  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.
