  In this article, we introduce a method to approximate solutions of some
variational mean field game problems with congestion, by finite sets of player
trajectories. These trajectories are obtained by solving a minimization problem
similar to the initial variational problem. In this discretized problem,
congestion is penalized by a Moreau envelop with the 2-Wasserstein distance.
Study of this envelop as well as efficient computation of its values and
variations is done using semi-discrete optimal transport. We show convergence
of the discrete sets of trajectories toward a solution of the mean field game,
under some conditions on the parameters of the discretization.
