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Báo cáo hóa học: " FIXED POINTS AS NASH EQUILIBRIA"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: FIXED POINTS AS NASH EQUILIBRIA | FIXED POINTS AS NASH EQUILIBRIA JUAN PABLO TORRES-MARTINEZ Received 27 March 2006 Revised 19 September 2006 Accepted 1 October 2006 The existence of fixed points for single or multivalued mappings is obtained as a corollary of Nash equilibrium existence in finitely many players games. Copyright 2006 Juan Pablo Torres-Martinez. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction In game theory the existence of equilibrium was uniformly obtained by the application of a fixed point theorem. In fact Nash 3 4 shows the existence of equilibria for noncooperative static games as a direct consequence of Brouwer 1 or Kakutani 2 theorems. More precisely under some regularity conditions given a game there always exists a correspondence whose fixed points coincide with the equilibrium points of the game. However it is natural to ask whether fixed points arguments are in fact necessary tools to guarantee the Nash equilibrium existence. In this direction Zhao 5 shows the equivalence between Nash equilibrium existence theorem and Kakutani or Brouwer fixed point theorem in an indirect way. However as he points out a constructive proof is preferable. In fact any pair of logical sentences A and B that are true will be equivalent in an indirect way . For instance to show that A implies B it is sufficient to repeat the proof of B. For this reason we study conditions to assure that fixed points of a continuous function or of a closed-graph correspondence can be attained as Nash equilibria of a noncooperative game. 2. Definitions Let Y c R be a convex set. A function v Y R is quasiconcave if for each A e 0 1 we have v Ay1 1 - A y2 min v y1 v y2 for all y1 y2 e Y X Y. Moreover if for each pair y1 y2 e Y X Y such that y1 y2 the inequality above is strict independently of the value of A e 0 1 we say that v is strictly