## Finding Correlated Equilibrium of Constrained Markov Game: A Primal-Dual Approach

Abstract: Constrained Markov game is a fundamental problem that covers many applications, where multiple players compete with each other under behavioral constraints. The existing literature has proved the existence of Nash equilibrium for constrained Markov games, which turns out to be PPAD-complete and cannot be computed in polynomial time. In this work, we propose a surrogate notion of correlated equilibrium (CE) for constrained Markov games that can be computed in polynomial time, and study its fundamental properties. We show that the modification structure of CE of constrained Markov games is fundamentally different from that of unconstrained Markov games. Moreover, we prove that the corresponding Lagrangian function has zero duality gap. Based on this result, we develop the first primal-dual algorithm that provably converges to CE of constrained Markov games. In particular, we prove that both the duality gap and the constraint violation of the output policy converge at the rate $\mathcal{O}(\frac{1}{\sqrt{T}})$. Moreover, when adopting the V-learning algorithm as the subroutine in the primal update, our algorithm achieves an approximate CE with $\epsilon$ duality gap with the sample complexity $\mathcal{O}(H^9|\mathcal{S}||\mathcal{A}|^{2} \epsilon^{-4})$.