Deterministic mean field games with jumps and mixed variational structure

We consider in this paper some mean field game problems where the agents have an individual cost composed of an action penalizing their motion and a running cost depending on the density of all players. This running cost involves two parts: one which is variational but possibly nonsmooth (typically, a local cost of the form f′(ρ(x)) where ρ(x) is the density of the distribution of players at point x) and one which is smooth but non-variational (typically, a non-local cost involving convolutions). In order to show that this is a very general framework, we consider both the case where the action is of kinetic energy type, i.e. it is an integral of a power of the velocity, and the case where it counts the number of jumps of the curve, more adapted to some real estate and urban planning models. We provide an existence result of a “formal” equilibrium by Kakutani’s fixed point theorem. “Formal” means that we do not prove that such an equilibrium is a measure on curves concentrated on optimal curves but only that it solves a variational problem whose optimality conditions formally correspond to this. We then rigorously prove that optimizers of this variational problem are indeed concentrated on optimal curves for an individual problem. Both the existence and these optimality conditions for the minimizers are studied in the kinetic and in the jumps case.We then prove that the optimization among measures on curves (the Lagrangian framework) can be reduced to a minimization among curves of measures (the Eulerian framework) by proving a representation of the action functional. This is classical in the kinetic case and involves the Wasserstein distances Wp, while it lets the total variation norm appear in the jump case. We then prove that, under some assumptions, the solution of the variational problem expressed in Eulerian language depends in a Lipschitz continuous way on the data, which can prove that the fixed point argument for the equilibrium can be reformulated as the fixed point of a Lipschitz uni-valued map. Under smallness assumptions on some data, this becomes a contraction and the equilibrium can be found by Banach fixed point. This allows for efficient numerical computations, based on the solution of a non-smooth convex optimization problem, which we present at the end of the paper.