A Cucker-smale Inspired Deterministic Mean Field Game With Velocity Interactions

We introduce a mean field game (MFG) model for pedestrians moving in a given domain and choosing their trajectories so as to minimize a cost including a penalization on the difference between their own velocity and that of the other agents they meet. We prove the existence of an equilibrium in a Lagrangian setting by using its variational structure and then study its properties, including regularity of the trajectories and monokineticity. We also discuss the MFG system ruling the equilibria and the possibility of obtaining equilibria in pure strategies, and we adapt the model to the case of multiple populations, a case which could give rise to a well-known phenomenon of lane formation.