Nash Equilibria via Duality and Homological Selection

TL;DR

This paper introduces a novel homological approach to prove the existence of Nash equilibria, relaxing traditional multilinearity assumptions using advanced topological tools like the Dold-Thom Theorem and intersection theory.

Contribution

It develops a homological selection theorem and applies it to establish Nash equilibria without requiring cost functions to be multilinear.

Findings

Established a homological selection theorem using the Dold-Thom Theorem.

Proved existence of Nash equilibria under relaxed conditions.

Applied intersection theory and Poincaré Duality in the analysis.

Abstract

Given a multifunction from $X$ to the $k-$fold symmetric product $Sym_k(X)$, we use the Dold-Thom Theorem to establish a homological selection Theorem. This is used to establish existence of Nash equilibria. Cost functions in problems concerning the existence of Nash Equilibria are traditionally multilinear in the mixed strategies. The main aim of this paper is to relax the hypothesis of multilinearity. We use basic intersection theory, Poincar\'e Duality in addition to the Dold-Thom Theorem.