Luce rule with limited consideration

We study a modification of the Luce rule for stochastic choice which admits the possibility of zero probabilities. In any given menu, the decision maker uses the Luce rule on a consideration set, potentially a strict subset of the menu. Without imposing any structure on how the consideration sets are formed, we characterize the resulting behavior using a single axiom. Our result offers insight into special cases where consideration sets are formed under various restrictions.