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.
Comparison to Utilitarian systems
Utilitarian systems have a different intent when choosing a winner. Under the philosophy of utilitarianism, choosing a majoritarian winner when a Utilitarian winner is available would be considered bad. Majoritarianism is viewed as an approximation of Utilitarianism.
All utilitarian systems are Cardinal voting methods because other ballot structures do not contain enough informaiton. All such systems fail the majority criterion by design. Common examples of such as Approval, Score, and STAR voting. Following from Utilitarian theory it is argued that the Utilitarian winner is preferred in situations where they are well-liked by all voters rather than a candidate who is narrowly preferred by a majority but loathed by the minority.
An illustrative score votingexample for 100 voters with candidates A, B, C ,D is
20 = A:5, B:2, C:2, D:0
20 = A:2, B:5, C:2, D:0
20 = A:2, B:2, C:5, D:0
40 = A:0, B:0, C:0, D:5
D is the score winner with 200 and all others have 180. D is the Utilitarian winner because they yield the most Utility. If only A and D were running the ballots would likely be.
60 = A:5, D:0
40 = A:0, D:5
Under Majoritarianism A would win so it looks as if B and C are spoilers for A. Systems which pass the Majority criteria, like IRV, would yield either A, B or C as the winner in the original case. Utilitarian philosophy would not view that B and C are spoilers for A. Score relies on there being enough candidates to properly get utilities. In the absence of a representative sample of candidates score would yield the majoritarian winner A. This does not mean that score gives the correct answer in the case of less candidates but that it does not work as intended because the voters do not have enough choice. It is important to then note that partisanship is fundamentally incompatible with score. If there is no party to put up candidates for a significant portion of the voters then score will not yield accurate utilities. In party based systems there are never parties for all groups. Majoritarian systems are in this way more compatible with partisanship which is why collectivists have always favoured majoritarian solutions and individualists favour utilitarian solutions.
Of course one could strategically vote under score and give
20 = A:5, B:5, C:5, D:0
20 = A:5, B:5, C:5, D:0
20 = A:5, B:5, C:5, D:0
40 = A:0, B:0, C:0, D:5
But this would not really happen if you look at it from a game theory perspective. The voters who favour A,B and C are in competition and are actually not so ideologically aligned. In the end some might give a little more and this might be enough to win in this scenario. For this reason some advocate that Approval Voting is the appropriate system because it forces voters into the Nash Equilibrium.
Another game theory perspective to consider is that elections do not exist in a vacuum. Utilitarianism leads to an equilibrium where more candidates run who are centrist to try to find the right balance to please the most. Majoritarianism leads to tyranny of the majority. Polling data does not show that voters are divided ideologically into incompatible factions. Ideologically people are Gaussian distributed around the center.
Majority criterion for Cardinal 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
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.
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.