  We obtain a necessary and sufficient condition under which random-coefficient
discrete choice models, such as mixed-logit models, are rich enough to
approximate any nonparametric random utility models arbitrarily well across
choice sets. The condition turns out to be the affine-independence of the set
of characteristic vectors. When the condition fails, resulting in some random
utility models that cannot be closely approximated, we identify preferences and
substitution patterns that are challenging to approximate accurately. We also
propose algorithms to quantify the magnitude of approximation errors.
