Variational Mean Field Games

Jean-David Benamou (1, 2) , Guillaume Carlier (1, 2) , Filippo Santambrogio (3)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 MOKAPLAN - Méthodes numériques pour le problème de Monge-Kantorovich et Applications en sciences sociales
3 LM-Orsay - Laboratoire de Mathématiques d'Orsay

This paper is a brief presentation of those Mean Field Games with congestion penalization which have a variational structure, starting from the deterministic dynamical framework. The stochastic framework (i.e. with diffusion) is also presented both in the stationary and dynamic case. The variational problems relevant for MFG are described via Eulerian and Lagrangian languages, and the connection with equilibria is explained by means of convex duality and of optimality conditions. The convex structure of the problem also allows for efficient numerical treatment, based on Augmented Lagrangian Algorithms, and some new simulations are shown at the end of the paper.

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

Jean-David Benamou, Guillaume Carlier, Filippo Santambrogio. Variational Mean Field Games. Nicola Bellomo, Pierre Degond, Eitan Tadmor. Active Particles, Volume 1, Springer, pp.141-171, 2017. ⟨hal-01295299⟩