Characterizing plurality using the majoritarian condition: a new proof and implications for other scoring rules

Plurality rule selects whichever alternative is most preferred by the greatest number of voters. The majoritarian principle states that if a simple majority of voters agree on the most preferred alternative, then it must be selected uniquely. Lepelley (RAIRO-Recherche Opérationnelle 26: 361–365, 1992) adopts a proof by contradiction approach to show that plurality is the only scoring rule satisfying the majoritarian principle. We make use of the relationship that majoritarianism implies faithfulness to present a new proof allowing us to derive limits on the size of the group for which a particular scoring rule will satisfy majoritarianism without restricting voter preferences. We then determine the limits for three specific faithful scoring rules where voters rank the alternatives: positive/negative voting, wherein one point is awarded to a voter’s top preference and one point is subtracted from a voter’s bottom preference; Borda, in which an equal increase in points is awarded to each successively higher rank; and Dowdall, for which rank points entail an harmonic sequence. Comparing these rules by the sizes of group and alternative set combinations for which they are majoritarian we find that Borda is dominated by positive/negative voting, and both are dominated by Dowdall. We also derive the relative point gaps between certain pairs of rankings beyond which a scoring rule will not be majoritarian for any group of more than two voters.