Nash Equilibrium and Minimax Theorems via Variational Tools of Convex Analysis

TL;DR

This paper presents a unified variational approach to proving the existence of Nash equilibria and minimax theorems in infinite-dimensional spaces using convex analysis and fixed-point theorems.

Contribution

It introduces a simple variational proof for Nash equilibrium existence in Hilbert spaces and extends minimax theorems to locally convex topological vector spaces.

Findings

Unified variational proof for Nash equilibrium in Hilbert spaces

Extension of minimax theorems to locally convex spaces

Application of convex analysis and fixed-point theory

Abstract

In this paper, we first provide a simple variational proof of the existence of Nash equilibrium in Hilbert spaces by using optimality conditions in convex minimization and Schauder's fixed-point theorem. Then applications of convex analysis and generalized differentiation are given to the existence of Nash equilibrium and extended versions of von Neumann's minimax theorem in locally convex topological vector spaces. Our analysis in this part combines generalized differential tools of convex analysis with elements of fixed point theory revolving around Kakutani's fixed-point theorem and related issues.