Teaching Nonzero Sum Games Using a Diagrammatic Determination of Equilibria

Abstract

We present a diagrammatic method that allows the determination of all Nash equilibria of 2 × M nonzero sum games, extending thus the well known diagrammatic techniques for 2 × M zero sum and 2 × 2 nonzero sum games. We show its appropriateness for teaching purposes by analyzing modified versions of the prisoners' dilemma, the battle of the sexes, as well as of the zero sum game of matching pennies. We then use the method to give simple proofs for the existence of Nash equilibria in all 2 × M games, for both the nonzero sum (Nash existence theorem) and the zero sum case (Minimax theorem). We also prove in the same spirit the remarkable general fact that in a nondegenerate 2 × M nonzero sum game there is an odd number of equilibria.