Direct Approach of Linear-Quadratic Stackelberg Mean Field Games of Backward-Forward Stochastic Systems

TL;DR

This paper develops a direct method to solve linear-quadratic Stackelberg mean field games involving backward-forward stochastic systems, providing decentralized strategies and equilibrium analysis.

Contribution

It introduces a novel direct approach to solve LQ Stackelberg mean field games with backward-forward systems, deriving decentralized strategies and equilibrium conditions.

Findings

Decentralized strategies form an $(ta_1, ta_2)$-Stackelberg equilibrium.

The approach effectively decouples high-dimensional FBSDEs using mean field approximations.

The method applies to systems with a leader and numerous followers in stochastic environments.

Abstract

This paper is concerned with a linear-quadratic (LQ) Stackelberg mean field games of backward-forward stochastic systems, involving a backward leader and a substantial number of forward followers. The leader initiates by providing its strategy, and subsequently, each follower optimizes its individual cost. A direct approach is applied to solve this game. Initially, we address a mean field game problem, determining the optimal response of followers to the leader's strategy. Following the implementation of followers' strategies, the leader faces an optimal control problem driven by high-dimensional forward-backward stochastic differential equations (FBSDEs). Through the decoupling of the high-dimensional Hamiltonian system using mean field approximations, we formulate a set of decentralized strategies for all players, demonstrated to be an $(\epsilon_1, \epsilon_2)$-Stackelberg equilibrium.