Silvia Bortot

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>=2 individuals, and on the other hand, the Mobius 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 Donaldsons 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.

Inconsistency and non-additive capacities: The Analytic Hierarchy Process in the framework of Choquet integration

We examine the AHP in the framework of Choquet integration and we propose an extension of the standard AHP aggregation scheme on the basis of the Shapley values associated with the criteria. In our model a measure of dominance inconsistency between criteria is defined in terms of the totally inconsistent matrix associated with the main pairwise comparison matrix of the AHP. The measure of dominance inconsistency is then used to construct a non-additive capacity whose associated Shapley values reduce to the standard AHP priority weights in the consistency case. In the general inconsistency case, however, the extended aggregation scheme based on the Shapley weighted mean tends to attenuate (resp. emphasize) the priority weights of the criteria with higher (resp. lower) average dominance inconsistency with respect to the other criteria.

The Generalized Gini Welfare Function in the Framework of Symmetric Choquet Integration

In the context of Social Welfare and Choquet integration, we briefly review the classical Gini inequality index for populations of n ≥ 2 individuals, including the associated Lorenz area formula, plus the k-additivity framework for Choquet integration introduced by Grabisch, particularly in the additive and 2-additive symmetric cases. We then show that any 2-additive symmetric Choquet integral can be written as the difference between the arithmetic mean and a multiple of the classical Gini inequality index, with a given interval constraint on the multi- plicative parameter. In the special case of positive parameter values, this result corresponds to the well-known Ben Porath and Gilboa’s formula for Weymark’s generalized Gini welfare functions, with linearly decreasing (inequality averse) weight distributions

The Soft Consensus Model in the Multidistance Framework

In the context of the soft consensus model due to (Fedrizzi et al. in Journal international journal of intelligent systems 14:63–77, 1999) [27], (Fedrizzi et al. in New mathematics and natural computation 3:219–237, 2007) [28], (Fedrizzi et al. in Preferences and Decisions: models and applications, studies in fuzziness and soft computing Springer, Heidelberg, pp. 159–182, 2010) [30], we investigate the reformulation of the soft dissensus measure in relation with the notion of multidistance, recently introduced by Martin and Mayor (Information processing and management of uncertainty in knowledge-based systems. Theory and methods, communications in computer and information science, springer, heidelberg, pp. 703–711 2010) [43], Martin and Mayor (Fuzzy sets and systems 167:92–100 2011) [44]. The concept of multidistance is as an extension of the classical concept of binary distance, obtained by means of a generalization of the triangular inequality. The new soft dissensus measure introduced in this paper is a particular form of sum-based multidistance. This multidistance is constructed on the basis of a binary distance defined by means of a subadditive scaling function, whose role is that of emphasizing small distances and attenuating large distances in preferences. We present a detailed study of the subadditive scaling function, which is analogous but not equivalent to the one used in the traditional form of the soft consensus model.

A Multidistance Approach to Consensus Modeling

We investigate the relationship between the soft measure of collective dissensus introduced in (Fedrizzi et al. Int J Intell Syst 14:63–77, 1999; Fedrizzi et al. New Math Nat Comput 3:219–237, 2007; Preferences and Decisions: Models and Applications, Springer, Heidelberg, 2010) and the multidistance approach to consensus evaluation described in (Brunelli et al. IPMU 2012, Part I, CCIS, Springer, Berlin, 2012). The novelty of the contribution consists in the introduction of a particular type of sum-based multidistance used as a measure of dissensus, closely related with the one introduced in (Fedrizzi et al. New Math Nat Comput 3:219–237, 2007). This multidistance is characterized by the application of a subadditive filtering function whose effect is that of emphasizing small distances and attenuating large ones. An illustrative example is then developed in order to compare the new dissensus measure with the OWA-based multidistance obtained assuming that the weights are linearly decreasing with respect to increasing distance values.

Choquet Integration and the AHP: Inconsistency and Non-additivity

We propose to extend the aggregation scheme of the AHP, from the standard weighted averaging to the more general Choquet integration. In our model, a measure of dominance inconsistency between criteria is derived from the main pairwise comparison matrix of the AHP and it is used to construct a non-additive capacity, whose associated Choquet integral reduces to the standard weighted mean of the AHP in the consistency case. In the general inconsistency case, however, the new AHP aggregation scheme based on Choquet integration tends to attenuate (resp. emphasize) the priority weights of the criteria with higher (resp. lower) average dominance inconsistency with the other criteria.

The Single Parameter Family of Gini Bonferroni Welfare Functions and the Binomial Decomposition, Transfer Sensitivity and Positional Transfer Sensitivity

We consider the binomial decomposition of generalized Gini welfare functions in terms of the binomial welfare functions \(C_j\), \(j=1,\ldots ,n\) and we examine the weighting structure of the latter, which progressively focus on the poorest part of the population. In relation with the generalized Gini welfare functions, we introduce measures of transfer sensitivity and positional transfer sensitivity and we illustrate the behaviour of the binomial welfare functions \(C_j\), \(j=1,\ldots ,n\) with respect to these measures. We investigate the binomial decomposition of the Gini Bonferroni welfare functions and we illustrate the dependence of the binomial decomposition coefficients in relation with the single parameter which describes the family. Moreover we examine the family of Gini Bonferroni welfare functions with respect to the transfer sensitivity and positional transfer sensitivity principles.

The Binomial Decomposition of the Single Parameter Family of GB Welfare Functions

We consider the binomial decomposition of generalized Gini welfare functions in terms of the binomial welfare functions \(C_j\), \(j=1,\ldots ,n\) and we examine the weighting structure of the binomial welfare functions \(C_j\), \(j=1,\ldots ,n\) which progressively focus on the poorest part of the population. We introduce a parametric family of income distributions and we illustrate the numerical behavior of the single parameter family of GB welfare functions with respect to those income distributions. Moreover, we investigate the binomial decomposition of the GB welfare functions and we illustrate the dependence of the binomial decomposition coefficients in relation with the single parameter which describes the family.

The binomial decomposition of OWA functions, the 2‐additive and 3‐additive cases in n dimensions

In the context of the binomial decomposition of ordered weighted averaging (OWA) functions, we investigate the constraints associated with the 2‐additive and 3‐additive cases in n dimensions. The 2‐additive case depends on one coefficient whose feasible region does not depend on the dimension n. On the other hand, the feasible region of the 3‐additive case depends on two coefficients and is explicitly dependent on the dimension n. This feasible region is a convex polygon with n vertices and n edges, which is strictly expanding in the dimension n. The orness of the OWA functions within the feasible region is linear in the two coefficients, and the vertices associated with maximum and minimum orness are identified. Finally, we discuss the 3‐additive binomial decomposition in the asymptotic infinite dimensional limit.

The binomial decomposition of generalized Gini welfare functions, the S-Gini and Lorenzen cases

Abstract We consider the binomial decomposition of generalized Gini welfare functions in terms of the binomial welfare functions and the associated binomial inequality indices. We examine in detail the weights of the binomial welfare functions and the coefficients of the associated binomial inequality indices which progressively focus on the poorest sector of the population, and we illustrate the numerical behavior of the binomial welfare functions and inequality indices in relation with a new parametric family of income distributions. The main contribution of the paper is to investigate the analogy between the binomial welfare functions and the S-Gini and Lorenzen parametric families of generalized Gini welfare functions, particularly in the context of the binomial decomposition. Finally, we examine the orness of the parametric S-Gini and Lorenzen families of generalized Gini welfare functions.