Well-posedness Analysis for the Master Equation of Mean Field Games of Controls

Abstract

The theory of mean field games of controls explores models where interaction occurs through the joint distribution of both state and control variables. In this thesis, we investigate the global well-posedness theory for the master equation of mean field games of controls with common noise. In standard mean field games, a specific monotonicity condition is essential for ensuring the uniqueness of mean field equilibria and the master equation. We extend the classical Lasry-Lions monotonicity and displacement monotonicity to the context of mean field games of controls, considering both differential and integral forms. On the other hand, even in the absence of the monotonicity condition, the well-posedness of the master equation can be analyzed under specific structures, such as the linear-quadratic problem. To exemplify this, we study a generalized linear-quadratic mean-field type games of controls and its master equation.

We first address the global well-posedness of the master equation for mean field games of controls, focusing on Lasry-Lions monotonicity and displacement monotonicity. The approach hinges on the propagation of the monotonicity property and an a priori W₁-Lipschitz estimate. We present two equivalent formulations for each type of monotonicity and demonstrate the propagation property through both the stochastic PDE system and the master equation. By combining this with the uniform W₁-Lipschitz estimate, we can extend a local solution to a global solution backward. As an application of the master equation, we also explore the N-player games of controls and establish the convergence.

Without monotonicity conditions, the master equation may become ill-posed. However, the presence of common noise may help restore the uniqueness of the master equation. As a typical example, we investigate linear-quadratic mean-field type games of controls with common noise. Unlike mean field games, these games incorporate both individual and population distributions as mean-field terms. Consequently, when the population's mean-field term is fixed, we must solve a McKean-Vlasov type control problem, introducing time inconsistency and a different structure to the master equation. Within the linear-quadratic framework, we analyze the well-posedness of the master equation and its convergence properties.

Date of Award	28 Aug 2025
Original language	English
Awarding Institution	City University of Hong Kong
Supervisor	Chenchen MOU (Supervisor)