Nash Equilibrium Existence

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´ 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 strict, independently of the value of λ ∈ (0,1), we say that v is strictly quasiconcave. Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2006, Article ID 36135, Pages 1–4 DOI 10.1155/FPTA/2006/36135


Fixed points as Nash equilibria

A mapping f : X ⊂ Rm → X has a fixed point if there is x ∈ X such that f (x) = x. A vector x ∈ X is a fixed point of a correspondence Φ : X X if x ∈ Φ(x). Given a game = {I,Si ,vi }, in which each player i ∈ I = {1,2,...,n} is characterized by a set of strategies Si ⊂ Rni , and by an objective function vi : n=1 S j → R, a Nash equij librium is a vector s = (s1 ,s2 ,...,sn ) ∈ Πn=1 Si , such that vi (s) ≥ vi (si ,s−i ), for all si ∈ Si , for i all i ∈ I, where s−i = (s1 ,...,si−1 ,si+1 ,... ,sn ). Finally, let = {S : ∃n ∈ N, S ⊂ Rn is nonempty, convex, and compact}. 3. Main Results Consider the following statements. [Nash-1]. Given = {I,Si ,vi }, suppose that each set Si ∈ and that objective functions are continuous in its domains and strictly quasiconcave in its own strategy. Then there is a Nash equilibrium for . [Nash-2]. Given = {I,Si ,vi }, suppose that each set Si ∈ and that objective functions are continuous in its domains and quasiconcave in its own strategy. Then there is a Nash equilibrium for . [Brouwer]. Given X ∈ , every continuous function f : X → X has a fixed point.

[Kakutani∗ ]. Given X ∈ , every closed-graph correspondence Φ : X X, with Φ(x) ∈ for all x ∈ X, has a fixed point, provided that Φ(x) = m 1 π m (Φ(x)) for each x ∈ X ⊂ j j= m . (For each j ∈ {1,...,m}, the projections π m : Rm → R are defined by π m (x) = x , R j j j where x = (x1 ,...,xm ) ∈ Rm .) (The last property, Φ(x) = m 1 π m (Φ(x)), is not necessary j j= to assure the existence of a fixed point, provided that the other assumptions hold. However, when objective functions are quasiconcave, [Kakutani∗ ] is sufficient to assure the existence of a Nash equilibrium.) Our results are [Nash-1] → [Brouwer]. Proof. Given a...
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