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 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 penalized OT problem, for which we derive novel finite-sample complexity guarantees. Our second strategy leverages bilevel optimization to learn a ground cost that induces a fair OT solution, and we establish a bound on the deviation of fairness when matching unseen data. Finally, we present empirical results illustrating the performance of our approaches and the trade-off between fairness and transport cost.

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。聚焦于最优传输（OT），我们引入了一种新颖的群体公平性概念，要求最优传输方案中任意两个给定群体的个体匹配概率满足预设目标。我们首先提出一种改进的Sinkhorn算法，以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最优传输问题，为此我们推导了新颖的有限样本复杂度保证。我们的第二种策略利用双层优化来学习一个能诱导出公平最优传输解的基础成本函数，并建立了在匹配未见数据时公平性偏差的界限。最后，我们展示了实证结果，说明了所提方法的性能以及公平性与传输成本之间的权衡关系。