A Stricter Canon: General Luce Models for Arbitrary Menu Sets

Abstract

Alós-Ferrer and Mihm (2025, Corollary 1) recently provided a characterisation the classical Luce model (Luce, 1959) when choices are observed for an arbitrarily restricted collection of menus, as is typical in experimental settings or when working with field data. They also characterise the general Luce model (ibid., Theorem 1), which allows choice probabilities to be zero, for the same setting. The latter characterisation involves a single axiom – the general product rule (GPR). An important special case of the general Luce model is obtained when the mapping from menus to the support of choice probabilities can be rationalised by a weak order. Cerreia-Vioglio et al. (2021) show that this special case is characterised by Luce’s (1959) choice axiom, provided choice is observed for all possible (finite) menus. The choice axiom is thus a fundamental “canon of probabilistic rationality”. We show that a natural – and surprisingly simple – strengthening of the GPR characterises the model of Cerreia-Vioglio et al. (2021) when the menu set is arbitrarily restricted. Our axiom implies the choice axiom, and is therefore a “stricter canon”.