Noncoercive and Noncontinuous Minimax Problems

Abstract

Techniques and principles of Minimax theory play a key role in many areas of research, including game theory, optimization. Arguably the most important result in zero-sum games, the Minimax Theorem was stated by John von Neumann in 1928 which was considered the starting point of game theory. Formally, von Neumann’s minimax theorem states: Let X ⊂ Rn and Y ⊂ Rm be compact convex sets. If f : X × Y → R is a continuous function that is concave-convex, i.e. (1) f (·, y) : X → R is concave for fixed y , and (2) f (x, ·) : Y → R is convex for fixed x. Then max x∈X min y∈Y f (x, y) = min y∈Y max x∈X f (x, y). In this paper, By using asymptotic function, as the main result, we prove Minimax Theorem under weaker assumptions of continuity and convexity, when the feasible set is an unbounded