On Some Mean Field Games and Master Equations through the lens of conservation laws

TL;DR

This paper introduces a new nonlinear transport equation on probability measures for mean field games, revealing a novel monotonicity condition that ensures uniqueness and well-posedness, and explores entropy solutions in non-monotone cases.

Contribution

It derives a new transport equation framework for deterministic mean field games, identifying a unique monotonicity condition and analyzing entropy solutions without monotonicity.

Findings

New monotonicity condition ensures uniqueness of MFG equilibria.

Transport equation framework links to well-posedness of master equations.

Entropy solutions may not always correspond to actual Nash equilibria.

Abstract

In this manuscript we derive a new nonlinear transport equation written on the space of probability measures that allows to study a class of deterministic mean field games and master equations, where the interaction of the agents happens only at the terminal time. The point of view via this transport equation has two important consequences. First, this equation reveals a new monotonicity condition that is sufficient both for the uniqueness of MFG Nash equilibria and for the global in time well-posedness of master equations. Interestingly, this condition is in general in dichotomy with both the Lasry--Lions and displacement monotonicity conditions, studied so far in the literature. Second, in the absence of monotonicity, the conservative form of the transport equation can be used to define weak entropy solutions to the master equation. We construct several concrete examples to demonstrate that MFG Nash equilibria, whether or not they actually exist, may not be selected by the entropy solutions of the master equation.