SFB 303 Discussion Paper No. A - 605
Author: Nehring,K. und C.Puppe
Title: A Theory of Diversity
Abstract: 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.
Keywords: Diveristy, endangered species, similarity, set function, conjugate
Moebius inversion
JEL-Classification-Number: D 11
Creation-Date: September 1999
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