Concavity and additivity in diversity measurement

Re-discovery of an unknown concept

Abstract

Concavity and additivity are two desirable properties of diversity measures that are derived from distribution based Shannon entropy. Unfortunately, both properties are traded-off against each other when generalizing Shannon entropy by deformation or Kolmogorov-Nagumo means. We analyze concavity and additivity properties of common Shannon entropy generalizations and drive a unifying framework for all of them. This framework, originally developed in information theory by Sharma and Mittal (1975), not only includes all common measures of distribution based diversity but also the classic "effective number" frequently used in ecology (Hill 1973), economics (Hannah and Kay 1977) and political sciences (Laakso and Taagepera 1979). Once having its properties revealed, the Sharma-Mittal formalism clearly allows decision makers to model diversity in a much more flexible and appropriate manner with regard to the given context and the properties required.