Inequality measurement with an ordinal and continuous variable

What would be the analogue of the Lorenz quasi-ordering when the variable of interest is continuous and of a purely ordinal nature? We argue that it is possible to derive such a criterion by substituting for the Pigou–Dalton transfer used in the standard inequality literature what we refer to as a Hammond progressive transfer. According to this criterion, one distribution of utilities is considered to be less unequal than another if it is judged better by both the lexicographic extensions of the maximin and the minimax, henceforth referred to as the leximin and the antileximax, respectively. If one imposes in addition that an increase in someone’s utility makes the society better off, then one is left with the leximin, while the requirement that society welfare increases as the result of a decrease of one person’s utility gives the antileximax criterion. Incidentally, the paper provides an alternative and simple characterisation of the leximin principle widely used in the social choice and welfare literature.