Mathematical Methods of Game and Economic Theory
2nd Edition
Published on 1. July 1982
613 pages
978-0-08-087528-6 (ISBN)
Description
This book presents a unified treatment of optimization theory, game theory and a general equilibrium theory in economics in the framework of nonlinear functional analysis. It not only provides powerful and versatile tools for solving specific problems in economics and the social sciences but also serves as a unifying theme in the mathematical theory of these subjects as well as in pure mathematics itself.
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07/1982
2nd Edition
North-Holland
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Content
- Front Cover
- Mathematical Methods of Game and Economic Theory
- Copyright Page
- CONTENTS
- PREFACE
- SUMMARY OF RESULTS: A GUIDELINE FOR THE READER
- CONTENTS OF OTHER POSSIBLE COURSES
- NOTATIONS
- PART I: OPTIMIZATION AND CONVEX ANALYSIS
- CHAPTER 1. MINIMIZATION PROBLEMS AND CONVEXITY
- 1.1. Strategy sets and loss functions
- 1.2. Decomposition principle
- 1.3. Mixed strategies and convexity
- 1.4. Indicators, support functions and gauges
- CHAPTER 2. EXISTENCE, UNIQUENESS AND STABILITY OF OPTIMAL SOLUTIONS
- 2.1. Existence and uniqueness of an optimal solution
- 2.2. Minimization of quadratic functionals on convex sets
- 2.3. Minimization of quadratic functionals on subspaces
- 2.4. Perturbation by linear forms: conjugate functions
- 2.5. Stability properties: an introduction to correspondences
- CHAPTER 3. COMPACTNESS AND CONTINUITY PROPERTIES
- 3.1. Lower semi-compact functions
- 3.2. Proper maps and preimages of compact subsets
- 3.3. Continuous convex functions
- 3.4. Continuous Convex functions (continuation)
- CHAPTER 4. DIFFERENTIABILITY AND SUBDIFFERENTIABILITY: CHARACTERIZATION OF OPTIMAL SOLUTIONS
- 4.1. Subdifferentiability
- 4.2. Differentiability and variational inequalities
- 4.3. Differentiability from the.right
- 4.4. Local e-subdifferentiability and perturbed minimization problems
- CHAPTER 5. INTRODUCTION TO DUALITY THEORY
- 5.1. Dual problem and Lagrange multipliers
- 5.2. Case of linear constraints: extremality relations
- 5.3. Existence of Lagrange multipliers in the case of a finite number of constraints
- PART II: GAME THEORY AND THE WALRAS MODEL OF ALLOCATION OF RESOURCES
- CHAPTER 6. TWO-PERSON GAMES: AN INTRODUCTION
- 6.1. Some solution concepts
- 6.2. Examples: some finite games
- 6.3. Example: Analysis of duopoly
- 6.4. Example: Edgeworth economic game
- 6.5. Two-person zero-sum games
- CHAPTER 7. TWO-PERSON ZERO-SUM GAMES: EXISTENCE THEOREMS
- 7.1. The fundamental existence theorems
- 7.2. Extension of games without and with exchange of informations
- 7.3. Iterated games
- CHAPTER 8. THE FUNDAMENTAL ECONOMIC MODEL: WALRAS EQUILIBRIA
- 8.1. Description of the model
- 8.2. Existence of a Walras equilibrium
- 8.3. Demand correspondences defined by loss functions
- 8.4. Economies with producers
- CHAPTER 9. NON-COOPERATIVE n-PERSON GAMES
- 9.1. Existence of a non-cooperative equilibrium
- 9.2. Case of quadratic loss functions
- application to Walras-Cournot equilibria
- 9.3. Constrained non cooperative games and fixed point theorems
- 9.4. Non-cooperative Walras equilibria
- CHAPTER 10. MAIN SOLUTION CONCEPTS OF COOPERATIVE GAMES
- 10.1. Behavior of the whole set of players: Pareto strategies
- 10.2. Selection of Pareto strategies and imputations
- 10.3. Behavior of coalitions of players: the core
- 10.4. Behavior of fuzzy coalitions: the fuzzy core
- 10.5. Selection of elements of the core: cooperative equilibrium and nucleolus
- CHAPTER 11. GAMES WITH SIDE-PAYMENTS
- 11.1. Core of a fuzzy game with side-payments
- 11.2. Core of a game with side-payments
- 11.3. Values of fuzzy games
- 11.4. Shapley value and nucleolus of games with side-payments
- CHAPTER 12. GAMES WITHOUT SIDE-PAYMENTS
- 12.1. Equivalence between the fuzzy core and the set of equilibria
- 12.2. Non-emptiness of the fuzzy core uf a balanced game
- 12.3. Qmvalence between the fuzzy core of an economy and the set of Walras allocations
- PART III: NON-LINEAR ANALYSIS AND OPTIMAL CONTROL THEORY
- CHAPTER 13. MINIMAX TYPE INEQUALITIES. MONOTONE CORRESPONDENCES AND ?-CONVEX FUNCTIONS
- 13.1. Relaxation of compactness assumptions
- 13.2. Relaxation of continuity assumptions: variational inequalities, for monotone correspondences
- 13.3. Relaxation of convexity assumptions
- CHAPTER 14. INTRODUCTION TO CALCULUS OF VARIATIONS AND OPTIMAL CONTROL
- 14.1. Duality in infinite dimensional spaces
- 14.2. Duality in the case of non-convex integral criterion and constraints
- 14.3 Duality in calculus of variations
- 14.4. Optimal control and impulsive control problems
- CHAPTER 15. FIXED POINT THEOREMS, QUASI-VARIATIONAL INEQUALITIES AND CORRESPONDENCES
- 15.1. Fixed point and surjectivity theorems for correspondences
- 15.2. Quasi-variational inequalities
- 15.3. Other properties and examples of upper and lower semi-continuous correspondences
- APPENDIX A. SUMMARY OF LINEAR FUNCTIONAL ANALYSIS
- 1. Hahn-Banach theorems
- 2. Paired spaces
- 3. Topologies of uniform convergence
- 4. Topologies associated with a duality pairing
- 5. The Banach-Steinhauss theorem
- APPENDIX B. THE KNASTER-KURATOWSKI-MAZURKIEWICZ LEMMA
- 1. Barycentric-subdivision of simplexes
- 2. Sequence of barycentric subdivisions
- 3. The Sperner lemma
- 4. The Knaster-Kuratowski-Mazurkiewicz lemma
- 5. The Brouwer theorem
- APPENDIX C. LYAPUNOV'S THEOREM ON THE RANGE OF A VECTOR VALUED MEASURE
- COMMENTS
- REFERENCES
- SUBJECT INDEX
