Before revision: 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.
After revision: Stochastic dominance is a crucial tool for the analysis of choice under risk. It is typically analyzed as a property of two gambles that are taken in isolation. We study how additional independent sources of risk (e.g. uninsurable labor risk, house price risk, etc.) can affect the ordering of gambles. We show that, perhaps surprisingly, background risk can be strong enough to render lotteries that are ranked by their expectation ranked in terms of second-order stochastic dominance.