We propose a robust method of discrete choice analysis when agents’ choice sets are unobserved. Our core model assumes nothing about agents’ choice sets apart from their minimum size. Importantly, it leaves unrestricted the dependence, conditional on observables, between agents’ choice sets and their preferences. We first characterize the sharp identification region of the model’s parameters by a finite set of conditional moment inequalities. We then apply our theoretical findings to learn about households’ risk preferences and choice sets from data on their deductible choices in auto collision insurance. We find that the data can be explained by expected utility theory with low levels of risk aversion and heterogeneous choice sets, and that more than three in four households require limited choice sets to explain their deductible choices. We also find that the data are consistent with some models of choice set formation, but not others.