Equity, hierarchy, and ordinal social choice

A social welfare ordering is a mixed hierarchy if and only if it satisfies (ordinal or cardinal) interpersonal noncomparability. In order to avoid this impossibility, we consider the set of social welfare orderings satisfying coordinality, which allows for interpersonal comparability in the ordinal sense and is nicely compatible with the requirement of equity. Under the assumption of coordinality, a social welfare ordering is a hierarchy if and only if it satisfies the strong Pareto principle and invariance with respect to individual changes of origin; it is a rank hierarchy if and only if it satisfies anonymity, weak separability, and the strong Pareto principle; it is the leximin rule if and only if it satisfies anonymity, the strong Pareto principle, and the Pigou-Dalton principle.