Markov-Nash equilibria in mean-field games with discounted cost

In this paper, we consider discrete-time dynamic games of the mean-field type with a finite number N of agents subject to an infinite-horizon discounted-cost optimality criterion. The state space of each agent is a Polish space. At each time, the agents are coupled through the empirical distribution of their states, which affects both the agents' individual costs and their state transition probabilities. We introduce a new solution concept of the Markov-Nash equilibrium, under which a policy is player-by-player optimal in the class of all Markov policies. Under mild assumptions, we demonstrate the existence of a mean-field equilibrium in the infinite-population limit N → ∞, and then show that the policy obtained from the mean-field equilibrium is approximately Markov-Nash when the number of agents N is sufficiently large.