We study the compatibility between the optimal statistical parity solutions and individual fairness. While individual fairness seeks to treat similar individuals similarly, optimal statistical parity aims to provide similar treatment to individuals who share relative similarity within their respective sensitive groups. The two fairness perspectives, while both desirable from a fairness perspective, often come into conflict in applications. Our goal in this work is to analyze the existence of this conflict and its potential solution. In particular, we establish sufficient (sharp) conditions for the compatibility between the optimal (post-processing) statistical parity $L^2$ learning and the ($K$-Lipschitz or $(\epsilon,\delta)$) individual fairness requirements. Furthermore, when there exists a conflict between the two, we first relax the former to the Pareto frontier (or equivalently the optimal trade-off) between $L^2$ error and statistical disparity, and then analyze the compatibility between the frontier and the individual fairness requirements. Our analysis identifies regions along the Pareto frontier that satisfy individual fairness requirements. (Lastly, we provide individual fairness guarantees for the composition of a trained model and the optimal post-processing step so that one can determine the compatibility of the post-processed model.) This provides practitioners with a valuable approach to attain Pareto optimality for statistical parity while adhering to the constraints of individual fairness.

翻译：我们研究最优统计奇偶性解与个体公平之间的兼容性。个体公平旨在对相似个体给予相似对待，而最优统计奇偶性则力求对各自敏感群体内具有相对相似性的个体提供相似对待。这两种公平视角虽然从公平性角度均具有可取性，但在实际应用中常相互冲突。本文旨在分析该冲突的存在性及其潜在解决方案。具体而言，我们建立了最优（后处理）统计奇偶性$L^2$学习与（$K$-利普希茨或$(\epsilon,\delta)$）个体公平要求之间兼容性的充分（尖锐）条件。进一步地，当两者存在冲突时，我们首先将前者松弛为$L^2$误差与统计差异之间的帕累托前沿（即最优权衡），继而分析该前沿与个体公平要求之间的兼容性。我们的分析识别出帕累托前沿上满足个体公平要求的区域。（最后，我们为训练模型与最优后处理步骤的组合提供个体公平性保证，从而可判定后处理模型的兼容性。）这为实践者提供了一种在遵守个体公平约束同时实现统计奇偶性帕累托最优性的有价值方法。