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.
We obtain a necessary and sufficient condition under which random-coefficient discrete choice models such as the mixed logit models are rich enough to approximate any nonparametric random utility models across choice sets. The condition turns out to be very simple and tractable. When the condition is not satisfied and, hence, there exists a random utility model that cannot be approximated by any random-coefficient discrete choice model, we provide algorithms to measure the approximation errors. After applying our theoretical results and the algorithms to real data, we find that the approximation errors can be large in practice.