Impossibility Theorems for Menu-Dependent Preference Functional

We consider functions that assign to each evaluation profile a preference system or a list of menu-dependent preferences. The rule by which such an assignment takes place is said to be a menu-dependent preference functional (MDPFL). We extend the concepts of invariance under individual cardinal transformations, weak Pareto, binary independence, weak dictatorship, and veto power from the context of social welfare functional to our framework of MDPFLs. We consider admissible sets of evaluation profiles that are slightly more general than necessarily requiring that all evaluation profiles be admissible. We introduce the concepts of nested and nested* MDPFLs. Our first result says that a nested MDPFL that is invariant under individual transformations, globally weakly Paretian, and satisfies global binary independence must be weakly dictatorial. Our second result says that a nested* MDPFL that is invariant under individual transformations, globally weakly Paretian, and satisfies global binary independence must have an individual/criterion that wields veto power.