Mixed Equilibrium in 2 × 2 Normal Form Games: When Burning Money Is Rational

Abstract

We discuss the rationality of burning money behavior from a new perspective: the mixed Nash equilibrium in 2 × 2 normal form games. We support our argument analyzing the first-order derivatives of the players' mixed equilibrium expected utility payoffs with respect to their own utility payoffs. We establish necessary and sufficient conditions that guarantee the existence of negative derivatives. In particular, games with negative derivatives are the ones that create incentives for burning money behavior since such behavior in these games improves the player's mixed equilibrium expected utility payoff. We show that a negative derivative of some player i's mixed equilibrium expected utility payoff occurs if, and only if, he has a strict preference for one of the strategies of the other player. Moreover, negative derivatives always occur when they are taken with respect to player i's highest and lowest game utility payoffs. We also present sufficient conditions that ensure that such derivatives are always non-negative in finite normal form games.