Master equation of discrete-time Stackelberg mean field games with multiple leaders

TL;DR

This paper develops a master equation framework for discrete-time Stackelberg mean field games involving multiple leaders, major followers, and minor followers, enabling the computation of equilibrium strategies in complex hierarchical settings.

Contribution

It introduces a master equation for Stackelberg mean field games with multiple leaders and extends it to infinite leader scenarios, advancing the theoretical understanding of hierarchical mean field games.

Findings

Derived a master equation for SMFE-ML.

Extended the framework to infinite leaders.

Provided a method to compute all equilibria.

Abstract

In this paper, we consider a discrete-time Stackelberg mean field game with a finite number of leaders, a finite number of major followers and an infinite number of minor followers. The leaders and the followers each observe types privately that evolve as conditionally independent controlled Markov processes. The leaders are of "Stackelberg" kind which means they commit to a dynamic policy. We consider two types of followers: major and minor, each with a private type. All the followers best respond to the policies of the Stackelberg leaders and each other. Knowing that the followers would play a mean field game (with major players) based on their policy, each (Stackelberg) leader chooses a policy that maximizes her reward. We refer to the resulting outcome as a Stackelberg mean field equilibrium with multiple leaders (SMFE-ML). In this paper, we provide a master equation of this game that allows one to compute all SMFE-ML. We further extend this notion to the case when there are infinite number of leaders.