First Order Mean Field Games with Density Constraints: Pressure Equals Price

Abstract

Abstract. In this paper we study Mean Field Game systems under density constraints as optimality conditions of two optimization problems in duality. A weak solution of the system contains an extra term, an additional price imposed on the saturated zones. We show that this price corresponds to the pressure field from the models of incompressible Euler's equationsà la Brenier. By this observation we manage to obtain a minimal regularity, which allows to write optimality conditions at the level of single agent trajectories and to define a weak notion of Nash equilibrium for our model.

New estimates on the regularity of the pressure in density‐constrained mean field games

“…However, to get existence of a solution of the dual problem, it is too restrictive to look only at smooth functions. As understood in , the right functional space is the following. Definition Let be the set of pairs where and is a positive measure, and the Hamilton Jacobi equation is understood in the distributional sense, provided that we set and that we take in account the possible jump from to in the temporal distributional derivative.…”

“…A particular potential MFG has been studied in , where the density is constrained to be below a certain threshold, which represents a capacity constraint of the transportation network or of the medium where agents move. In this case, a pressure appears: according to what we know from fluid mechanics, the pressure is a scalar field, vanishing where the density does not saturate the constraint, and its gradient affects the acceleration of the particles.…”

“…We start with the MFG system, which can be obtained as a consequence of the primal–dual optimality conditions of a variational problem, see . These conditions read, for functions depending on time and space , where is a measure concentrated on the set .…”

Global-in-time regularity via duality for congestion-penalized Mean Field Games

“…However, with J(m) = m q/2 and using the injection H 1 ⊂ L 2 * , we also get m ∈ L q2 * /2 . This is also important in the case G(m) = m log m − m, where one obtains m ∈ L 2 * /2 : this is especially useful when one needs to exit the space L 1 , where many properties lack (see, for instance, [6] or [13], to see applications where integrability properties of the maximal function are required). Note, however, that the exponent 2 * should be computed here w.r.t.…”

“…Later, in Section 4, we will present a recent method which uses duality to prove regularity results in a certain kind of variational problems. This method has, to the authors' knowledge, first been used in a paper by Y. Brenier on the Incompressible Euler equation in fluid mechanics ( [7]: later the results have been improved in [1]) and then adapted in [13] to the case of MFG with density constraints. It is however much more general, and [22] explains how to use it in order to recover, for instance, standard results in elliptic regularity.…”