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. [10]. We show that this equation, which is a non-linear version of the Kimura type equation studied in Epstein and Mazzeo [28], 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 Hölder type for the corresponding Kimura operator when the drift therein is merely continuous.

通过添加Wright-Fisher公共噪声，我们在有限状态平均场游戏中强制唯一性。我们通过分析该游戏的主方程来实现这一目标，该主方程是在单纯形上设置的退化抛物型二阶偏微分方程，其特征是解决与均值场博弈相关的随机向前-向后系统；参见Cardaliaguet等。[10]。我们证明，这个方程是在爱泼斯坦和马泽奥[28]中研究的木村型方程的非线性版本，只要边界处漂移的法向分量足够强，就具有唯一的平滑解。其中，当其中的漂移只是连续的时，这需要对相应的Kimura算子进行Hölder类型的先验估计。