Stackelberg equilibrium with social optima in linear-quadratic-Gaussian mean-field system

A linear quadratic social optimal problem with a leader and a vast amount of weakly coupled followers is investigated in this paper. To be specific, the followers cooperate to minimize the social cost after the leader first announces his/her strategy, while the leader will then optimize his/her own cost functional on consideration of the followers' reactions. We assume that the controls are involved in both the followers' and the leader's state equations. Meanwhile, the followers' average control and average state terms enter into the drift term of the leader's state equation, reflecting that the leader's state is influenced by the followers' decisions and states. In order to obtain the asymptotic optimal strategies, two corresponding auxiliary problems are present using variational and person-by-person technologies. Furthermore, we construct the decentralized strategies for this leader-follower system by virtue of a solvable consistency condition system and the corresponding asymptotic Stackelberg equilibrium is demonstrated rigorously. Eventually, a numerical example is presented to illustrate the theoretical results.