Existence for general monotone graph mean field games

Establish existence for general monotone graph mean field games without relying on the variational structure available for potential mean field games.

Background

The paper proves existence of the discrete solution in the potential case by identifying the time-staggered scheme with the Karush–Kuhn–Tucker system of a convex discrete action. For general monotone Hamiltonian and transport coefficients, however, the paper establishes only uniqueness and estimates for every interior discrete solution; it does not provide an existence theory because a corresponding scalar variational formulation need not exist. The authors explicitly identify existence for this broader class of monotone graph mean field games as unresolved.