Bivariate scoring rules: Unifying the characterizations of positional scoring rules and Kemeny's rule

This paper studies social preference functions (SPFs), which map the voters' ordinal preferences over a set of alternatives to a non-empty set of strict rankings over the alternatives. Maybe the most prominent SPFs are positional scoring rules and Kemeny's rule. While these two types of rules behave intuitively quite differently, they are axiomatically surprisingly similar and we thus provide a joint axiomatic characterization of these SPFs. To this end, we introduce the class of bivariate scoring rules which generalize Kemeny's rule by weighting comparisons between alternatives depending on their positions in the voters' preference relations. In particular, this class contains both Kemeny's rule and all positional scoring rules. As our main result, we then characterize the set of bivariate scoring rules as the only SPFs that satisfy—aside from some standard axioms—mild consistency conditions for variable electorates and variable agendas. As corollaries of this result, we also derive variants of the well-known characterizations of positional scoring rules by Smith (1973) and of Kemeny's rule by Young (1988). Thus, our result unifies the independent streams of research on positional scoring rules and Kemeny's rule by giving a joint axiomatic basis of these SPFs.