Discrete-time indefinite linear-quadratic mean field games and control: The finite-population case

In classical mean field games, the asymptotically optimal decentralized strategies are concerned for large-population systems. The computational complexity is reduced but a relatively large optimality loss may be caused in small-population systems. This paper is concerned with the optimal decentralized strategies for finite-population discrete-time linear quadratic mean field games and social control problems. The weighting matrices in the cost function are not assumed to be definite. We adopt the conditional expectations to transform the forward-backward stochastic difference equations (FBSDEs) of the finite-population system into the mean field FBSDEs of a representative player. For the noncooperative games, the solvability of decentralized Nash equilibria is characterized by the existence of a solution to a system of FBSDEs, together with a convexity condition. For the cooperative social control, the solvability of decentralized social optimal control is characterized by the existence of regular solutions to two Riccati equations. The proposed decentralized strategies are optimal in the case of finite-population, and also consistent with the classical asymptotic optimal strategies in the case of infinite-population. Therefore the computational complexity is significantly reduced without additional optimality loss.