## Optima and Equilibria: An Introduction to Nonlinear AnalysisAdvances 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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### Contents

II | 9 |

III | 10 |

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### Common terms and phrases

A C L(X Algebraic associate assumptions belongs Brouwer's Fixed-point Theorem C C(x closed set closed values conjugate function consider continuous linear operator continuous mapping converges convex and closed convex and lower convex closed cone convex closed subset convex compact subset convex function Corollary decision rules Deduce defined Definition denote df(x differentiable Emil exists a solution f is convex finite fixed point function f graph Graph(F Hilbert space implies inclusion locally Lipschitz loss functions lower semi lower semi-continuous function map F mathematics minimisation problem multilosses NK(x non-cooperative equilibrium non-empty nontrivial function normal cone Pareto Pareto optimal partition of unity players positively homogeneous problem h(y satisfying set-valued map Shapley value Show strategy pair subdifferential sup inf Suppose that f surjective tangential condition theory TK(x topology upper semi-continuous variational inequality Vx C K Walrasian equilibrium whence

### Popular passages

Page 2 - The next subject to be mentioned concerns the static or dynamic nature of the theory. We repeat most emphatically that our theory is thoroughly static. A dynamic theory would unquestionably be more complete and therefore preferable. But there is ample evidence from other branches of science that it is futile to try to build one as long as the static side is not thoroughly understood. On the other hand, the reader may object to some definitely dynamic arguments which were made in the course of our...

Page 2 - Finally let us note a point at which the theory of social phenomena will presumably take a very definite turn away from the existing patterns of mathematical physics.

Page 2 - Our static theory specifies equilibria— ie solutions . . . which are sets of imputations. A dynamic theory— when one is found— will probably describe the changes in terms of simpler concepts: of a single imputation — valid at the moment under consideration— or something similar. This indicates that the formal structure of this part of the theory — the relationship between statics and dynamics — may be generically different from that of the classical physical theories.