Fuzzy Inequality Ranking Relations and Their Crisp Approximations

This paper is concerned with inequality ranking of income distributions. Ranking a pair of distributions in terms of inequality is a difficult exercise excepting in the special case where one of these Lorenz dominates the other. To get around the difficulty, the notion of a fuzzy inequality ranking relation (FIRR) has been fruitfully used in the literature. However, it is natural to ask how, from a given FIRR, we can derive a crisp approximation that would give us an overall crisp (i.e., non-fuzzy) judgment regarding the inequality ranking of any pair of distributions. It turns out that this can be done in many different ways (leading to different crisp judgments) and, therefore, there arises the question which of these alternative ways is to be chosen. It is seen that this question is closely linked to the more general one as to how we can decide which income vectors in any given finite set A of such vectors can be considered to be least unequal in an overall crisp assessment. In this paper, we first axiomatically characterize a specific procedure for making this decision. A particular procedure for obtaining a crisp approximation of an FIRR then follows as a corollary. We also construct an illustrative example of an FIRR that does not seem to have appeared in the literature before and compare it with some other known examples.