  We propose a new viewpoint on variational mean-field games with diffusion and
quadratic Hamiltonian. We show the equivalence of such mean-field games with a
relative entropy minimization at the level of probabilities on curves. We also
address the time-discretization of such problems, establish
$\Gamma$-convergence results as the time step vanishes and propose an efficient
algorithm relying on this entropic interpretation as well as on the Sinkhorn
scaling algorithm.
