Luce's choice axiom

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.