Consistency without neutrality in voting rules: When is a vote an average?

Smith [J.H. Smith, Aggregation of preferences with variable electorate, Econometrica 41 (1973) 1027–1041] and Young [H.P. Young, A note on preference aggregation, Econometrica 42 (1974) 1129–1131; H.P. Young, Social choice scoring functions, SIAM J. Appl. Math. 28 (1975) 824–838] characterized scoring rules via four axioms: consistency, continuity, anonymity, and neutrality. In their context a ballot consists of a strict ranking of alternatives, and an election outcome is either a set of (winning) alternatives (Young) or a weak ordering of alternatives (Smith). Many rules fail to fit this context, yet intuitively satisfy one’s notion of a generalized scoring rule; this very broad class GSR includes the Kemeny rule, approval voting, and certain grading systems. We show that GSR is identical with the class MPR of mean proximity rules loosely, rules in MPR are those for which the “average voter” determines the outcome. The techniques in the proof allow us to make some surprisingly direct comparisons between rules (for example, between Kemeny and Borda) that might initially seem to be of completely different sorts. The abstract anonymous voting rules provide the context for GSR, which is of necessity too general to admit a neutrality axiom. A natural question arises: “What happens to the Smith and Young characterizations in the absence of neutrality?” We discuss one answer in the form of a characterization of the rational mean neat voting rules (a class closely related to GSR) as those that are consistent and connected. Connectedness is a strong form of continuity that implies a discrete analogue to the Intermediate Value Theorem.