On Fixed Points of Locally and Pointwise Contractive Set-Valued Maps with an Application to the Existence of Nash Equilibrium in Games

TL;DR

This paper proves fixed point theorems for set-valued maps with local or pointwise contraction conditions and applies these results to establish the existence of Nash equilibria in strategic games on metric spaces.

Contribution

It introduces new fixed point theorems for set-valued maps with local and pointwise contraction conditions and applies them to game theory.

Findings

Fixed points exist for locally and pointwise contractive set-valued maps.

Nash equilibria are guaranteed in a broad class of strategic games.

The results extend classical fixed point theorems to more general settings.

Abstract

We establish the existence of fixed points for set-valued maps defined on metric spaces and satisfying a pointwise or a local version of Banach's contraction property. As an application, we demonstrate the existence of Nash equilibrium in a general class of strategic games played on metric spaces of strategies.