Consistent and inconsistent inequality indices for ordinal variables

Should inequality comparisons with ordinal variables be sensitive to alternative sorting of the categories? We introduce the consistency property whereby an inequality or bipolarisation index regards frequency distribution r less unequal than s if and only if it ranks the reverse-ordered distribution r′ less unequal than s′ for every pair of comparable frequency distributions. We characterise the class of consistent indices with a functional equation that serves as a useful test of the property. Applying the test to the most popular indices in the literature, we identify the respective consistent and inconsistent sets.