In the assignment problem, a set of items must be allocated to unit-demand agents who express ordinal preferences (rankings) over the items. In the assignment problem with priorities, agents with higher priority are entitled to their preferred goods with respect to lower priority agents. A priority can be naturally represented as a ranking and an uncertain priority as a distribution over rankings. For example, this models the problem of assigning student applicants to university seats or job applicants to job openings when the admitting body is uncertain about the true priority over applicants. This uncertainty can express the possibility of bias in the generation of the priority ranking. We believe we are the first to explicitly formulate and study the assignment problem with uncertain priorities. We introduce two natural notions of fairness in this problem: stochastic envy-freeness (SEF) and likelihood envy-freeness (LEF). We show that SEF and LEF are incompatible and that LEF is incompatible with ordinal efficiency. We describe two algorithms, Cycle Elimination (CE) and Unit-Time Eating (UTE) that satisfy ordinal efficiency (a form of ex-ante Pareto optimality) and SEF; the well known random serial dictatorship algorithm satisfies LEF and the weaker efficiency guarantee of ex-post Pareto optimality. We also show that CE satisfies a relaxation of LEF that we term 1-LEF which applies only to certain comparisons of priority, while UTE satisfies a version of proportional allocations with ranks. We conclude by demonstrating how a mediator can model a problem of school admission in the face of bias as an assignment problem with uncertain priority.

翻译：在分配问题中，一组物品需分配给具有单位需求且对物品表达序数偏好（排序）的代理人。在具有优先权的分配问题中，优先级较高的代理人相较于优先级较低的代理人有权获得其偏好的物品。优先权可自然地表示为一种排序，而不确定优先权则表示为排序上的分布。例如，当招生机构对申请者真实优先权存在不确定性时，该模型可描述学生申请大学名额或求职者申请职位空缺的问题。这种不确定性可表达优先权排序生成过程中可能存在的偏差。我们认为本研究首次明确阐述并研究了具有不确定优先权的分配问题。我们引入了该问题中的两种自然公平概念：随机无嫉妒性（SEF）和似然无嫉妒性（LEF）。我们证明了SEF与LEF不相容，且LEF与序数效率不相容。我们描述了两种算法：循环消除算法（CE）和单位时间消耗算法（UTE），它们满足序数效率（一种事前帕累托最优形式）和SEF；著名的随机序列独裁算法满足LEF以及较弱的效率保证——事后帕累托最优。我们还证明了CE满足LEF的松弛版本（我们称之为1-LEF，仅适用于某些特定优先权比较），而UTE满足基于排名的比例分配版本。最后，我们展示了中介机构如何将存在偏差的学校招生问题建模为具有不确定优先权的分配问题。