Measuring Development pp 85-108 | Cite as

# Unidimensional Development Ranking and Fuzzy Lorenz Dominance

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## Abstract

This chapter is devoted to an attempt at extending the unidimensional theory of development ranking so as to reduce the possibility of ranking failures. For this purpose, we borrow from the theory of fuzzy sets in mathematics and use the notion of *fuzzy Lorenz dominance relation*. The approach of modelling Lorenz dominance by a fuzzy binary relation was proposed in the 1980s. The idea does not seem to have been followed up actively in the subsequent literature. The fuzzy dominance relation proposed in the chapter is a follow-up on this line of research. It is, however, different from the earlier suggestions. Moreover, if the idea of Lorenz dominance is “fuzzified”, it would be natural to fuzzify the notion of development ranking itself. Indeed, this is what we do in this chapter. It is seen that such fuzzy development rankings can also be used to induce *crisp* (i.e. non-fuzzy) development rankings. The ranking methods developed in this chapter seem to be able to *reduce* the preponderance of the problem of ranking failures that arises frequently under the *crisp* (i.e. non-fuzzy) approach. In particular, it is shown that if two economies have the same per capita income, we shall now *always* be able to rank them in terms of development. While that is not the case when per capita incomes differ, completeness of the ranking is achieved under *weaker* conditions than in crisp theory.

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