This scheme is naturally derived from the approximation of the fractional integral as a Riemann sum and long known to be consistent (for an concise illustration we refer to section 3 of [@tt]). Moreover, L1 approximation scheme stands out by being able to preserve at the discrete level certain desirable features of the origin PDEs, such as maximum principle [@glm] and energy stability [@du; @Shen_2019; @tt]. Therefore, approximations can be constructed to converge to suitable notions of weak solutions, which may be viscosity solutions to Hamilton-Jacobi-Bellman equations [@glm], or distributional solutions in Sobolev spaces to diffusion (phase field) equations [@du; @Shen_2019; @tt].\
If fractional derivatives are simply replaced by first order derivatives, system becomes the classical Mean Field Games system introduced in [@ll] to characterize the Nash equilibria for differential games with a large number of agents. Achdou et al. [@acd; @acc] introduced an finite difference type discretization of mean field games system where the scheme for the Fokker-Planck equation is deduced via a duality argument, reproducing at a discrete level a crucial property of the continuous system. This type of schemes has been further developed in [@afg; @bks; @cs].\
In this paper, we present a finite differences approximation of 