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Fundamental Limits of Testing the Independence of Irrelevant Alternatives in Discrete Choice

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Document pages: 32 pages

Abstract: The Multinomial Logit (MNL) model and the axiom it satisfies, theIndependence of Irrelevant Alternatives (IIA), are together the most widelyused tools of discrete choice. The MNL model serves as the workhorse model fora variety of fields, but is also widely criticized, with a large body ofexperimental literature claiming to document real-world settings where IIAfails to hold. Statistical tests of IIA as a modelling assumption have been thesubject of many practical tests focusing on specific deviations from IIA overthe past several decades, but the formal size properties of hypothesis testingIIA are still not well understood. In this work we replace some of theambiguity in this literature with rigorous pessimism, demonstrating that anygeneral test for IIA with low worst-case error would require a number ofsamples exponential in the number of alternatives of the choice problem. Amajor benefit of our analysis over previous work is that it lies entirely inthe finite-sample domain, a feature crucial to understanding the behavior oftests in the common data-poor settings of discrete choice. Our lower bounds arestructure-dependent, and as a potential cause for optimism, we find that if onerestricts the test of IIA to violations that can occur in a specific collectionof choice sets (e.g., pairs), one obtains structure-dependent lower bounds thatare much less pessimistic. Our analysis of this testing problem is unorthodoxin being highly combinatorial, counting Eulerian orientations of cycledecompositions of a particular bipartite graph constructed from a data set ofchoices. By identifying fundamental relationships between the comparisonstructure of a given testing problem and its sample efficiency, we hope theserelationships will help lay the groundwork for a rigorous rethinking of the IIAtesting problem as well as other testing problems in discrete choice.

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