Discrete-time mean field games in multi-agent systems

In this paper, we investigate the behavior of agents in mean field games where each agent evolves according to a dynamic equation containing the input average and seeks to minimize its long time average (LTA) cost encompassing a population state average (PSA), which is also known as the mean field term. Due to the informational burden resulting from the PSA coupling to the states of all agents, our idea is to find a deterministic function φ to approximate it. It is shown that φ is an approximation of the PSA as the population size N goes to infinity. The resulting decentralized mean field control laws lead the system to achieve mean-consensus asymptotically as time goes to infinity. Furthermore, the optimal controls generate an almost sure asymptotic Nash equilibrium, which implies that the LTA cost of each agent can reach its minimal value as the number of agents increases to infinity. Finally, we consider the socially optimal case where the basic objective is to minimize the social cost as the sum of the individual LTA cost containing the PSA. In this case, it is shown that the decentralized mean field social control strategies are the same as the mean field Nash controls for infinite population systems.