Backward-forward linear-quadratic mean-field Stackelberg games

Kehan Si,Zhen Wu

doi:10.1186/s13662-021-03236-9

Abstract

This paper studies a controlled backward-forward linear-quadratic-Gaussian (LQG) large population system in Stackelberg games. The leader agent is of backward state and follower agents are of forward state. The leader agent is dominating as its state enters those of follower agents. On the other hand, the state-average of all follower agents affects the cost functional of the leader agent. In reality, the leader and the followers may represent two typical types of participants involved in market price formation: the supplier and producers. This differs from standard MFG literature and is mainly due to the Stackelberg structure here. By variational analysis, the consistency condition system can be represented by some fully-coupled backward-forward stochastic differential equations (BFSDEs) with high dimensional block structure in an open-loop sense. Next, we discuss the well-posedness of such a BFSDE system by virtue of the contraction mapping method. Consequently, we obtain the decentralized strategies for the leader and follower agents which are proved to satisfy the ε-Nash equilibrium property.

Highlights

The dynamic optimization of a large-population system has attracted extensive research attention from academic communities
Its most significant feature is the existence of numerous insignificant agents, denoted by {Ai}Ni=1, whose dynamics and cost functionals are coupled via their state-average
Step 1 (SOC-F): Consider the Nash equilibrium response functional of Problem (II) for the representative follower agent denoted by ui[·, ·]

Summary

Introduction

The dynamic optimization of a (linear) large-population system has attracted extensive research attention from academic communities. Step 2: Given the response functional of followers x, solve the decentralized stochastic control problem of the leader A0, and denote the optimal solution pair by (x0, u 0) = (x0(x), u 0(x)). Problem (II) For given xi0, Ftw0 -measurable functions x(·), and the control u0(·) of the leader agent A0, find the optimal response functional ui[·] : U0[0, T] × L2Fw0 (0, T; R) → Ui[0, T] of the following differential games among followers: Ji xi0, x(·), u0(·); ui u0(·), x(·) = inf Ji xi0, x(·), u0(·); ui(·) .

Objectives

Results

Conclusion