On Noncooperative Games, Minimax Theorems, and Equilibrium Problems

In this chapter, we give an overview on the theory of noncooperative games. In the first part, we consider in detail zero-sum (constant-sum) games with two players having arbitrary strategy sets, under which necessary and sufficient conditions on the payoff function and the different (extended strategy) sets an equilibrium (saddle-point) strategy (for both players) exists. The existence of such an equilibrium strategy is equivalent to whether a so-called minimax theorem for the payoff function holds. The proof of such a result uses either the separation result for disjoint convex sets in finite dimensional linear spaces or the strong duality theorem for linear programming in combination with some elementary properties of compact sets. Both proof techniques are given together with a discussion of the most well-known minimax theorems that appeared in the literature. Also for the most famous minimax result given by Sion, we separately show an elementary proof avoiding the KKM lemma and using only the definition of connectedness. In the final part, we also consider n-person nonzero-sum noncooperative games. It is shown by a simple application of the same KKM lemma that a Nash equilibrium strategy (a generalization of a (saddle-point) equilibrium strategy for two players) exists under certain conditions on the payoff functions and the strategy sets. The main goal of this chapter is to discuss in detail and full generality the most elementary mathematical techniques for proving the existence of equilibrium points in noncooperative games (with an emphasis on two players).