Measuring diversity with an ordinal conception of similarity: an Axiomatic approach

This paper provides an axiomatic characterization of two rules for comparing alternative sets of objects on the basis of the diversity that they offer. The framework considered assumes a Þnite universe of objects and an a priori given ordinal quadernary relation that compares alternative pairs of objects on the basis of their ordinal dissimilarity. Very few properties of this quadernary relation are assumed (beside completeness, transitivity and a very natural form of symmetry). The two rules that we characterize are the maxi max criterion and the leximax criterion. The maxi max criterion considers that a set is more diverse than another if and only if the two objects that are the most dissimilar in the former are weakly as dissimilar as the two most dissimilar objects in the later. The lexi-max criterion is deÞned as usual as the lexicographic extension of the maximax criterion.