A further generalization of the Kakutani fixed theorem, with application to Nash equilibrium points

Abstract

Introduction.Kakutani's fixed point theorem [3]1 states that in Euclidean «-space a closed point to (nonvoid) convex set map of a convex compact set into itself has a fixed point.Kakutani showed that this implied the minimax theorem for finite games.The object of this note is to point out that Kakutani's theorem may be extended to convex linear topological spaces, and implies the minimax theorem for continuous games with continuous payoff as well as the existence of Nash equilibrium points.