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 ≥ 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 [14] 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, . . . , n, associated with the distributional judgements involved. The j-th generalized Gini pair focuses on the (n − j + 1)/n poorest fraction of the population and is insensitive to income transfers within the complementary richest fraction of the population.
We consider a set N = {1, ..., n} of interacting agents whose individual opinions are denoted by x i , i ∈ N in some domain D ⊆ R. The interaction among the agents is expressed by a symmetric interaction matrix with null diagonal and off-diagonal coefficients in the open unit interval. The interacting network structure is thus that of a complete graph with edge values in (0, 1).In the Choquet framework, the interacting network structure is the basis for the construction of a consensus capacity µ, where the capacity value µ(S) of a coalition of agents S ⊆ N is defined to be proportional to the sum of the edge interaction values contained in the subgraph associated to S. The capacity µ is obtained in terms of its 2-additive Möbius transform m µ , and the corresponding Shapley power and interaction indices are identified.We then discuss two types of consensus dynamics, both of which refer significantly to the notion of context opinion. The second type converges simply the plain mean, whereas the first type produces the Shapley mean as the asymptotic consensual opinion. In this way it provides a dynamical realization of Shapley aggregation.
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