The use of the entropic regularization and the relative Sinkhorn algorithm in order to calculate optimal transport problems is nowadays widely used.

We propose a new viewpoint on application of this regularization technique to variational mean-field games with quadratic Hamiltonian. In particular we focus on the fact that this regularization amounts to solving second order variational mean-field games, dropping in particular the requirement for the regularization parameter to be small, whenever we are interested in the case with diffusion.

Based on a joint work with J.-D. Benamou, G. Carlier and L. Nenna.