Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An Extension of Major-Minor Mean Field Game Theory

Agustín Muñoz González

Abstract

This work extends the theory presented in Mean Field Games with a Dominating Player by Bensoussan, Chau and Yam on mean field games with a dominating player, to the case in which the utility and cost functions depend not only on the law of the states, but on the joint state--control law. We incorporate the conditional distribution of the state--control pair of the representative agent given the common noise of the dominating player. In addition, we generalize the role of the dominating player to include the direct impact of its controls $u_0$ on the dynamics and functionals of the system. The optimization problems are reformulated in terms of the conditional distribution of the state--control pair, the necessary optimality conditions are established via stochastic maximum principles, and a coupled SHJB--FP system of equations is obtained that synthesizes the equilibrium conditions. This framework provides a significant extension of the existing literature on MFG with a dominating player.

An Extension of Major-Minor Mean Field Game Theory

Abstract

This work extends the theory presented in Mean Field Games with a Dominating Player by Bensoussan, Chau and Yam on mean field games with a dominating player, to the case in which the utility and cost functions depend not only on the law of the states, but on the joint state--control law. We incorporate the conditional distribution of the state--control pair of the representative agent given the common noise of the dominating player. In addition, we generalize the role of the dominating player to include the direct impact of its controls on the dynamics and functionals of the system. The optimization problems are reformulated in terms of the conditional distribution of the state--control pair, the necessary optimality conditions are established via stochastic maximum principles, and a coupled SHJB--FP system of equations is obtained that synthesizes the equilibrium conditions. This framework provides a significant extension of the existing literature on MFG with a dominating player.
Paper Structure (20 sections, 4 theorems, 55 equations)

This paper contains 20 sections, 4 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Working spaces and assumptions
  3. Preliminaries
  4. Derivatives on the space of measures: Lions, linear, and Wasserstein gradient
  5. Lions derivative (via lift).
  6. Linear derivative.
  7. Wasserstein gradient (intrinsic).
  8. Notation for joint state--control laws.
  9. Fokker--Planck equation
  10. Problems
  11. Problem 1: Control of the Representative Agent
  12. Problem 2: Equilibrium Condition
  13. Problem 3: Control of the Dominating Player
  14. Main results
  15. Extension of the results
  16. ...and 5 more sections

Key Result

Lemma 5.1

Given $x_0, u_0$ and an exogenous flow of joint measures $\Pi$ as in Problem1, the optimal control $\hat{u}_1 \in \mathcal{A}_1$ is optimal if and only if it satisfies the following SHJB equation: where The infimum is attained uniquely at $\hat{u}_1$, that is,

Theorems & Definitions (14)

  • Remark
  • Lemma 5.1: Necessary condition for Problem 1
  • Remark : Martingale process $K_\Psi$
  • proof
  • Corollary 5.2: Necessary condition for Problems 1 and 2
  • Proposition 6.1: Necessary condition for Problem 3 in the extended case
  • Remark
  • proof : Sketch of proof
  • Remark
  • Remark
  • ...and 4 more