By Jean-Pierre Aubin (Eds.)

ISBN-10: 0444851844

ISBN-13: 9780444851840

This e-book offers a unified therapy of optimization idea, video game conception and a common equilibrium concept in economics within the framework of nonlinear sensible research. It not just presents robust and flexible instruments for fixing particular difficulties in economics and the social sciences but additionally serves as a unifying subject matter within the mathematical idea of those topics in addition to in natural arithmetic itself.

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**Extra resources for Mathematical Methods of Game and Economic Theory**

**Example text**

The sections S(f,A) are contained in X = Dom fxand thus are relatively compact when X is relatively compact. 0 Of course, there are examples where fx is lower semi-continuous on U when Ch. 11 EXISTENCE AND UNIQUENESS OF AN OPTIMAL SOLUTION 45 X is not closed in U and examples where f x is lower semi-compact when X is not relatively compact. 2. Existence of an optimal solution From ( 5 ) and Definition I we deduce the following theorem. Theorem 1. (Weierstrass). Let U be a topological space and let f be a function mapping U into 1- m, -1.

For this purpose, we use the bipolar theorem: Theorem 3. Suppose that X is a closed convex set containing 0 and that the interior of its recession cone is non-empty (in the Mackey topology). Then its barrier cone P(X") is closed and (33) P(X*) = P,(X)-. -(X)-. Ch. 41 INDICATORS, SUPPORT FUNCTIONS A&D GAUGES 35 Proof. Suppose that I belongs to the interior of the recession cone P J X ) . Then there exists a neighborhood K* of a such that Z+KX c P&). Such a neighborhood is the polar subset of a ball of positive radius K(which is compact) when U is a finite dimensional space or the polar subset of a weak compact convex subset containing 0 when U is supplied with the Mackey topology.

Either I p = 0 E P(X*) or p f 0. In this case, we can suppose that p,, # 0 for all p. Then q,, = -p,,/(p,,, 2 ) belongs to the compact subset P. Hence a subsequence q, converges to an element q E P. On the other hand, the subsequence (pv,3) converges to ( p , I)=s 0. Therefore the subsequence p y = (p,,, 2)q, converges to p =- (p, 2) q which belongs to P(X#). 5. Interpretation :production sets and projt functions Let U be a commodity space. We shall represent each particular realization of the pcoduction of a firm by a pair {x, y } consisting of an output vector x E U and an input vector y E U.

### Mathematical Methods of Game and Economic Theory by Jean-Pierre Aubin (Eds.)

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