Robust Correlated Equilibrium: Definition and Computation

We study N-player finite games with costs perturbed due to time-varying disturbances in the underlying system and to that end, we propose the concept of Robust Correlated Equilibrium that generalizes the definition of Correlated Equilibrium. Conditions under which the Robust Correlated Equilibrium exists are specified, and a decentralized algorithm for learning strategies that are optimal in the sense of Robust Correlated Equilibrium is proposed. The primary contribution of the paper is the convergence analysis of the algorithm and to that end, we propose a modification of the celebrated Blackwell's Approachability theorem to games with costs that are not just time-average, as in the original Blackwell's Approachability Theorem, but also include the time-average of previous algorithm iterates. The designed algorithm is applied to a practical water distribution network with pumps being the controllers and their costs being perturbed by uncertain consumption due to the consumers. Simulation results show that each controller achieves no regret, and empirical distributions converge to the Robust Correlated Equilibrium.