A Mean Field Game approach for multi-lane traffic management

In this work, we discuss a Mean Field Game approach to traffic management on multi-lane roads. The control is related to the optimal choice to change lane to reach a desired configuration of the system. Such approach is particularly indicated to model self-driven vehicles with complete information on the state of the system. The mathematical interest of the problem is that the system of partial differential equations obtained is not in the classic form, but it consists of some continuity equations (one for each lane) and a variational inequality, coming from the Hamilton-Jacobi theory of the hybrid control. We propose a consistent semi-Lagrangian scheme for the approximation of the system and we discuss how to improve its efficiency with the use of a policy iteration technique. We finally present a numerical test which shows the potential of our approach.