Linear Quadratic Mean Field Stackelberg Games: Open-loop and Feedback Solutions

TL;DR

The paper tackles a linear-quadratic mean-field Stackelberg game with a single leader and a large population of followers. It develops open-loop solutions by solving a centralized social control problem for followers via MF forward-backward SDEs and characterizes the leader's optimal response, establishing an asymptotic Stackelberg equilibrium. It also derives decentralized feedback strategies for both followers and the leader using the matrix maximum principle and Riccati-type equations, with costs expressed in closed form. A numerical study compares the two solution concepts, showing that feedback often improves follower performance while open-loop may benefit the leader depending on the interaction parameter, highlighting trade-offs in hierarchical MF control.

Abstract

This paper investigates open-loop and feedback solutions of linear quadratic mean field (MF) games with a leader and a large number of followers. The leader first gives its strategy and then all the followers cooperate to optimize the social cost as the sum of their costs. By variational analysis with MF approximations, we obtain a set of open-loop controls of players in terms of solutions to MF forward-backward stochastic differential equations (FBSDEs), which is further shown be to an asymptotic Stackelberg equilibrium. By applying the matrix maximum principle, a set of decentralized feedback strategies is constructed for all the players. For open-loop and feedback solutions, the corresponding optimal costs of all players are explicitly given by virtue of the solutions to two Riccati equations, respectively. The performances of two solutions are compared by the numerical simulation.

Figures (7)

• Figure 1: The curves of $\Pi$, $\bar{\Pi}$ and $M$.
• Figure 2: The curves of $X$ and $Y$.
• Figure 3: The curves of $K,\bar{K},K_0,$$\Lambda_0$, $\bar{\Lambda}$, $\Psi_1,\Psi_2$ and $\Psi_3$.
• Figure 4: State averages and MF effects of followers under open-loop and feedback controls.
• Figure 5: State trajectories of the leader under open-loop and feedback controls.