Ordinal Distance, Dominance, and the Measurement of Diversity
The purpose of this chapter is to consider a class of rules for comparing sets of objects1 in terms of the degrees of diversity that they offer. Such comparisons of sets are important for many purposes. For example, in discussing biodiversity of different
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1 Introduction The purpose of this chapter is to consider a class of rules for comparing sets of objects1 in terms of the degrees of diversity that they offer. Such comparisons of sets are important for many purposes. For example, in discussing biodiversity of different ecosystems, one is interested in knowing whether or not one ecosystem is more diverse than another. Similarly, when discussing issues relating to cultural diversities of various communities, one may be interested in knowing how these communities compare with each other in terms of cultural diversity. In the economics literature, there have been several contributions to the measurement of diversity. Weitzman (1992, 1993, 1998) develops a measure of diversity based on cardinal distances between objects. Among other things, Nehring and Puppe (2002) provide a conceptual foundation for cardinal distances in Weitzman’s framework. Weikard (2002) discusses an alternative measure of diversity; Weikard’s measure is based on the sum of cardinal distances between all objects contained in a set. Underlying much of our everyday discussion of diversity, we have some intuition regarding the extent to which objects are dissimilar,2 though, in its coarsest form, this intuition may distinguish between only two degrees of similarity by declaring that two objects are either similar or dissimilar. It is difficult to see how one can compare the diversity of one group of objects with that of another without some P.K. Pattanaik Department of Economics, University of California, Riverside, CA 92521, USA e-mail: [email protected] Y. Xu Department of Economics, Andrew Young School of Policy Studies, Georgia State University, Atlanta, GA 30303, USA e-mail: [email protected] 1
We use the term “objects” rather broadly so that people and animals can also be objects. The basis for assessing dissimilarity or similarity of two objects will, of course, vary depending on the specific notion of diversity in which one is interested.
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P.K. Pattanaik et al. (eds.) Rational Choice and Social Welfare: Theory and Applications, c Springer-Verlag Berlin Heidelberg 2008 Studies in Choice and Welfare.
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notion, however coarse, of the distances or the degrees of dissimilarity between different objects in these sets. The requirement that distances be cardinally measurable does, however, seem rather restrictive in many contexts. Thus, in considering linguistic diversity, we may not be able to compare the extent to which the (linguistic) dissimilarity between an English person and a Chinese person exceeds the difference between a Chinese person and a Hindi-speaking person, on the one hand, and the extent to which the difference between an Italian person and a Hindi-speaking person exceeds the dissimilarity between a Spanish-speaking person and a Hindispeaking person, on the other. This is not to claim that cardinal distance functions never have sound intuitive foundations. In general, however, the requirement of a cardinal distance function for the measurement of div
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