Stochastic Dominance Under Independent Noise

We ask the following question: Given two random variables, $X$ and $Y$, under what conditions is it possible to find a random variable $Z$, independent from $X$ and $Y$, so that $X+Z$ first-order stochastically dominates $Y+Z$ ? We show that such a $Z$ exists whenever $X$ has higher expectation than $Y$. In addition, if $X$ and $Y$ have equal mean, but the first has lower variance, then $Z$ can be chosen so that $X + Z$ dominates $Y + Z$ in terms of second-order stochastic dominance. We present applications to choice under risk and mechanism design.