Optima and Equilibria: An Introduction to Nonlinear Analysis

Jean-Pierre Aubin (auth.)9783540649830, 3540649832

Advances in game theory and economic theory have proceeded hand in hand with that of nonlinear analysis and in particular, convex analysis. These theories motivated mathematicians to provide mathematical tools to deal with optima and equilibria. Jean-Pierre Aubin, one of the leading specialists in nonlinear analysis and its applications to economics and game theory, has written a rigorous and concise-yet still elementary and self-contained- text-book to present mathematical tools needed to solve problems motivated by economics, management sciences, operations research, cooperative and noncooperative games, fuzzy games, etc. It begins with convex and nonsmooth analysis,the foundations of optimization theory and mathematical programming. Nonlinear analysis is next presented in the context of zero-sum games and then, in the framework of set-valued analysis. These results are applied to the main classes of economic equilibria. The text continues with game theory: noncooperative (Nash) equilibria, Pareto optima, core and finally, fuzzy games. The book contains numerous exercises and problems: the latter allow the reader to venture into areas of nonlinear analysis that lie beyond the scope of the book and of most graduate courses. -(See cont. News remarks)

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Table of contents :
Front Matter….Pages I-XVII
Introduction….Pages 1-6
Front Matter….Pages 7-7
Minimisation Problems: General Theorems….Pages 9-19
Convex Functions and Proximation, Projection and Separation Theorems….Pages 21-34
Conjugate Functions and Convex Minimisation Problems….Pages 35-56
Subdifferentials of Convex Functions….Pages 57-73
Marginal Properties of Solutions of Convex Minimisation Problems….Pages 75-86
Generalised Gradients of Locally Lipschitz Functions….Pages 87-99
Two-person Games. Fundamental Concepts and Examples….Pages 101-123
Two-person Zero-sum Games: Theorems of Von Neumann and Ky Fan….Pages 125-142
Solution of Nonlinear Equations and Inclusions….Pages 143-166
Introduction to the Theory of Economic Equilibrium….Pages 167-178
The Von Neumann Growth Model….Pages 179-188
n -person Games….Pages 189-209
Cooperative Games and Fuzzy Games….Pages 211-233
Front Matter….Pages 235-235
Exercises….Pages 237-302
Statements of Problems….Pages 303-347
Solutions to Problems….Pages 349-400
Back Matter….Pages 401-437