On the Measurement of Dissimilarity and Related Orders

We consider populations partitioned into groups, whose members are distributed across a finite number of classes such as, for instance, types of occupation, residential locations, social status of fathers, levels of education, health or income. Our aim is to assess the dissimilarity between the patterns of distributions of the different groups. These evaluations are relevant for the analysis of multi-group segregation, socioeconomic mobility, equalization of opportunity and discrimination. We conceptualize the notion of dissimilarity making use of reasonable transformations of the groups' distributions, based on sequences of transfers and exchanges of population masses across classes and/or groups. Our analysis clarifies the substantial differences underlying the concept of dissimilarity when applied to ordered or to permutable classes. In both settings, we illustrate the logical connections of dissimilarity evaluations with matrix majorization preorders, and provide equivalent implementable criteria to test unambiguous reductions in dissimilarity. Furthermore, we show that inequality evaluations can be interpreted as special cases of dissimilarity assessments and discuss relations with concepts of segregation and discrimination.