The Complexity of Markov Equilibrium in Stochastic Games

TL;DR

This paper proves that computing approximate stationary Markov CCEs in general-sum stochastic games is computationally intractable, highlighting fundamental differences from normal-form games and single-agent RL, and offers a decentralized algorithm for nonstationary CCEs.

Contribution

It establishes the intractability of stationary Markov CCE computation in stochastic games and provides a polynomial-time decentralized algorithm for nonstationary CCE learning.

Findings

Computing approximate stationary Markov CCEs is intractable in general-sum stochastic games.

Normal-form games allow efficient computation of exact CCEs, unlike stochastic games.

A decentralized polynomial-time algorithm can learn nonstationary Markov CCE policies.

Abstract

We show that computing approximate stationary Markov coarse correlated equilibria (CCE) in general-sum stochastic games is computationally intractable, even when there are two players, the game is turn-based, the discount factor is an absolute constant, and the approximation is an absolute constant. Our intractability results stand in sharp contrast to normal-form games where exact CCEs are efficiently computable. A fortiori, our results imply that there are no efficient algorithms for learning stationary Markov CCE policies in multi-agent reinforcement learning (MARL), even when the interaction is two-player and turn-based, and both the discount factor and the desired approximation of the learned policies is an absolute constant. In turn, these results stand in sharp contrast to single-agent reinforcement learning (RL) where near-optimal stationary Markov policies can be efficiently learned. Complementing our intractability results for stationary Markov CCEs, we provide a decentralized algorithm (assuming shared randomness among players) for learning a nonstationary Markov CCE policy with polynomial time and sample complexity in all problem parameters. Previous work for learning Markov CCE policies all required exponential time and sample complexity in the number of players.