Approximating Choice Data by Discrete Choice Models

TL;DR

This paper establishes conditions under which discrete choice models can approximate any nonparametric random utility model, providing theoretical insights and algorithms to measure approximation errors.

Contribution

It introduces a necessary and sufficient condition based on affine-independence for the approximation capability of discrete choice models and proposes algorithms to quantify errors.

Findings

Affine-independence is key for approximation success.

Some models cannot be closely approximated when the condition fails.

Algorithms are provided to measure approximation errors.

Abstract

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.