In the setup of selling one or more goods, various papers have shown, in various forms and for various purposes, that a small change in the distribution of a buyer's valuations may cause only a small change in the possible revenue that can be extracted. We prove a simple, clean, convenient, and general statement to this effect: let $X$ and $Y$ be random valuations on $k$ additive goods, and let $W(X,Y)$ be the Wasserstein (or "earth mover's") distance between them; then $$\left\vert \sqrt{Rev(X)}-\sqrt{Rev(Y)}\right\vert \le \sqrt{W(X,Y)}.$$ This further implies that a simple explicit modification of any optimal mechanism for $X$, namely, "uniform discounting," is guaranteed to be almost optimal for any $Y$ that is close to $X$ in the Wasserstein distance.

翻译：在销售一种或多种商品的设定中，已有诸多文献以不同形式、出于不同目的证明：买家估值分布的微小变化仅可能导致可提取收益的微小变化。我们针对这一效应提出了一个简洁、清晰、便利且普适的表述：设$X$和$Y$为$k$种可加性商品的随机估值，$W(X,Y)$表示两者之间的Wasserstein距离（或称“推土机距离”），则有$$\left\vert \sqrt{Rev(X)}-\sqrt{Rev(Y)}\right\vert \le \sqrt{W(X,Y)}.$$ 这进一步表明，对$X$的最优机制进行简单的显式修正——即“均匀折扣”——可保证对任何在Wasserstein距离上接近$X$的$Y$几乎保持最优性。