Ranking is an important part of several areas of contemporary research, including social sciences, decision theory, data analysis, and information retrieval. The goal of this paper is to align developments in quantitative social sciences and decision theory with the current thought in Computer Science, including a few novel results. Specifically, we consider binary preference relations, the so-called weak orders that are in one-to-one correspondence with rankings. We show that the conventional symmetric difference distance between weak orders, considered as sets of ordered pairs, coincides with the celebrated Kemeny distance between the corresponding rankings, despite the seemingly much simpler structure of the former. Based on this, we review several properties of the geometric space of weak orders involving the ternary relation “between,” and contingency tables for cross-partitions. Next, we reformulate the consensus ranking problem as a variant of finding an optimal linear ordering, given a correspondingly defined consensus matrix. The difference is in a subtracted term, the partition concentration that depends only on the distribution of the objects in the individual parts. We apply our results to the conventional Likert scale to show that the Kemeny consensus rule is rather insensitive to the data under consideration and, therefore, should be supplemented with more sensitive consensus schemes.

中文翻译：

---

有序分区对应的偏好关系的距离与共识

排名是当代研究多个领域的重要组成部分，包括社会科学、决策理论、数据分析和信息检索。本文的目标是将定量社会科学和决策理论的发展与计算机科学的当前思想保持一致，包括一些新的结果。具体来说，我们考虑二元偏好关系，即与排名一一对应的所谓弱顺序。我们表明，弱顺序之间的传统对称差异距离（被视为有序对的集合）与相应排名之间著名的 Kemeny 距离重合，尽管前者的结构看似简单得多。在此基础上，我们回顾了弱阶几何空间的几个性质，涉及三元关系“之间，”和跨分区的列联表。接下来，在给定相应定义的共识矩阵的情况下，我们将共识排序问题重新表述为寻找最佳线性排序的变体。区别在于减去项，分配浓度仅取决于各个部分中对象的分布。我们将我们的结果应用于传统的李克特量表，以表明 Kemeny 共识规则对所考虑的数据相当不敏感，因此，应该补充更敏感的共识方案。分配浓度仅取决于对象在各个部分的分布。我们将我们的结果应用于传统的李克特量表，以表明 Kemeny 共识规则对所考虑的数据相当不敏感，因此，应该补充更敏感的共识方案。分配浓度仅取决于对象在各个部分的分布。我们将我们的结果应用于传统的李克特量表，以表明 Kemeny 共识规则对所考虑的数据相当不敏感，因此，应该补充更敏感的共识方案。