  This paper provides a mathematical study of the well-posedness of master
equation on finite state space involving terms modelling common noise. In this
setting, the solution of the master equation depends on an additional variable
modelling the value of a stochastic process impacting all players. Using
technique from viscosity solutions, we give sufficient conditions for the
existence of a Lipschitz continuous solution on any time interval. Under some
structural assumptions, we are even able to treat cases in which the dynamics
of this stochastic process depend on the state of the game.
