Nash's Existence Theorem for Non-compact Strategy Sets

Abstract

This paper generalizes the Fan-Knaster-Kuratowski-Mazurkiewicz (FKKM) lemma to the case of weak topology, and obtains the Ky Fan minimax inequality defined on non-empty non-compact convex subsets in reflexive Banach spaces, then we apply it to game theory and obtain Nash's existence theorem for non-compact strategy sets, together with John von Neumann's existence theorem in two-player zero-sum games.