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The goal of this paper is to provide a selection principle for potential meanfield games on a finite state space and, in this respect, to show thatequilibria that do not minimize the corresponding mean field control problemshould be ruled out. Our strategy is a tailored-made version of the vanishingviscosity method for partial differential equations. Here, the viscosity has tobe understood as the square intensity of a common noise that is inserted in themean field game or, equivalently, as the diffusivity parameter in the relatedparabolic version of the master equation. As established recently, the randomlyforced mean field game becomes indeed uniquely solvable for a relevant choiceof a Wright-Fisher common noise, the counterpart of which in the masterequation is a Kimura operator on the simplex. We here elaborate on this idea tomake the mean field game with common noise both uniquely solvable andpotential, meaning that its unique solution is in fact equal to the uniqueminimizer of a suitable stochastic mean field control problem. Taking the limitas the intensity of the common noise vanishes, we obtain a rigorous proof ofthe aforementioned selection principle. As a byproduct, we get that theclassical solution to the viscous master equation associated with the meanfield game with common noise converges to the gradient of the value function ofthe mean field control problem without common noise; we hence select aparticular weak solution of the master equation of the original mean fieldgame. Lastly, we establish an intrinsic uniqueness criterion for this solutionwithin a suitable class.

Selection by vanishing common noise for potential finite state mean field games