Minimax Theorem

Abstract

This chapter deals with the Minimax Theorem and its proof, which is based on elementary results from convex analysis. The theorem states that for every matrix A, the average security levels of both players coincide. In a mixed policy, the min and max always commute. For every constant c, at least one of the players can guarantee a security level of c. The chapter first considers the statement of the Minimax Theorem before discussing the convex hull and the Separating Hyperplane Theorem, one of the key results in convex analysis. It then demonstrates how to prove the Minimax Theorem and presents the proof. It also shows the consequences of the Minimax Theorem and concludes with a practice exercise related to symmetric games and the corresponding solution.