  We force uniqueness in finite state mean field games by adding a
Wright-Fisher common noise. We achieve this by analyzing the master equation of
this game, which is a degenerate parabolic second-order partial differential
equation set on the simplex whose characteristics solve the stochastic
forward-backward system associated with the mean field game; see Cardaliaguet
et al. (2019). We show that this equation, which is a non-linear version of the
Kimura type equation studied in Epstein and Mazzeo (2013), has a unique smooth
solution whenever the normal component of the drift at the boundary is strong
enough. Among others, this requires a priori estimates of H\"older type for the
corresponding Kimura operator when the drift therein is merely continuous.
