  We study mean-field control problems in discrete-time under the infinite
horizon average cost optimality criteria. We focus on both the finite
population and the infinite population setups. We show the existence of a
solution to the average cost optimality equation (ACOE) and the existence of
optimal stationary Markov policies for finite population problems under (i) a
minorization condition that provides geometric ergodicity on the collective
state process of the agents, and (ii) under standard Lipschitz continuity
assumptions on the stage-wise cost and transition function of the agents when
the Lipschitz constant of the transition function satisfies a certain bound.
For the infinite population problem, we establish the existence of a solution
to the ACOE, and the existence of optimal policies under the continuity
assumptions on the cost and the transition functions. Finally, we relate the
finite population and infinite population control problems: (i) we prove that
the optimal value of the finite population problem converges to the optimal
value of the infinite population problem as the number of agents grows to
infinity; (ii) we show that the accumulation points of the finite population
optimal solution corresponds to an optimal solution for the infinite population
problem, and finally (iii), we show that one can use the solution of the
infinite population problem for the finite population problem symmetrically
across the agents to achieve near optimal performance when the population is
sufficiently large.
