Actor-Critic Provably Finds Nash Equilibria of Linear-Quadratic Mean-Field Games
This provides a provably convergent model-free reinforcement learning method for multi-agent systems, addressing a foundational challenge in mean-field game theory with potential applications in economics and control.
The paper tackles the problem of finding Nash equilibria in discrete-time mean-field Markov games with linear-quadratic structures, proposing a mean-field actor-critic algorithm that converges to the equilibrium at a linear rate without requiring knowledge of the dynamics model.
We study discrete-time mean-field Markov games with infinite numbers of agents where each agent aims to minimize its ergodic cost. We consider the setting where the agents have identical linear state transitions and quadratic cost functions, while the aggregated effect of the agents is captured by the population mean of their states, namely, the mean-field state. For such a game, based on the Nash certainty equivalence principle, we provide sufficient conditions for the existence and uniqueness of its Nash equilibrium. Moreover, to find the Nash equilibrium, we propose a mean-field actor-critic algorithm with linear function approximation, which does not require knowing the model of dynamics. Specifically, at each iteration of our algorithm, we use the single-agent actor-critic algorithm to approximately obtain the optimal policy of the each agent given the current mean-field state, and then update the mean-field state. In particular, we prove that our algorithm converges to the Nash equilibrium at a linear rate. To the best of our knowledge, this is the first success of applying model-free reinforcement learning with function approximation to discrete-time mean-field Markov games with provable non-asymptotic global convergence guarantees.