Nonlinear Open-Loop Mean field Stackelberg Stochastic Differential Game

TL;DR

This paper develops a probabilistic framework using FBSDEs to analyze nonlinear open-loop mean field Stackelberg stochastic differential games, deriving maximum principles and proving approximate equilibrium solutions.

Contribution

It introduces a novel approach to solve nonlinear mean field Stackelberg games via conditional mean-field FBSDEs and establishes existence, uniqueness, and equilibrium properties.

Findings

Derived maximum principles for leader and followers.

Proved existence and uniqueness of solutions for the conditional mean-field FBSDE.

Applied the theory to a robot control problem involving swarm robots.

Abstract

This paper studies a nonlinear open-loop mean field Stackelberg stochastic differential game by using the probabilistic method through the FBSDE system and the idea of taking control as the fixed point. We successively construct the decentralized optimal control problems for the followers and the leader, among which the leader's decentralized optimal control problem is a partial information optimal control problem with the fully coupled conditional mean-field forward-backward stochastic differential equation (FBSDE, in short) as the state equation. We successively derive the maximum principles for the corresponding decentralized optimal control problems of the followers and the leader. To obtain the existence, uniqueness and estimations of solutions of the state equation, the variational equation and the adjoint equation for the leader's decentralized optimal control problem, we study the well-posedness of a new form of conditional mean-field FBSDE. And the decentralized optimal controls of the leader and followers are proved to be the approximate Stackelberg equilibrium of the nonlinear mean field Stackelberg game. Finally, we apply the theoretical results developed in this paper to solve a nonlinear mean field Stackelberg game problem between a robot control center and unicycle-type swarm robots.