Distributionally Robust Nash Equilibrium Seeking with Partial Observations and Distributed Communication

Abstract: In this work, we study stochastic one-shot games where agents' utilities depend on the collective strategy profiles of other agents as well as on some well-behaved randomness. While each decision-maker is agnostic to the random variable's underlying distribution, they have access to finitely many i.i.d. samples generated from it. We consider two cases: one where samples are shared; and another, more special one, where samples are individually accessible. To hedge against the unknown uncertainty, each agent plays a distributionally robust game and aims to maximize the worst-case expected utility over a Wasserstein ball around the sample average distribution. In this setting, we provide conditions under which the game has a non-empty set of distributionally robust Nash equilibria (DRoNE) and then characterize the closeness of the DRoNE set to the Nash equilibria (NE) of the associated stochastic game. We then propose an inertial, supported, better response, ascending supergradient dynamics ISBRAG that seeks the DRoNE's when the distributionally robust game possesses what we term as amicable supergradients. This forms the basis of a distributed version (d-ISBRAG) where agents estimate others' strategies by means of a dynamic consensus subroutine over a directed communication network. While initially the distributed algorithm works in the case where agents have individual samples, we later extend this to the case of shared observations under certain simplifying assumptions. This involves analyzing a tractable reformulation of the distributionally robust optimization problem and solving it in a distributed manner to compute the required supergradients. Simulations illustrate our results.