Stochastic Adaptive Nash Certainty Equivalence Control: Self-Identification Case

For noncooperative games the Nash Certainty Equivalence (NCE), or Mean Field (MF) methodology developed in previous work provides decentralized strategies which asymptotically yield Nash equilibria. The NCE (MF) control laws use only the local information of each agent on its own state evolution and knowledge of its own dynamical parameters, while the behaviour of the mass is precomputable from knowledge of the distribution of dynamical parameters throughout the mass population. Relaxing the a priori information condition introduces the methods of parameter estimation and stochastic adaptive control (SAC) into MF control theory. In particular one may consider incrementally the problems where the agents must estimate: (i) its own dynamical parameters, (ii) the distribution of the population’s dynamical parameters [1], and (iii) the distribution of the population’s cost function parameters [2]. In this paper we treat the first problem. Each agent estimates its own dynamical parameters via the recursive weighted least squares (RWLS) algorithm. Under reasonable conditions on the population dynamical parameter distribution, we establish: (i) the strong consistency of the selfparameter estimates; and that (ii) all agent systems are long run average L stable; (iii) the set of controls yields a (strong) -Nash equilibrium for all ; and (iv) in the population limit the long run average cost obtained is equal to the non-adaptive long run average cost.