zbMATH — the first resource for mathematics

A theory of diversity. (English) Zbl 1121.91414
Summary: How can diversity be measured? What does it mean to value biodiversity? Can we assist Noah in constructing his preferences? To address these questions, 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 flexible yet tractable manner.
A natural starting point is to think of the diversity of a set as an aggregate of the pairwise dissimilarities between its elements. The multi-attribute framework allows one to make this program formally precise. It is shown that the program can be realized if and only if the family of relevant attributes is well-ordered (“acyclic”). Moreover, there is a unique functional form aggregating dissimilarity into diversity, the length of a minimum spanning tree. Examples are taxonomic hierarchies and lines representing uni-dimensional qualities. In multi-dimensional settings, pairwise dissimilarity information among elements is insufficient to determine their diversity. By consequence, the qualitative and quantitative behavior of diversity differs fundamentally.

91D10 Models of societies, social and urban evolution
Full Text: DOI