Chance-constrained games: A mathematical programming approach

We consider a two player bimatrix game where the entries of the payoff matrices are random variables. We formulate this problem as a chance-constrained game by considering that the payoff of each player is defined using a chance constraint. We consider the case where the entries of the payoff matrices are independent normal/Cauchy random variables. We show a one-to-one correspondence between a Nash equilibrium of a chance-constrained game corresponding to normal distribution and a global maximum of a certain mathematical program. Further as a special case where the payoffs are independent and identically distributed normal random variables, we show that a strategy pair, where each player’s strategy is a uniform distribution over his action set, is a Nash equilibrium. We show a one-to-one correspondence between a Nash equilibrium of a chance-constrained game corresponding to Cauchy distribution and a global maximum of a certain quadratic program.