Mutual majority criterion

The mutual majority criterion is a criterion used to compare voting systems. It is also known as the majority criterion for solid coalitions and the generalized majority criterion. The criterion states that if there is a subset S of the candidates, such that more than half of the voters strictly prefer every member of S to every candidate outside of S, this majority voting sincerely, the winner must come from S. This is similar to but stricter than the majority criterion, where the requirement applies only to the case that S contains a single candidate.[1] This is also stricter than the majority loser criterion, where the requirement applies only to the case that S contains all but one candidate. The mutual majority criterion is the single-winner case of the Droop proportionality criterion.

The Schulze method, ranked pairs, instant-runoff voting, Nanson's method, and Bucklin voting pass this criterion. All Smith-efficient Condorcet methods pass the mutual majority criterion.[2]

The plurality vote, two-round system, contingent vote, Black's method, and minimax satisfy the majority criterion but fail the mutual majority criterion.[3]

The anti-plurality voting, approval voting, range voting, and the Borda count fail the majority criterion and hence fail the mutual majority criterion.

Methods which pass mutual majority but fail the Condorcet criterion can nullify the voting power of voters outside the mutual majority. Instant runoff voting is notable for excluding up to half of voters by this combination.

Methods which pass the majority criterion but fail mutual majority can have a spoiler effect, since if a non-mutual majority-preferred candidates wins instead of a mutual majority-preferred candidate, then if all but one of the candidates in the mutual majority-preferred set drop out, the remaining mutual majority-preferred candidate will win, which is an improvement from the perspective of all voters in the majority.