Mean Field Stochastic Adaptive Control

Abstract: For noncooperative games the mean field (MF) methodology provides decentralized strategies which yield Nash equilibria for large population systems in the asymptotic limit of an infinite (mass) population. The MF control laws use only the local information of each agent on its own state and own dynamical parameters, while the mass effect is calculated offline using the distribution function of (i) the population's dynamical parameters, and (ii) the population's cost function parameters, for the infinite population case. These laws yield approximate equilibria when applied in the finite population. In this paper, these a priori information conditions are relaxed, and incrementally the cases are considered where, first, the agents estimate their own dynamical parameters, and, second, estimate the distribution parameter in (i) and (ii) above. An MF stochastic adaptive control (SAC) law in which each agent observes a random subset of the population of agents is specified, where the ratio of the cardinality of the observed set to that of the number of agents decays to zero as the population size tends to infinity. Each agent estimates its own dynamical parameters via the recursive weighted least squares (RWLS) algorithm and the distribution of the population's dynamical parameters via maximum likelihood estimation (MLE). Under reasonable conditions on the population dynamical parameter distribution, the MF-SAC Law applied by each agent results in (i) the strong consistency of the self parameter estimates and the strong consistency of the population distribution function parameters; (ii) the long run average stability of all agent systems; (iii) a (strong) e-Nash equilibrium for the population of agents; and (iv) the a.s. equality of the long run average cost and the non-adaptive cost in the population limit.