Robust linear quadratic mean field social control: A direct approach

Abstract

This paper investigates a robust linear quadratic mean field team control problem. The model involves a global uncertainty drift which is common for a large number of weakly-coupled interactive agents. All agents treat the uncertainty as an adversarial agent to obtain a "worst case'' disturbance. The direct approach is applied to solve the robust social control problem, where the state weight is allowed to be indefinite. Using variational analysis, we first obtain a set of forward-backward stochastic differential equations (FBSDEs) and the centralized controls which contain the population state average. Then the decentralized feedback-type controls are designed by mean field heuristics. Finally, the relevant asymptotically social optimality is further proved under proper conditions.

A class of mean‐field games with optimal stopping and its applications

“…After solving the problem, the decentralized strategies can be designed. Another route is called the direct approach (see previous works [8, 9, 37, 38]), which starts by solving the $$$ N $$$‐player game problem and obtain the centralized optimal strategies in a high dimensional environment. After that, the decentralized optimal controls can be derived by letting $$$ N $$$ goes to infinity.…”

“…We write the above relative performance functional by  i (𝜏 i , 𝜏 −i ) to emphasize its dependence on both 𝜏 i and 𝜏 −i due to the weakly coupling structure in payoff functionals. In (9),…”

“…Remark 5. The functional (31) can be viewed as a special case of the payoff functional (9) given in Problem (I) by setting 𝜃 = 0, l 1 = 0, l 2 = 1. Moreover, in (31), the state-average excludes the individual state of  i .…”

“…As a consequence, one may design a set of decentralized strategies for the large, however, finite population system originally analyzed, where each agent needs only to know its own state information and mass-effect computed off-line. Some recent literature can be found in previous studies [2][3][4][5] for the study of mean-field games; previous works [6][7][8][9] for mean-field teams; other works [10,11] for models with a major player, and the literature [6,12] for models with a leader-follower structure. The mean-field game can be found their applications in various different domains (e.g., engineering, social science, economics and finance, and medicine).…”