Wasserstein Distributionally Robust Nash Equilibrium Seeking with Heterogeneous Data: A Lagrangian Approach

We study a class of distributionally robust games where agents are allowed to heterogeneously choose their risk aversion with respect to distributional shifts of the uncertainty. In our formulation, heterogeneous Wasserstein ball constraints on each distribution are enforced through a penalty function leveraging a Lagrangian formulation. We then formulate the distributionally robust game as a variational inequality problem, and show that under certain assumptions the original seemingly infinite-dimensional Nash equilibrium problem is equivalent to a multi-agent but finite-dimensional variational inequality problem with a strongly monotone mapping. Due to the inner maximization problem, it is however still challenging to calculate a distributionally robust Nash equilibrium. To this end, we design an approximate Nash equilibrium seeking algorithm and prove convergence of the average regret to a quantity that diminishes with the number of iterations, thus learning the desired equilibrium up to an a priori specified accuracy. Numerical simulations corroborate our theoretical findings.