This article establishes an equivalence between four incomplete rankings of distributions of income among agents who are vertically differentiated with respect to some nonincome characteristic (health, household size, etc.). The first ranking is the possibility of going from one distribution to the other by a finite sequence of income transfers from richer and more highly ranked agents to poorer and less highly ranked ones. The second ranking is the unanimity among utilitarian planners who assume that agents' marginal utility of income is decreasing with respect to both income and the source of vertical differentiation. The third ranking is the Bourguignon (Journal of Econometrics, 42 (1989), 67–80) Ordered Poverty Gap dominance criterion. The fourth ranking is a new dominance criterion based on cumulative lowest incomes.

中文翻译：

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减少不平等之间的不平等

本文建立了四个不完全排名之间的等价关系，这些排名在某些非收入特征（健康状况，家庭人数等）之间存在纵向差异。排名靠前的是一种可能性，即按收入从富裕程度较高的代理商到贫困程度较低的代理商的有限转移顺序，从一种分配转到另一种分配。第二个排名是功利计划者之间的一致，他们认为代理人的边际收入效用相对于收入和垂直差异的来源都在下降。第三名是布尔吉尼翁（计量经济学杂志），42（1989），67–80）有序的贫困差距主导标准。第四位是基于累计最低收入的新主导标准。