Distributionally robust games with an application to supply chain

In this paper, we propose a distributionally robust optimization approach for N-player, nonzero sum finite state/action games with incomplete information where the payoff matrix is stochastic with an imprecise distribution which is assumed to be attached to an a-prior known set. Our model is different from the robust game theory which presents a robust optimization approach to game theory with the uncertain payoff matrix in a compact convex set without probabilistic information which can lead to overly conservative solutions. A distributionally robust approach is used to cope with our setting in the games by combining the stochastic optimization approach and the robust optimization approach which can be called the distributionally robust games. We show that the existence of the equilibria for the distributionally robust games. The computation method for equilibrium point, with the first- and second information about the uncertain payoff matrix, can be reformulated as semidefinite programming problems which can be tractably realized. A two-echelon supply chain competition with demand uncertainty is analyzed by applying the distributionally robust game theory.