Linear-quadratic mixed Stackelberg-zero-sum game for mean-field regime switching system

TL;DR

This paper develops a theoretical framework for a linear-quadratic Stackelberg zero-sum game in a regime switching system with mean-field terms, providing explicit strategies and applying them to a product pricing problem.

Contribution

It introduces a novel approach combining continuation, induction, and stochastic maximum principle to solve a complex mean-field Stackelberg game with regime switching.

Findings

Established existence and uniqueness of the mean-field FBSDE with Markovian switching.

Proved unique solvability of Hamiltonian systems for all players.

Derived explicit optimal feedback strategies for the leader and followers.

Abstract

Motivated by a product pricing problem, a linear-quadratic Stackelberg differential game for a regime switching system involving one leader and two followers is studied. The two followers engage in a zero-sum differential game, and both the state system and the cost functional incorporate a conditional mean-field term. Applying continuation method and induction method, we first establish the existence and uniqueness of a conditional mean-field forward-backward stochastic differential equation with Markovian switching. Based on it, we prove the unique solvability of Hamiltonian systems for the two followers and the leader. Moreover, utilizing stochastic maximum principle, decoupling approach and optimal filtering technique, the optimal feedback strategies of two followers and leader are obtained. Employing the theoretical results, we solve a product pricing problem with some numerical simulations.