From infinity to one: The reduction of some mean field games to a global control problem

This paper presents recent results from Mean Field Game theory underlying the introduction of common noise that imposes to incorporate the distribution of the agents as a state variable. Starting from the usual mean field games equations introduced in [11 , 12 , 13 ] and adapting them to games on graphs, we introduce a partial differential equation, often referred to as the Master equation (see [14]), from which the MFG equations can be deduced. Then, this Master equation can be reinterpreted using a global control problem inducing the same behaviors as in the non-cooperative initial mean field game.