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."
We prove that a random choice rule satisfies Luce's Choice Axiom if and only if its support is a choice correspondence that satisfies the Weak Axiom of Revealed Preference, thus it consists of alternatives that are optimal according to some preference, and random choice then occurs according to a tie breaking among such alternatives 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."
D
, the set of "alternatives that can be chosen,"
A
thus it consists of alternatives that are optimal according to some preference,
A
then
D
stochastic
R
optimizers
such alternatives