I will present a new class of MFG motivated by models for the real estate market, where agents choose a trajectory which is not continuous in time but has jumps. Each jump corresponds to moving from an address to another, and has a fixed cost. In some cases, the equilibrium is variational and it s possible to state an eulerian problem on curves of densities, but the distance to be considered on densities is no more the Wassrestein distance as it is often the case in MFG but the L^1 (or total variation) norm, and the velocity will be penalized via its L^1 norm in time as well. I will then study variational problems of this form (with L^1 costs for moving) with the aim to see whether the solutions evolve smoothly or also have jumps, and prove some regularity results which seem relevant for the MFG analysis of these models. This is a joint work with Annette Dumas, who starts a PhD in Lyon 1 on this topic.