The accepted manuscript version (last revised 5 Jan 2018 (v8)) has 118 pages, 3 tables, 11 figures, and includes supplementary appendix. This version corrects some typos in Example 2 of the published version. This supplement contains 11 appendices with additional results and some omitted proofs. Appendices F-J include additional results for Sections 2-7, respectively. Appendix K gathers auxiliary results on algebra of covering entropies. Appendices L and M contain the proofs of Sections 4 and 5 omitted from the main text. Appendix N contains the proofs of Sections 6 omitted from the main text, together with the proofs of the additional results for Section 6 in Appendix I. Appendix O reports the results of a simulation experiment.

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 choice sets and 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 non‐singleton choice sets, and that more than three in four households require limited choice sets to explain their deductible choices. We also provide simulation evidence on the computational tractability of our method in applications with larger feasible sets or higher‐dimensional unobserved heterogeneity.