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

Jean-David Benamou (1) , Guillaume Carlier (2) , Simone Di Marino (3) , Luca Nenna (2)

1 MOKAPLAN - Méthodes numériques pour le problème de Monge-Kantorovich et Applications en sciences sociales
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
3 SNS - Scuola Normale Superiore di Pisa

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 Γ-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.

Mots clés: Mean-Field Games; Fokker-Planck equation; entropy minimization; Schrödinger bridges; Sinkhorn algorithm

Domaines: Analyse numérique [math.NA]; Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]; Optimisation et contrôle [math.OC]

Jean-David Benamou, Guillaume Carlier, Simone Di Marino, Luca Nenna. An entropy minimization approach to second-order variational mean-field games. 2018. ⟨hal-01848370v1⟩