Decentralised dynamic games for large population stochastic multi-agent systems

In this study, the authors consider the large population dynamic games where each agent evolves according to a dynamic equation containing the input average of all agents. The long time average (LTA) cost that each agent aims to minimise is coupled with other agents’ states via a population state average (PSA), which is also known as the mean field term. In order to design decentralised controls, the Nash certainty equivalence is introduced. It is shown that the resulting decentralised mean field control laws lead the system to mean-consensus asymptotically as time goes to infinity. The stability property of the mass behaviour and the almost sure asymptotic Nash equilibrium property of the optimal controls are also guaranteed, and the case with non-linear system dynamics is also discussed. In addition, the influence of inaccurate mean field information on individual agent is analysed. Finally, they investigate the socially cooperative formulation where the objective is to minimise the social cost as the sum of all individual LTA costs containing the PSA. In this case, they show that the decentralised mean field social controls are the same as the mean field Nash controls for infinite population systems.