用于排名聚合的平方凯梅尼（Squared Kemeny）规则
The Squared Kemeny Rule for Averaging Rankings
AI总结：
针对传统凯梅尼规则的多数主义特性在多标准加权排名聚合中不合理的问题，提出平方凯梅尼规则，证明其权重比例性，给出公理刻画并讨论计算与仿真表现。
摘要：
针对将多个排名聚合为一个排名的问题，Kemeny（1959）提出了两种方法：中位数规则，即选择与输入排名总交换距离最小的排名；均值规则，即最小化与输入排名的交换距离平方。此后中位数规则得到了广泛研究，如今被简称为凯梅尼规则（Kemeny's rule）。它具有多数主义特性，例如，如果超过一半的输入排名相同，那么该规则的输出就是这个相同的排名。\n 我们发现这种行为在许多排名聚合场景中是不可取的。例如，当我们根据不同标准（质量、价格等）对对象进行排名，并希望按照指定的标准权重进行聚合时，权重为51%的标准应该对输出产生51%的影响，而非100%。我们通过建立输出排名与任意输入排名之间的距离界（作为其权重的函数），证明了平方凯梅尼规则（即均值规则）具有这种特性。此外，我们给出了平方凯梅尼规则的公理刻画，它与现有的凯梅尼规则刻画相呼应，但将多数主义的孔多塞（Condorcet）公理替换为比例性公理。最后，我们讨论了该规则的计算方法，并通过仿真研究展示了其行为表现。
英文摘要：
For the problem of aggregating several rankings into one ranking, Kemeny (1959) proposed two methods: the median rule which selects the ranking with the smallest total swap distance to the input rankings, and the mean rule which minimizes the squared swap distances to the input rankings. The median rule has been extensively studied since and is now known simply as Kemeny's rule. It exhibits majoritarian properties, so for example if more than half of the input rankings are the same, then the output of the rule is the same ranking. We observe that this behavior is undesirable in many rank aggregation settings. For example, when we rank objects by different criteria (quality, price, etc.) and want to aggregate them with specified weights for the criteria, then a criterion with weight 51% should have 51% influence on the output instead of 100%. We show that the Squared Kemeny rule (i.e., the mean rule) behaves this way, by establishing a bound on the distance of the output ranking to any input rankings, as a function of their weights. Furthermore, we give an axiomatic characterization of the Squared Kemeny rule, which mirrors the existing characterization of the Kemeny rule but replaces the majoritarian Condorcet axiom by a proportionality axiom. Finally, we discuss the computation of the rule and show its behavior in a simulation study.
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