  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 by J.M. Lasry and P.L. Lions and adapting them to games on
graphs, we introduce a partial differential equation, often referred to as the
Master equation, 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.
