  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.
