Nash Equilibrium Existence without Convexity

TL;DR

This paper establishes the existence of pure-strategy Nash equilibria in certain nonconvex games by using topological methods, extending classical results to broader strategy spaces and connecting to the fundamental theorem of algebra.

Contribution

It introduces topological conditions under which pure Nash equilibria exist in nonconvex strategy spaces, a significant extension of classical convexity-based results.

Findings

Pure-strategy Nash equilibria exist under topological conditions.

Strategy spaces as Euclidean neighborhood retracts ensure equilibrium existence.

Application to prove the fundamental theorem of algebra.

Abstract

In this paper, I prove the existence of a pure-strategy Nash equilibrium for a large class of games with nonconvex strategy spaces. Specifically, if each player's strategies form a compact, connected Euclidean neighborhood retract and if all best-response correspondences are null-homotopic, then the game has a pure-strategy Nash equilibrium. As an application, I show how this result can prove the fundamental theorem of algebra.