We study a family of distance functions on rankings that allow for asymmetric treatments of alternatives and consider the distinct relevance of the top and bottom positions for ordered lists. We provide a full axiomatic characterization of our distance. In doing so, we retrieve new characterizations of existing axioms and show how to effectively weaken them for our purposes. This analysis highlights the generality of our distance as it embeds many (semi)metrics previously proposed in the literature. Subsequently, we show that, notwithstanding its level of generality, our distance is still readily applicable. We apply it to preference aggregation, studying the features of the associated median voting rule. It is shown how the derived preference function satisfies many desirable features in the context of voting rules, ranging from fairness to majority and Pareto-related properties. We show how to compute consensus rankings exactly, and provide generalized Diaconis-Graham inequalities that can be leveraged to obtain approximation algorithms. Finally, we propose some truncation ideas for our distances inspired by Lu and Boutilier (2010). These can be leveraged to devise a Polynomial-Time-Approximation Scheme for the corresponding rank aggregation problem.

翻译：我们研究了一种排序上的距离函数族，该函数允许对备选方案进行非对称处理，并考虑有序列表中顶部和底部位置的不同相关性。我们给出了该距离的完整公理刻画。在此过程中，我们重新获取了已有公理的新刻画，并展示了如何为我们的目的有效弱化这些公理。分析表明，该距离具有高度普适性，因为它包含了文献中先前提出的许多（半）度量。随后，我们证明尽管具有这种通用性，该距离仍然易于应用。我们将它应用于偏好聚合，研究了相关中位数投票规则的特征。结果表明，导出的偏好函数在投票规则背景下满足许多理想特性，涵盖公平性、多数原则以及帕累托相关性质。我们展示了如何精确计算共识排序，并提供了可被用于获得近似算法的广义Diaconis-Graham不等式。最后，我们提出了受Lu和Boutilier（2010）启发的距离截断思想。这些思想可用于为相应的排序聚合问题设计多项式时间近似方案。