Consistent and Inconsistent Inequality Indices for Ordinal Variables

Should inequality comparisons with ordinal variables be sensitive to alternative sorting of the categories, i.e. ascending versus descending order? 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 possible pair of comparable frequency distributions. We characterise the class of consistent inequality and bipolarisation indices with a functional equation that serves as a useful test of the property. Applying the test to the most popular inequality and bipolarisation indices proposed in the literature, we identify the respective consistent and inconsistent sets