A characterization of Nash equilibrium for the games with random payoffs

We consider a two player random bimatrix game where each player is interested in the payoffs which can be obtained with certain confidence. The players’ payoff functions in such game theoretic problems are defined using chance constraints. We consider the case where the entries of each player’s random payoff matrix jointly follow a multivariate elliptically symmetric distribution. We show an equivalence between a Nash equilibrium problem and a global maximization of a certain mathematical program. The case where the entries of the payoff matrices are independent normal/Cauchy random variables are also considered. The case of independent normally distributed random payoffs can be viewed as a special case of a multivariate elliptically symmetric distributed random payoffs. For the Cauchy distribution case, we show that a Nash equilibrium problem is equivalent to a global maximization of a certain quadratic program. Our theoretical results are illustrated by considering a random bimatrix game between two manufacturing firms acting on the same market.