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.”
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This paper combines and supersedes two independent works: Paper 6 ( Lindberg, 2012b ) of the thesis of Lindberg (2012a) and a working paper of Cerreia-Vioglio et al. (2016) . We thank Sean Horan, Marco Pavan (the editor), an anonymous associate editor, two anonymous referees for very helpful comments, Giulio Principi for research assistance, as well as the ERC (grants SDDM-TEA and INDIMACRO ) and a PRIN grant ( 2017CY2NCA ) for financial support. Aldo Rustichini thanks the US Army for financial support, contract W911NF2010242 .
© 2021 The Authors
- Choice axiom
- Random choice
- Weak axiom of revealed preferences