A stricter canon: general Luce models for arbitrary menu sets

The classical Luce model (Luce, 1959) assumes positivity of random choice: each available alternative is chosen with strictly positive probability. The model is characterised by Luce's choice axiom. Ahumada and Ulku (2018) and (independently) Echenique and Saito (2019) define the general Luce model (GLM), which relaxes the positivity assumption, and show that it is characterised by a cyclical independence (CI) axiom. Cerreia-Vioglio et al. (2021) subsequently proved that the choice axiom characterises an important special case of the GLM in which a rational choice function (i.e., one that may be rationalised by a weak order) first selects the acceptable alternatives from the given menu, with any residual indifference resolved randomly in Luce fashion. The choice axiom is thus revealed as a fundamental "canon of probabilistic rationality". This result assumes that choice behaviour is specified for all non-empty, finite menus that can be constructed from a given universe, X, of alternatives. We relax this assumption by allowing choice behaviour to be specified for an arbitrary collection of non-empty, finite menus. In this context, we show that the Cerreia-Vioglio et al. (2021) result obtains when the choice axiom is replaced with a mild strengthening of CI. The latter condition implies the choice axiom, thus providing a "stricter canon".