The Complexity of Markov Equilibrium in Stochastic Games
This work addresses the computational hardness of equilibrium concepts in multi-agent reinforcement learning, showing fundamental limitations for practitioners in game theory and AI.
The paper proves that computing approximate stationary Markov coarse correlated equilibria (CCE) in general-sum stochastic games is computationally intractable, even under simplified conditions like two players and turn-based interactions, contrasting with efficient methods in normal-form games and single-agent RL. It also provides a decentralized algorithm for learning nonstationary Markov CCE policies with polynomial complexity, improving upon previous exponential requirements.
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.