Quantifying redundancies and synergies with measures of inequality

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.