The Relationship between Independent Race Models and Luce's Choice Axiom

Abstract

Four theorems concerning the relationship between independent race models and the choice axiom of R. D. Luce (1959, Individual choice behavior, New York: Wiley) are developed. In race models, selection (choice) is determined by a parallel processing race between elements in the choice set. Under weak assumptions, the choice axiom is equivalent to the condition that the hazard functions of the processing times of the elements are mutually proportional functions of time (Theorem 1). When this condition is satisfied, k th choices are equivalent to first choices from the reduced choice set (i.e., the original choice set minus the previously selected elements) (Theorem 2). Theorems 3 and 4 are general limit theorems. Whether or not the hazard functions are mutually proportional, the choice axiom describes the asymptotic choice probabilities approached under uniform expansion of the choice sets (Theorem 3). Moreover, in the limit, kth-choice probabilities are identical to first-choice probabilities (Theorem 4).