Equivalence of inequality indices: Three dimensions of impact revisited

Inequality is an inherent part of our lives: we see it in the distribution of incomes, talents, resources, and citations, amongst many others. Its intensity varies across different environments: from relatively evenly distributed ones, to where a small group of stakeholders controls the majority of the available resources. We would like to understand why inequality naturally arises as a consequence of the natural evolution of any system. Studying simple mathematical models governed by intuitive assumptions can bring many insights into this problem. In particular, we recently observed (Siudem et al., PNAS 117:13896-13900, 2020) that impact distribution might be modelled accurately by a time-dependent agent-based model involving a mixture of the rich-get-richer and sheer chance components. Here we point out its relationship to an iterative process that generates rank distributions of any length and a predefined level of inequality, as measured by the Gini index. Many indices quantifying the degree of inequality have been proposed. Which of them is the most informative? We show that, under our model, indices such as the Bonferroni, De Vergottini, and Hoover ones are equivalent. Given one of them, we can recreate the value of any other measure using the derived functional relationships. Also, thanks to the obtained formulae, we can understand how they depend on the sample size. An empirical analysis of a large sample of citation records in economics (RePEc) as well as countrywise family income data, confirms our theoretical observations. Therefore, we can safely and effectively remain faithful to the simplest measure: the Gini index.