  We consider the variational structure of a time-fractional second order Mean
Field Games (MFG) system with local coupling. The MFG system consists of
time-fractional Fokker-Planck and Hamilton-Jacobi-Bellman equations. In such a
situation the individual agent follows a non-Markovian dynamics given by a
subdiffusion process. Hence, the results of this paper extend the theory of
variational MFG to the subdiffusive situation, providing an Eulerian
interpretation of time-fractional MFG systems.
