An entropy minimization approach to second-order variational mean-field games

TL;DR

This paper introduces an entropy minimization framework for second-order variational mean-field games with diffusion and quadratic Hamiltonian, providing theoretical insights and an efficient computational algorithm.

Contribution

It establishes the equivalence between mean-field games and entropy minimization, and develops a time-discretization scheme with convergence analysis and a practical Sinkhorn-based algorithm.

Findings

Proves the equivalence between mean-field games and entropy minimization.

Establishes $ ext{Gamma}$-convergence of the discretized problems.

Proposes an efficient Sinkhorn-based algorithm for computation.

Abstract

We propose a new viewpoint on variational mean-field games with diffusion and quadratic Hamiltonian. We show the equivalence of such mean-field games with a relative entropy minimization at the level of probabilities on curves. We also address the time-discretization of such problems, establish $\Gamma$-convergence results as the time step vanishes and propose an efficient algorithm relying on this entropic interpretation as well as on the Sinkhorn scaling algorithm.