  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.
