Ensuring fairness in matching algorithms is a key challenge in allocating scarce resources and positions. Focusing on Optimal Transport (OT), we introduce a novel notion of group fairness requiring that the probability of matching two individuals from any two given groups in the OT plan satisfies a predefined target. We first propose \texttt{FairSinkhorn}, a modified Sinkhorn algorithm to compute perfectly fair transport plans efficiently. Since exact fairness can significantly degrade matching quality in practice, we then develop two relaxation strategies. The first one involves solving a penalised OT problem, for which we derive novel finite-sample complexity guarantees. This result is of independent interest as it can be generalized to arbitrary convex penalties. Our second strategy leverages bilevel optimization to learn a ground cost that induces a fair OT solution, and we establish a bound guaranteeing that the learned cost yields fair matchings on unseen data. Finally, we present empirical results that illustrate the trade-offs between fairness and performance.

翻译：确保匹配算法的公平性是分配稀缺资源和职位的关键挑战。本文聚焦于最优传输（OT），引入了一种新颖的群体公平性概念，要求OT方案中任意两个给定群体的个体匹配概率满足预设目标。我们首先提出了\texttt{FairSinkhorn}，一种改进的Sinkhorn算法，用于高效计算完全公平的传输方案。由于严格的公平性在实践中可能显著降低匹配质量，我们随后开发了两种松弛策略。第一种策略涉及求解一个惩罚化的OT问题，为此我们推导了新颖的有限样本复杂度保证。该结果具有独立意义，因其可推广至任意凸惩罚函数。第二种策略利用双层优化学习一个能诱导公平OT解的基代价，并建立了一个保证性边界，确保所学代价在未见数据上产生公平匹配。最后，我们展示了实证结果，以说明公平性与性能之间的权衡关系。