by combining the classical approximation scheme for mean field games system developed in [@acd; @acc] with L1 schemes for the fractional derivatives. The duality structure of the time-fractional PDE system and the energy stable structure of the advection diffusion equation are shown to be preserved. Under appropriate monotonicity and convexity assumptions, we prove existence, uniqueness of the solution to the discrete system and convergence to the solution of the original PDE system. Some numerical tests are performed to illustrate the different behavior of the time-fractional case with respect to the standard one.\
The paper is organized as follows. In Section \[sec:scheme\], we introduce the finite differences scheme and prove some preliminary results. Section \[sec:well\_posed\] is devoted to show the well posedness of the discrete problem. In Section \[sec:convergence\], we prove the convergence of the scheme and in Section \[sec:numerics\] we carry out some numerical simulations. Finally we discuss some possible directions for future works.

The numerical scheme {#sec:scheme}
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In this section we introduce a numerical scheme for and we prove some preliminary results to show its well-posedness.\
For simplicity of notations, we assume that the dimension $d$ is equal to $2$, since the extension for general $d$ will be 