Majority criterion

The majority criterion is a criterion for evaluating voting systems. It can be most simply thought of as "if a majority prefers a candidate as their unique 1st choice (i.e. they prefer this candidate above all other candidates), then the majority's 1st choice must win."

The mutual majority criterion, which is sometimes simply called the majority criterion, generalizes the constraint to sets of candidates.

The Condorcet criterion implies the majority criterion. Practically every serious ranked voting method passes the majority criterion. Most rated voting methods fail the majority criterion, such as Approval, Score, and STAR voting, though this is argued to be a good thing in situations where those methods elect a candidate who is well-liked by all voters rather than a candidate who is narrowly preferred by a majority but loathed by the minority.

Example:

51 A

25 B>C

24 C>B

51 voters prefer A over all others (B and C), therefore A must win by the majority criterion.

It can be stated as follows:

If a majority of the voters vote a given candidate X ahead of everybody else, then X must win.

with the likely most general interpretation of "vote ahead of" being "ranked or rated higher than".

The majority criterion for rated ballots is a weaker, separate criterion which says that a candidate given a perfect (maximal) rating by a majority of voters must win if no other candidate received a perfect rating from that majority.

The difference between the two versions can be seen with this example:

51 A:1

49 B:5

If the highest score is a 5, then the majority criterion for rated ballots allows either A or B to win. This is in contrast to the regular majority criterion, which requires A to win. Arguably, the majority criterion for rated ballots is more appropriate in the context of rated ballots, since a voter who doesn't give their 1st choice a perfect score is essentially choosing not to use all of their voting power, and thus their preference need not be (or even perhaps, shouldn't) be maximally respected or enforced.

Notes

See the mutual majority criterion#Notes article for an example where a candidate preferred by a plurality of voters as their 1st choice who pairwise beat all other candidates wasn't guaranteed to win under the majority criterion. The Condorcet criterion guarantees the election of such a candidate, by virtue of them pairwise beating all others.

The very minimum a voting method must do in order to be considered "majoritarian" is to pass the majority criterion for at least the two-candidate case.

Some voting methods (most rated voting methods) pass a weaker form of the majority criterion, which only requires that a majority be able to force their 1st choice to win by voting strategically. Note that it is not always the case that the majority will have the ability to safely vote strategically I.e. if they're unsure as to whether there is or who their collective 1st choice is.

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