Characterization of Scoring Rules with Distances: Application to the Clustering of Rankings / 104
Positional scoring rules are often used for rank aggregation. In this work we study how scoring rules can be formulated as the minimization of some distance measures between rankings, and we also consider a new family of aggregation methods, called biased scoring rules. This work extends a previous known observation connecting Borda count with the minimization of the sum of the Spearman distances (calculated with respect to a set of input rankings). In particular we consider generalizations of the Spearman distance that can give different weights to items and positions; we also handle the case of incomplete rank data. This has applications in the clustering of rank data, where two main steps need to be performed: aggregating rankings of the same cluster into a representative ranking (the cluster's centroid) and assigning each ranking to its closest centroid. Using the proper combination of scoring rules (for aggregation) and distances (for assignment), it is possible to perform clustering in a computationally efficient way and as well account for specific desired behaviors (give more weight to top positions, bias the centroids in favor of particular items).