The binomial Gini inequality indices and the binomial decomposition of welfare functions

In the context of Social Welfare and Choquet integration, we briefly review, on the one hand, the generalized Gini welfare functions and inequality indices for populations of n≥2n≥2 individuals, and on the other hand, the Möbius representation framework for Choquet integration, particularly in the case of k  -additive symmetric capacities. We recall the binomial decomposition of OWA functions due to Calvo and De Baets and we examine it in the restricted context of generalized Gini welfare functions, with the addition of appropriate S-concavity conditions. We show that the original expression of the binomial decomposition can be formulated in terms of two equivalent functional bases, the binomial Gini welfare functions and the Atkinson–Kolm–Sen (AKS) associated binomial Gini inequality indices, according to Blackorby and Donaldson's correspondence formula. The binomial Gini pairs of welfare functions and inequality indices are described by a parameter j=1,…,nj=1,…,n, associated with the distributional judgements involved. The j  -th generalized Gini pair focuses on the (n−j+1)/n(n−j+1)/n poorest fraction of the population and is insensitive to income transfers within the complementary richest fraction of the population.