Linear-Quadratic Stackelberg Mean Field Games and Teams with Arbitrary Population Sizes

Abstract: This paper addresses a linear-quadratic Stackelberg mean field (MF) games and teams problem with arbitrary population sizes, where the game among the followers is further categorized into two types: non-cooperative and cooperative, and the number of followers can be finite or infinite. The leader commences by providing its strategy, and subsequently, each follower optimizes its individual cost or social cost. A new de-aggregation method is applied to solve the problem, which is instrumental in determining the optimal strategy of followers to the leader's strategy. Unlike previous studies that focus on MF games and social optima, and yield decentralized asymptotically optimal strategies relative to the centralized strategy set, the strategies presented here are exact decentralized optimal strategies relative to the decentralized strategy set. This distinction is crucial as it highlights a shift in the approach to MF systems, emphasizing the precision and direct applicability of the strategies to the decentralized context. In the wake of the implementation of followers' strategies, the leader is confronted with an optimal control problem driven by high-dimensional forward-backward stochastic differential equations (FBSDEs). By variational analysis, we obtain the decentralized strategy for the leader. By applying the de-aggregation method and employing dimension expansion to decouple the high-dimensional FBSDEs, we are able to derive a set of decentralized Stackelberg-Nash or Stackelberg-team equilibrium solution for all players.