Before revision:  In the finite horizon case or with a discounted cost and an infinite horizon, we show that when the number of particles becomes large, the optimal cost of the system converges almost surely to the optimal cost of a discrete deterministic system (the " optimal mean field " ). Convergence also holds for optimal policies.
After revision:  In the finite horizon case or with a discounted cost and an infinite horizon, we show that when the number of particles becomes large, the optimal cost of the system converges almost surely to the optimal cost of a discrete deterministic system (the " optimal mean field '' ). Convergence also holds for optimal policies.