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We study mean field games and corresponding N -player games in continuous time over a finite time horizon where the position of each agent belongs to a finite state space. As opposed to previous works on finite state mean field games, we use a probabilistic representation of the system dynamics in terms of stochastic differential equations driven by Poisson random measures. Under mild assumptions, we prove existence of solutions to the mean field game in relaxed open-loop as well as relaxed feedback controls. Relying on the probabilistic representation and a coupling argument, we show that mean field game solutions provide symmetric ε N -Nash equilibria for the N -player game, both in open-loop and in feedback strategies (not relaxed), with ε N ≤ constant √ N . Under stronger assumptions, we also find solutions of the mean field game in ordinary feedback controls and prove uniqueness either in case of a small time horizon or under monotonicity. Date

We show the convergence of finite state symmetric N -player differential games, where players control their transition rates from state to state, to a limiting dynamics given by a finite state Mean Field Game system made of two coupled forward-backward ODEs. We exploit the so-called Master Equation, which in this finite-dimensional framework is a first order PDE in the simplex of probability measures, obtaining the convergence of the feedback Nash equilibria, the value functions and the optimal trajectories. The convergence argument requires only the regularity of a solution to the Master Equation. Moreover, we employ the convergence results to prove a Central Limit Theorem and a Large Deviation Principle for the evolution of the N -player empirical measures. The well-posedness and regularity of solution to the Master Equation are also studied, under monotonicity assumptions.Date: February 9, 2018. 1991 Mathematics Subject Classification. 60F05, 60F10, 60J27, 60K35, 91A10, 93E20.