Social Optima in Leader-Follower Mean Field Linear Quadratic Control

TL;DR

This paper develops a framework for social optimality in leader-follower mean field linear quadratic control problems, providing decentralized strategies and establishing conditions for Stackelberg equilibrium.

Contribution

It introduces a novel approach using variational analysis and auxiliary control problems to derive decentralized social optimal strategies in leader-follower mean field games.

Findings

Derived decentralized social optimal strategies.

Established conditions for Stackelberg equilibrium.

Solved auxiliary control problems with mean field approximations.

Abstract

This paper investigates a linear quadratic mean field leader-follower team problem, where the model involves one leader and a large number of weakly-coupled interactive followers. The leader and the followers cooperate to optimize the social cost. Specifically, for any strategy provided first by the leader, the followers would like to choose a strategy to minimize social cost functional. Using variational analysis and person-by-person optimality, we construct two auxiliary control problems. By solving sequentially the auxiliary control problems with consistent mean field approximations, we can obtain a set of decentralized social optimality strategy with help of a class of forward-backward consistency systems. The relevant Stackelberg equilibrium is further proved under some proper conditions.