Entropic optimal planning for path-dependent mean field games

TL;DR

This paper addresses a path-dependent planning problem in mean field games, reducing it to an embedding problem involving McKean-Vlasov dynamics, and characterizes minimal entropy solutions under certain conditions.

Contribution

It introduces a novel reduction of the path-dependent planning problem to an embedding problem and characterizes minimal entropy solutions, extending mean field game theory.

Findings

Reduction of planning problem to embedding problem

Conditions for existence of solutions

Characterization of minimal entropy solutions

Abstract

In the context of mean field games, with possible control of the diffusion coefficient, we consider a path-dependent version of the planning problem introduced by P.L. Lions: given a pair of marginal distributions $(\mu_0, \mu_1)$, find a specification of the game problem starting from the initial distribution $\mu_0$, and inducing the target distribution $\mu_1$ at the mean field game equilibrium. Our main result reduces the path-dependent planning problem into an embedding problem, that is, constructing a McKean-Vlasov dynamics with given marginals $(\mu_0,\mu_1)$. Some sufficient conditions on $(\mu_0,\mu_1)$ are provided to guarantee the existence of solutions. We also characterize, up to integrability, the minimum entropy solution of the planning problem. In particular, as uniqueness does not hold anymore in our path-dependent setting, one can naturally introduce an optimal planning problem which would be reduced to an optimal transport problem along with controlled McKean-Vlasov dynamics.