This paper establishes the existence of relaxed solutions to mean eld games (MFGs for short) with singular controls. As a by-product, we obtain an existence of relaxed solutions results for McKean-Vlasov stochastic singular control problems. Finally, we prove approximations of solutions results for a particular class of MFGs with singular controls by solutions, respectively control rules, for MFGs with purely regular controls. Our existence and approximation results strongly hinge on the use of the Skorokhod M1 topology on the space of cadlag functions.
mean field game singular control relaxed control skorokhod m1 topology