The profile structure for Luce's choice axiom

Abstract

A geometric approach is introduced to explain phenomena that can arise with Luce's choice axiom; e.g., differences occur when determining the likelihood of a ranking by starting with the “best-first,” or “worst-first” alternative. As shown, the problem is caused by the way we compute pairwise probabilities: it forces “best-first” and “worst-first” computations to use different information from a profile. Thus agreement holds only should the different information agree: this happens only with complete indifference. An alternative “best-first” and “worst-first” comparison, which always holds, is developed. Ways to increase the applicability of the choice axiom are introduced; e.g., profiles admitted by Luce's formulation for ten alternatives have nine degrees of freedom; the approach described here allows millions of degrees of freedom. New ways to compute probabilities, which combine “best-first” and “worst-first” computations, are given: their properties are identified with a profile decomposition. A new way to compute pairwise probabilities, which eliminates all profile restrictions and problems associated with the choice axiom, is introduced; e.g., “best-first” and “worst-first” computations now agree. Three and four alternatives are emphasized for reasons of exposition, but most results extend to any number of alternatives.