We study voting rules with respect to how they allow or limit a majority from dominating minorities: whether a voting rule makes a majority powerful and whether minorities can veto the candidates they do not prefer. For a given voting rule, the minimal share of voters that guarantees a victory to one of the majority’s most preferred candidates is the measure of majority power; and the minimal share of voters that allows the minority to veto each of their least preferred candidates is the measure of veto power. We find tight bounds on such minimal shares for voting rules that are popular in the literature and used in real elections. We order the rules according to majority power and veto power. Instant-runoff voting has both the highest majority power and the highest veto power; plurality rule has the lowest. In general, the greater is the majority power of a voting rule, the greater its veto power. The three exceptions are: voting with proportional veto power, Black’s rule and Borda’s rule, which have relatively weak majority power and strong veto power, thus providing minority protection. Our results can shed light on how voting rules provide different incentives for voter participation and candidate nomination.
This paper considers a voting problem in which the individual preferences of electors are defined by the ranked lists of candidates. For single-winner elections, we apply the criterion of weak positional dominance (WPD, PD), which is closely related to the positional scoring rules. Also we formulate the criterion of weak mutual majority (WMM), which is stronger than the majority criterion but weaker than the criterion of mutual majority (MM). Then we construct two modifications for the median voting rule that satisfy the Condorcet loser criterion. As shown below, WPD and WMM are satisfied for the first modification while PD and MM for the second modification. We prove that there is no rule satisfying WPD and MM simultaneously. Finally, we check a list of 37 criteria for the constructed rules.