Variational Mean Field Games for Market Competition

Schauder Estimates for a Class of Potential Mean Field Games of Controls

“…In this section we prove the uniqueness of the solution (u, m, v, P) to (MFGC). We also prove that (P, v) is the solution to a dual problem to (17). Both results are obtained under the following additional monotonicity assumption of f : There exists a measurable mapping F(t, m) :…”

“…) be a feasible pair, i.e., it satisfies the constraint in (17). For all (x, t) ∈ Q, we have v τ = −H p (∇u τ + φ P τ ).…”

“…We have proved that the pair (m τ , v τ ) is the solution to an optimal control problem. Therefore, for all t, v τ (•, t) minimizes the Hamiltonian associated with problem (17). Let us introduce some notation, in order to exploit this property.…”

“…The existence of a classical solution of (MFGC) is established with the Leray-Schauder theorem and classical estimates for parabolic equations. A similar approach has been employed in [16], [17], and [18] for the analysis of a mean field game problem proposed by Chan and Sircar [9]. In this model, each agent exploits an exhaustible resource and fixes its price.…”

“…These bounds are obtained in particular with a potential formulation of the mean field game problem: we prove that all solutions to (MFGC) are also solutions to an optimal control problem of the Fokker-Planck equation. We are not aware of any other publication making use of such a potential formulation for a mean field game of controls, with the exception of [17] for the Chan and Sircar model. Let us mention that besides the derivation of a priori bounds, the potential formulation of the problem can be very helpful for the numerical resolution of the problem and the analysis of learning procedures (which are out of the scope of the present work).…”