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.

翻译：不平等是我们生活中固有的一部分：我们在收入、才能、资源和引用等的分布中都能看到它。其强度在不同环境中变化：从相对均匀的分布，到一小部分利益相关者控制大部分可用资源的情况。我们想理解不平等为何会自然产生，作为任何系统自然演化的结果。研究基于直觉假设的简单数学模型能为这一问题带来许多洞见。特别是，我们最近观察到（Siudem等，PNAS 117:13896-13900, 2020），影响分布可以通过一个含有时变因素的基于智能体的模型精确建模，该模型融合了“富者愈富”和纯粹随机成分。在此，我们指出其与一个迭代过程的关系，该过程能生成任意长度的等级分布和由基尼指数衡量的预定不平等水平。许多量化不平等程度的指数已被提出。其中哪一个最具信息量？我们表明，在我们的模型下，Bonferroni、De Vergottini和Hoover等指数是等价的。给定其中一个，我们可以利用推导出的函数关系重建任何其他测度的值。同时，凭借所得公式，我们能理解它们如何依赖于样本容量。对经济学中大量引文记录（RePEc）以及按国家划分的家庭收入数据的实证分析，证实了我们的理论观察。因此，我们可以安全有效地坚持使用最简单的测度：基尼指数。