Weak Solutions for Potential Mean Field Games of Controls

TL;DR

This paper studies a class of potential mean field games of controls, establishing existence and uniqueness of weak solutions for a degenerate second order PDE system with local coupling, using variational methods.

Contribution

It introduces a variational approach to prove existence and uniqueness of weak solutions for potential MFGC systems with degenerate diffusion.

Findings

Unique weak solution established for the PDE system

Additional regularity results obtained under certain conditions

Handling of state distribution-dependent coupling in analysis

Abstract

We analyze a system of partial differential equations that model a potential mean field game of controls, briefly MFGC. Such a game describes the interaction of infinitely many negligible players competing to optimize a personal value function that depends in aggregate on the state and, most notably, control choice of all other players. A solution of the system corresponds to a Nash Equilibrium, a group optimal strategy for which no one player can improve by altering only their own action. We investigate the second order, possibly degenerate, case with non-strictly elliptic diffusion operator and local coupling function. The main result exploits potentiality to employ variational techniques to provide a unique weak solution to the system, with additional space and time regularity results under additional assumptions. New analytical subtleties occur in obtaining a priori estimates with the introduction of an additional coupling that depends on the state distribution as well as feedback.