A Theory of Diversity
How can diversity be measured? What does it mean to value biodiversity? Can we assist Noah in constructing his preferences? To address these questions following Weitzman (1992,1998), we propose a multi-attribute approach under which the diversity of a set of species is the sum of the values of all attributes possessed by some species in the set. We develop the basic intuitions and requirements for a theory of diversity and show that the multi-attribute approach satisfies them in a highly flexible yet tractable manner. Conjugate Moebius inversion serves as the unifying mathematical tool. A basic starting point is to think of the diversity of a set as an aggregate of the dissimilarities between its elements. This intuition is made formally precise, and the exact conditions of itsapplicability are characterized: the family of relevant attributes must satisfy a condition of acyclicity. The two most important attribute structures satisfying acyclicity, taxonomic hierarchies and lines representing uni-dimensional qualities, are studied in depth, and the entailed restrictions on the dissimilarity metric are characterized. In multi-dimensional settings, pairwise dissimilarity information among elements is typically insufficient to determine the diversity of their set. Using a parametrization of the hypercube as the simplest high-dimensional model, we discuss the new issues and phenomena that arise. Even simple instances of Noah's choice problem become combinatorially complex, and the quantitative behaviour of diversity differs fundamentally.
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|Date of creation:||Sep 1999|
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