The Multinomial Logit (MNL) model and the axiom it satisfies, the Independence of Irrelevant Alternatives (IIA), are together the most widely used tools of discrete choice. The MNL model serves as the workhorse model for a variety of fields, but is also widely criticized, with a large body of experimental literature claiming to document real-world settings where IIA fails to hold. Statistical tests of IIA as a modelling assumption have been the subject of many practical tests focusing on specific deviations from IIA over the past several decades, but the formal size properties of hypothesis testing IIA are still not well understood. In this work we replace some of the ambiguity in this literature with rigorous pessimism, demonstrating that any general test for IIA with low worst-case error would require a number of samples exponential in the number of alternatives of the choice problem. A major benefit of our analysis over previous work is that it lies entirely in the finite-sample domain, a feature crucial to understanding the behavior of tests in the common data-poor settings of discrete choice. Our lower bounds are structure-dependent, and as a potential cause for optimism, we find that if one restricts the test of IIA to violations that can occur in a specific collection of choice sets (e.g., pairs), one obtains structure-dependent lower bounds that are much less pessimistic. Our analysis of this testing problem is unorthodox in being highly combinatorial, counting Eulerian orientations of cycle decompositions of a particular bipartite graph constructed from a data set of choices. By identifying fundamental relationships between the comparison structure of a given testing problem and its sample efficiency, we hope these relationships will help lay the groundwork for a rigorous rethinking of the IIA testing problem as well as other testing problems in discrete choice.

Many applications in preference learning assume that decisions come from the maximization of a stable utility function. Yet a large experimental literature shows that individual choices and judgements can be affected by "irrelevant" aspects of the context in which they are made. An important class of such contexts is the composition of the choice set. In this work, our goal is to discover such choice set effects from raw choice data. We introduce an extension of the Multinomial Logit (MNL) model, called the context dependent random utility model (CDM), which allows for a particular class of choice set effects. We show that the CDM can be thought of as a second-order approximation to a general choice system, can be inferred optimally using maximum likelihood and, importantly, is easily interpretable. We apply the CDM to both real and simulated choice data to perform principled exploratory analyses for the presence of choice set effects.