Abstract: The talk will be devoted to a joint work with W. Shim (Seoul) where we introduced a mean field game 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. Existence of an equilibrium is proven in a Lagrangian setting by using its variational structure, and it is possible to prove some properties and regularity of the optimal trajectories.