Preference data are a particular type of ranking data where some subjects (voters, judges,...) express their preferences over a set of alternatives (items). In most real life cases, some items receive the same preference by a judge, thus giving rise to a ranking with ties. An important issue involving rankings concerns the aggregation of the preferences into a “consensus”. The purpose of this paper is to investigate the consensus between rankings with ties, taking into account the importance of swapping elements belonging to the top (or to the bottom) of the ordering (position weights). By combining the structure of \(\tau _x\) proposed by Emond and Mason (J Multi-Criteria Decis Anal 11(1):17–28, 2002) with the class of weighted Kemeny-Snell distances, a position weighted rank correlation coefficient is proposed for comparing rankings with ties. The one-to-one correspondence between the weighted distance and the rank correlation coefficient is proved, analytically speaking, using both equal and decreasing weights.

偏好数据是一种特殊类型的排名数据，其中某些主体（选民、法官……）表达了他们对一组备选方案（项目）的偏好。在大多数现实生活中，有些项目会受到法官的相同偏好，从而产生并列排名。涉及排名的一个重要问题涉及将偏好聚合为“共识”。本文的目的是研究具有关系的排名之间的共识，考虑到交换属于排序（位置权重）顶部（或底部）的元素的重要性。通过结合\(\tau _x\)的结构由 Emond 和 Mason (J Multi-Criteria Decis Anal 11(1):17-28, 2002) 提出的加权 Kemeny-Snell 距离类，提出了位置加权等级相关系数来比较排名与关系。加权距离和秩相关系数之间的一一对应关系被证明，从分析上讲，使用相等权重和递减权重。