Fuzzy measurement of income inequality: Some possibility results on the fuzzification of the Lorenz ordering

This paper starts from the premise that the concept of income inequality is ill-defined, and hence, it studies the measurement of income inequality from a fuzzy set theoretical point of view. It is argued that the standard (fuzzy) transitivity concepts are not compatible with fuzzy inequality orderings which respect Lorenz ordering. For instance, we show that there does not exist a max-min transitive fuzzy relation on a given income distribution space which ranks distributions unambiguously according to the Lorenz criterion whenever they can actually be ranked by it. Weakening the imposed transitivity concept, it is possible to escape from the noted impossibility theorems. We introduce some alternative transitivity concepts for fuzzy relations, and subsequently, construct a class of fuzzy orderings which preserve Lorenz ordering and satisfy these alternative transitivities. It is also shown that fuzzy measurement can be used to construct confidence intervals for the crisp conclusions of inequality indices.