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# Learning in Multi-Player Stochastic Games

*Proceedings of the Thirty-Seventh Conference on Uncertainty in Artificial Intelligence*, PMLR 161:1927-1937, 2021.

#### Abstract

We consider the problem of simultaneous learning in stochastic games with many players in the finite-horizon setting. While the typical target solution for a stochastic game is a Nash equilibrium, this is intractable with many players. We instead focus on variants of

*correlated equilibria*, such as those studied for extensive-form games. We begin with a hardness result for the adversarial MDP problem: even for a horizon of 3, obtaining sublinear regret against the best non-stationary policy is NP-hard when both rewards and transitions are adversarial. This implies that convergence to even the weakest natural solution concept—normal-form coarse correlated equilibrium—is not possible via black-box reduction to a no-regret algorithm even in stochastic games with constant horizon (unless $NP\subseteqBPP$). Instead, we turn to a different target: algorithms which*generate*an equilibrium when they are used by all players. Our main result is algorithm which generates an*extensive-form*correlated equilibrium, whose runtime is exponential in the horizon but polynomial in all other parameters. We give a similar algorithm which is polynomial in all parameters for “fast-mixing” stochastic games. We also show a method for efficiently reaching normal-form coarse correlated equilibria in “single-controller” stochastic games which follows the traditional no-regret approach. When shared randomness is available, the two generative algorithms can be extended to give simultaneous regret bounds and converge in the traditional sense.