ON CHOICE-PROBABILITIES DERIVED FROM RANKING DISTRIBUTIONS

We present two axioms for a probability distribution P on the rankings of a finite set that are necessary and sufficient to induce choice probabilities that satisfy Luce's choice axiom. One axiom, a split-and-splice independence condition which says that P(rs) P(r's') = P(rs') P(r's) when r and r' are rankings of a subset and s and s' are rankings of the complementary subset, is tantamount to the L-decomposability condition in Critchlow, Fligner, and Verducci (1991, Journal of Mathematical Psychology, 35, 294-318). This axiom by itself allows P to be written as a product of ''choice'' probabilities for successively smaller sets, but if the second axiom fails then these probabilities should not be interpreted in the traditional manner of random utility maximization. The second axiom says that if x is a member of a non-singleton proper subset B of the basic set, then there is a ranking s of the objects not in B such that the unconditional probability that x is ranked first in B equals the conditional probability that x ranks first in B given that the objects not in B have ranking s and are ranked ahead of all objects in B. We note also that the conjunction of the two axioms is equivalent to a strong version of the second axiom. (C) 1994 Academic Press, Inc.