Optimality of adaption based Mean Field control laws in leader-follower stochastic collective dynamics

We study the optimality properties of maximum likelihood ratio estimation based Mean Field (Nash Certainty Equivalence) control laws in a leader-follower stochastic collective dynamics model. In this formulation the leaders track a convex combination of their centroid together with a certain reference trajectory which is unknown to the followers, and each follower reacts by tracking the centroid of the leaders. The followers use a maximum likelihood estimator (based on a fixed ratio sample of the population of the leaders´ trajectories) to identify the member of a given finite class of models which is generating the reference trajectory of the leaders. Subject to reasonable conditions, it is shown that each adaptive follower identifies the true reference trajectory model in finite time with probability one as the leaders´ population goes to infinity. It is also shown that the leaders´ control laws possess an almost sure ε-Nash equilibrium property with respect to all other leaders. In this paper we show that the system performance for the adaptive followers is almost surely ε-optimal with respect to the leaders.