  The goal of this paper is to provide a selection principle for potential mean
field games on a finite state space and, in this respect, to show that
equilibria that do not minimize the corresponding mean field control problem
should be ruled out. Our strategy is a tailored-made version of the vanishing
viscosity method for partial differential equations. Here, the viscosity has to
be understood as the square intensity of a common noise that is inserted in the
mean field game or, equivalently, as the diffusivity parameter in the related
parabolic version of the master equation. As established recently, the randomly
forced mean field game becomes indeed uniquely solvable for a relevant choice
of a Wright-Fisher common noise, the counterpart of which in the master
equation is a Kimura operator on the simplex. We here elaborate on this idea to
make the mean field game with common noise both uniquely solvable and
potential, meaning that its unique solution is in fact equal to the unique
minimizer of a suitable stochastic mean field control problem. Taking the limit
as the intensity of the common noise vanishes, we obtain a rigorous proof of
the aforementioned selection principle. As a byproduct, we get that the
classical solution to the viscous master equation associated with the mean
field game with common noise converges to the gradient of the value function of
the mean field control problem without common noise; we hence select a
particular weak solution of the master equation of the original mean field
game. Lastly, we establish an intrinsic uniqueness criterion for this solution
within a suitable class.
