibrium and Exchange
8
8.4 The Theory of Production
8
8.5 Efficient Points of a Leontief-Type Model
8
8.6 The Theory of Consumer Choice
8
8.7 Nonlinear Models of Equilibrium
8
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
9.1 Welfare Economics
9
9.2 The Stability of a Competitive Equilibrium
9
9.3 Local Stability
9
9.4 Global Stability of Price Adjustment Processes
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9.5 Global Stability (Continued)
9
9.6 A Difference-Equations Formulation of Global Stability
9
9.7 Stability and Expectations (Model I)
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9.8 Stability and Expectations (Model II)
9
9.9 The Von Neumann Model of an Expanding Economy
9
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
A.1 Euclidean and Unitary Spaces
A
A.2 Subspaces, Linear Independent, Basis, Direct Sums, Orthogonal Complements
A
A.3 Linear Transformations, Matrices, and Linear Equations
A
A.4 Eigenvalues, Eigenvectors, and the Jordan Canonical Form
A
A.5 Transposed, Normal, and Hermitian Matrices; Orthogonal Complement
A
A.6 Quadratic Form
A
A.7 Matrix-Valued Functions
A
A.8 Determinants; Minors, Cofactors
A
A.9 Some Identities
A
A.10 Compound Matrices
Appendix B. Convex Sets and Convex Functions
B B.1 Convex Sets in En
B
B.2 Convex Hulls of Sets and Extreme Points of Convex Sets
B
B.3 Convex Cones
B
B.4 Convex and Concave Functions
Appendix C. Miscellaneous Topics
C
C.1 Semicontinuous and Equicontinuous Functions
C
C.2 Fixed-Point Theorems
C
C.3 Set Functions and Probability Distributions
Bibliography
Index
