Luce's Choice Axiom
Luce’s Choice Axiom (hereafter LCA) is a quantitative hypothesis about choice behavior first proposed and analysed by Luce (1959). It envisions a setting in which an individual makes repeated choices from a set A containing N alternatives: A (a , a , …, aN) (e.g., N restaurants). 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). On each occasion exactly one alternative is chosen. Choice is assumed to be probabilistic: faced with the same set of alternatives on different occasions, the individual may make different choices.P(i; S) denotes the probability that ai is chosen when the set of available alternatives is S; e.g., P(1; A) is the probability that a is chosen when all N alternatives are available, and if S a , a , a , P(1; S) is the probability that a is chosen when only a , a , and a are available. The special case of 2- alternative choice (known as ‘paired comparison’) arises often and has its own notation: P(i, j) is the probability of choosing ai when the available set is ai, aj .