Distance rationalizability of scoring rules

Collective decision making problems can be seen as finding an outcome that is "closest" to a
concept of "consensus". Nitzan (1981) 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 Kemeny (1959) distance). Meskanen and Nurmi (2008) shows that the Dodgson rule is the closest to Condorcet under swap distance. Elkind et al. (2009, 2012) 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 Can (2012). Therefore, the results herein generalizes, partly, the results in Nitzan (1981) and complements the extensive findings in Elkind et al. (2009).