Nash Equilibrium Existence
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. ı Copyright © 2006 Juan Pablo Torres-Mart´nez. 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 ⊂ Rn be a convex set. A function v : Y → R is quasiconcave if, for each λ ∈ (0,1), we have v(λy1 + (1 − λ)y2 ) ≥ min{v(y1 ),v(y2 )}, for all (y1 , y2 ) ∈ Y × Y . Moreover, if for each pair (y1 , y2 ) ∈ Y × Y such that y1 = y2 the inequality above is
´ 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. ı Copyright © 2006 Juan Pablo Torres-Mart´nez. 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 ⊂ Rn be a convex set. A function v : Y → R is quasiconcave if, for each λ ∈ (0,1), we have v(λy1 + (1 − λ)y2 ) ≥ min{v(y1 ),v(y2 )}, for all (y1 , y2 ) ∈ Y × Y . Moreover, if for each pair (y1 , y2 ) ∈ Y × Y such that y1 = y2 the inequality above is
References: ¨ [1] L. E. J. Brouwer, Uber Abbildung von Mannigfaltigkeiten, Mathematische Annalen 71 (1912), no. 4, 598. [2] S. Kakutani, A generalization of Brouwer’s fixed point theorem, Duke Mathematical Journal 8 (1941), no. 3, 457–459. [3] J. F. Nash, Equilibrium points in n-person games, Proceedings of the National Academy of Sciences of the United States of America 36 (1950), no. 1, 48–49. , Non-cooperative games, Annals of Mathematics. Second Series 54 (1951), 286–295. [4] [5] J. Zhao, The equivalence between four economic theorems and Brouwer’s fixed point theorem, Working Paper, Departament of Economics, Iowa State University, Iowa, 2002. ´ Juan Pablo Torres-Mart´nez: Department of Economics, Pontif´cia Universidade Catolica do Rio de ı ı Janeiro (PUC-Rio), Marquˆ s de S˜o Vicente 225, Rio de Janeiro 22453-900, Brazil e a E-mail address: jptorres martinez@econ.puc-rio.br