A Canon of Probabilistic Rationality

We prove that a random choice rule satisfies Luce's Choice Axiom if and only if its support, the set of "alternatives that can be chosen," is a choice correspondence that satisfies the Weak Axiom of Revealed Preference, and random choice occurs according to a stochastic tie breaking among optimizers that satisfies Renyi's Conditioning Axiom. Our result shows that the Choice Axiom is, in a precise formal sense, a probabilistic version of the Weak Axiom. It thus supports Luce's view of his own axiom as a "canon of probabilistic rationality."