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."
It can be stated as follows:
If a majority of the voters endorse a given candidate X more than any other candidate, then X must win.
Or in plain English as
If one candidate is preferred by a majority (more than 50%) of voters, then that candidate 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.
Example:
51 A
25 B>C
24 C>B
51 voters out of 100 prefer A over all others (B and C), therefore A must win by the majority criterion.
Weaker forms of the criterion
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 coordinating and 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.
Majority criterion for rated ballots
There are some Cardinal systems which are designed to fulfil Majoritarianism not Utilitarianism. 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
Notes
For both the majority and mutual majority criterion, the size of the majority may either be an absolute majority of all voters, or an absolute majority of voters who have any preference between the candidates, depending on how it's defined. For example:
30 A>B
20 B>A
5 C>A
50 A=B=C
A is the 1st choice of the majority of voters who have any preference between the three candidates, but not a majority of all voters.
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.