Adaptive Mean Field Games for Large Population Coupled ARX Systems with Unknown Coupling Strength

This paper is concerned with decentralized tracking-type games for large population multi-agent systems with mean-field coupling. The individual dynamics are described by stochastic discrete-time auto-regressive models with exogenous inputs (ARX models), and coupled by terms of the unknown population state average (PSA) with unknown coupling strength. A two-level decentralized adaptive control law is designed. On the high level, the PSA is estimated based on the Nash certainty equivalence (NCE) principle. On the low level, the coupling strength is identified based on decentralized least squares algorithms and the estimate of the PSA. The decentralized control law is constructed by combining the NCE principle and Certainty equivalence (CE) principle. By probability limit theory, under mild conditions, it is shown that: (a) the closed-loop system is stable almost surely; (b) as the number of agents increases to infinity, the estimates of both the PSA and the coupling strength are asymptotically strongly consistent and the decentralized control law is an almost sure asymptotic Nash-equilibrium.