Before revision: 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.
After revision: 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 such alternatives that satisfies Renyi's Conditioning Axiom.