Approximate fixed point theorems in Banach spaces with applications in game theory

Abstract

In this paper some new approximate fixed point theorems for multifunctions in Banach spaces are presented and a method is developed indicating how to use approximate fixed point theorems in proving the existence of approximate Nash equilibria for non-cooperative games.

On the existence of ɛ-fixed points

“…The basic idea of the new definition is that in order to obtain an ε-fixed point it is sufficient to require an "approximate partial closeness" (see Theorems 3.1, 3.7 and Corollaries 3.2, 3.8). We remark that the multimaps in this article are defined on a not necessarily bounded set, in contrast to [1]. Moreover, we prove that even in the class of multimaps defined on bounded and convex domains, there exist examples of multimaps satisfying all conditions of Corollary 3.2 but which are not -closed, as required in [1, Theorem 2.1].…”

“…Without this property a Nash equilibrium need not exist. This fact has recently been studied by Brânzei, Morgan, Scalzo and Tijs in [1] where the existence of approximate fixed points for multimaps in a Banach space is obtained. The aforementioned authors introduce (for fixed ε > 0) the notion of ε-equilibrium (i.e.…”

“…Moreover, Corollary 3.5 considers multimaps defined in a set that is not necessarily bounded, whereas boundedness is required in [1]. Now we turn our attention to the existence of approximate fixed points in a Banach space that is not necessarily reflexive.…”

“…We present new approximate fixed point results for multimaps that extend, in a broad sense, those proved in [1]. To achieve these goals, we introduce the notion of a β-partially closed multimap and use the classic Glebov fixed point theorem [6] for partially closed multimaps.…”

“…Therefore, in the class of the multimaps examined in our fixed point propositions, the Glebov theorem does not apply. Moreover, by using a key proposition proved in [1] we describe how our results can be applied to obtain the existence of approximate Nash equilibria for -person strategic games. We end with an open problem (see Problem 6.3).…”