On the Relevance of Irrelevant Alternatives

Abstract

Multinomial logistic regression is a powerful tool to model choice from a finite set of alternatives, but it comes with an underlying model assumption called the independence of irrelevant alternatives, stating that any item added to the set of choices will decrease all other items' likelihood by an equal fraction. We perform statistical tests of this assumption across a variety of datasets and give results showing how often it is violated. When this axiom is violated, choice theorists will often invoke a richer model known as nested logistic regression, in which information about competition among items is encoded in a tree structure known as a nest. However, to our knowledge there are no known algorithms to induce the correct nest structure. We present the first such algorithm, which runs in quadratic time under an oracle model, and we pair it with a matching lower bound. We then perform experiments on synthetic and real datasets to validate the algorithm, and show that nested logit over learned nests outperforms traditional multinomial regression. Finally, in addition to automatically learning nests, we show how nests may be constructed by hand to test hypotheses about the data, and evaluated by their explanatory power.