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 then random choice occurs according to a tie-breaking mechanism 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.”

中文翻译：

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概率理性的经典

我们证明了一个随机选择规则满足 Luce 选择公理当且仅当它的支持是一个满足显示偏好弱公理的选择对应关系，因此它由根据某种偏好最优的替代方案组成，然后随机选择发生根据满足 Renyi 的条件公理的此类替代方案之间的平局机制。

我们的结果表明，在精确的形式意义上，选择公理是弱公理的概率版本。因此，它支持 Luce 将他自己的公理视为“概率理性标准”的观点。