Independence of alternatives in ranking models

When Luce (1959) introduced his Choice Axiom, this raised immediate criticism by Debreu (1960), pointing out inconsistencies when items are ranked from inferior to superior (instead of ranking them from superior to inferior). As recently shown by Breitmoser (2019), Luce’s Independence of Irrelevant Alternatives (IIA) is equivalent to Luce’s Choice Axiom when positivity holds. This fact seems to have escaped attention so far and might suggest that Debreu’s critique also applies to the notion of IIA, which is widely used in the literature. Furthermore, this notion could potentially be intuitively misleading, as the consequences of this axiom seem to be different than the name suggests. This might spill over to the intuitive interpretation of theoretical results that build on this axiom. This paper motivates the introduction of the notion of Independece of Alternatives (IoA) in the context of ranking models. IoA postulates a property of independence which seems intuitively reasonable (as it exactly captures what Luce himself describes when speaking about IIA), but does not exclusively hold in models where Luce’s Choice Axiom applies. Assuming IoA, expected ranks in the ranking of multiple alternatives can be determined from pairwise comparisons. The result holds in many models which do not satisfy IIA (e.g. certain Thurstone V models, Thurstone (1927)), can significantly simplify the calculation of expected ranks in practice and potentially facilitate analytic methods that build on more general approaches to model the ranking of multiple alternatives.