Luce's Choice Axiom

Abstract

Luce's Choice Axiom (LCA) is a hypothesis about probabilistic choice behavior (leading to a mathematical model) due to R. D. Luce. It envisions a situation in which an individual makes repeated choices from a set A containing N alternatives: A={ a 1, …, a N } (e.g., N restaurants). On each occasion exactly one alternative is selected. Sometimes all N alternatives are available for selection (all the restaurants are open); on other occasions only subsets of A are available (some restaurants are closed). P( i; A) denotes the probability that a i is chosen when all of A is available; P( i; S) is the probability that a i is chosen when the available set of alternatives is S⊆ A. What is the relationship between P(i; S) and P( i; A)? LCA is the assumption that P( i; S) equals the conditional probability that a i is chosen from the full set A, given that the choice from A belongs to subset S. The article deals with: (a) testable predictions of LCA (e.g., the constant ratio rule: for all i≠ j and S⊆ A, P( i; S)/ P( j; S)= P( i; A) /P( j; A)), (b) the empirical validity of LCA, (c) relationships between LCA and other models for choice behavior: Thurstone's ‘Law of Comparative Judgement,’ Tversky's ‘Choice by Elimination’ model, McFadden's ‘Multinomial Logit’ and ‘Generalized Extreme Value’ models, and (d) extension of LCA to preferences expressed by rank ordering.