By Aimé Lachapelle & Marie-Therese Wolfram (Cahier de la Chaire n°40)
In this paper we present a new class of pedestrian crowd models based on the mean field games theory introduced by Lasry and Lions in 2006. This macroscopic approach is based on a microscopic model, that considers smart pedestrians who rationally interact and anticipate the future. This leads to a forward-backward structure in time. We focus on two-population interactions and validate the modeling with simple examples such as self-organization behavior as for instance lane formation. Two complementary classes of problems are addressed, namely the case of crowd aversion and the one of congestion. In both cases we describe the model, build a numerical solver (respectively based on optimization formulation and partial differential equations), and finally provide some numerical tests involving complex group behaviors such as symmetry breaking and lane formation.
Keywords: Mean field games, interacting populations, Nash equilibrium, rational expectations, flow of pedestrians, lane formation, numerical approximation