Distance rationalizability of scoring rules

Collective decision making problems can be seen as finding an outcome that is "closest" to a concept of "consensus". [1] introduced "Closeness to Unanimity Procedure" as a first example to this approach and showed that the Borda rule is the closest to unanimity under swap distance (a.k.a the [2] distance). [3] shows that the Dodgson rule is the closest to Condorcet under swap distance. [4, 5] generalized this concept as distance-rationalizability, where being close is measured via various distance functions and with many concepts of consensus, e.g., unanimity, Condorcet etc. In this paper, we show that all non-degenerate scoring rules can be distance-rationalized as "Closeness to Unanimity" procedures under a class of weighted distance functions introduced in [6]. Therefore, the results herein generalizes [1] and builds a connection between scoring rules and a generalization of the Kemeny distance, i.e. weighted distances.