Quantifying redundancies and synergies with measures of inequality

TL;DR

This paper addresses how inequality arises from interactions among individual attributes by introducing a family of $f$-inequality measures derived from $f$-divergences, which unifies Pietra, Generalized Entropy, and Atkinson indices. It develops a set-theoretic decomposition on a lattice of attribute sets using Möbius inversion, enabling additive quantification of redundant, unique, and synergetic contributions to inequality, and shows how transformations yield Atkinson-type decompositions. The framework is tied to Lorenz curves and zonogon geometry, allowing a geometric interpretation of orderings and a practical, measure-consistent decomposition that complements subgroup-based analyses. It also connects inequality decomposition to information theory (PID), mapping $f$-information to $f$-inequality and enabling cross-pollination of ideas between these domains. The practical impact lies in providing a principled method to dissect how attribute interactions shape inequality and to support multi-layer or multi-criteria analyses in engineering, economics, and fairness-oriented applications, with an available implementation at the referenced repository.

Abstract

Inequality measures provide a valuable tool for the analysis, comparison, and optimization based on system models. This work studies the relation between attributes or features of an individual to understand how redundant, unique, and synergetic interactions between attributes construct inequality. For this purpose, we define a family of inequality measures (f-inequality) from f-divergences. Special cases of this family are, among others, the Pietra index and the Generalized Entropy index. We present a decomposition for any f-inequality with intuitive set-theoretic behavior that enables studying the dynamics between attributes. Moreover, we use the Atkinson index as an example to demonstrate how the decomposition can be transformed to measures beyond f-inequality. The presented decomposition provides practical insights for system analyses and complements subgroup decompositions. Additionally, the results present an interesting interpretation of Shapley values and demonstrate the close relation between decomposing measures of inequality and information.

Figures (9)

• Figure 1: Intuition for the relation between a subgroup decomposition and the proposed attribute decomposition. Consider a set of companies with industry type and region as attributes: (a) A subgroup decomposition provides detailed insights for the possible values of an attribute, such as region 1 or 2. However, it does not provide insights into the dynamics between attributes. (b) An attribute decomposition provides detailed insights into the interaction between attributes, such as redundant and synergetic effects between industries and regions. However, it does not provide insights for particular attribute values, such as region 1 or 2. Therefore, subgroup and attribute decompositions complement each other.
• Figure 2: Zonogon construction, the meaning of its boundary and their ordering. (a) The zonogon of a population is a symmetric convex polygon containing the line from (0,0) to (1,1). Sorting the vectors of a normalized population matrix by increasing slope provides the lower boundary of the zonogon, which is the Lorenz curve. (b) Each zonogon boundary segment corresponds to one subgroup of the partition, and its slope is the expected normalized indicator value of its individuals. The subgroups for the partition on attribute $\mathbb{A}_1$ are labeled in the upper triangle and those for the partition on $\mathbb{A}_2$ are labeled in the lower triangle. The example was constructed such that the join of both attributes (Figure \ref{['fig:example-zonogon']}b) equals their joint distribution (Figure \ref{['fig:example-zonogon']}a). For any other attribute dependence, the zonogon of their joint distribution is a superset of Figure \ref{['fig:example-zonogon']}a.
• Figure 3: Redundancy example. Fully redundant contribution by both attributes.
• Figure 4: Unique example. Fully unique contribution by attribute $\mathbb{A}_1$.
• Figure 5: Synergetic example. Fully synergetic contribution by both attributes.

Theorems & Definitions (51)

• Remark
• Definition 1
• Example 1
• Definition 2
• Definition 3: Stochastic matrix
• Definition 4: Normalized population matrix
• Example 2
• Definition 5: Zonogon Mosler1996Mosler2007blackwell-no-lattice
• Definition 6: Zonogon order blackwell-no-lattice
• Definition 7: Population equivalence