We investigate the possibility that a decision-maker prefers to avoid making a decision and instead delegates it to an external device, e.g., a coin flip. In a series of experiments our participants often choose a stochastically dominated lottery between outcomes, contradicting most theories of choice such as expected utility. A large data set on university applications in Germany shows a choice pattern that is consistent with a preference for randomization, entailing substantial allocative consequences. The findings are consistent with our theory of responsibility aversion. (author's abstract)