Mutual majority criterion
The Mutual majority criterion is a criterion for evaluating voting systems. Most simply, it can be thought of as requiring that whenever a majority of voters prefer a set of candidates (often candidates from the same political party) above all others (i.e. when choosing among ice cream flavors, a majority of voters are split between several variants of chocolate ice cream, but agree that any of the chocolate-type flavors are better than any of the other ice cream flavors), someone from that set must win (i.e. one of the chocolate-type flavors must win). It is the single-winner case of Droop-Proportionality for Solid Coalitions.
It is an extension of (and also implies) the majority criterion for sets of candidates. Thus, it is often called the Majority criterion for solid coalitions.
The mutual majority criterion is implied by the Smith criterion.
Example for candidates A, B, C, D and E (scores are shown for each candidate, with the implicit ranked preferences in parentheses, and the unscored candidates assumed to be ranked last):
17 A:10 B:9 C:8 (A>B>C >D=E)
17 B:10 C:9 A:8 (B>C>A >D=E)
18 C:10 A:9 B:8 (C>A>B >D=E)
49 D:10 E:10 (D>E >A=B=C)
A, B, and C are preferred by a mutual majority, because a group of 52 voters (out of 100), an absolute majority, scored all of them higher than (preferred them over) all other candidates (D and E). So the mutual majority criterion requires that one of A, B, and C win the election.
- Systems which pass
- Borda-Elimination, Bucklin, Coombs, IRV, Kemeny-Young, Nanson (original), Pairwise-Elimination, Ranked Pairs, Schulze, Smith//Minmax, Descending Solid Coalitions, Majority Choice Approval, any Smith-efficient Condorcet method, most Condorcet-IRV hybrid methods
- Systems which fail
- most rated methods (such as Approval voting, Score voting, and STAR voting), Black, Borda, Dodgson, Minmax, Sum of Defeats
Alternative Definitions
It can be stated as follows:
If there is a majority of voters for which it is true that they all rank a set of candidates above all others, then one of these candidates must win.
A generalized form that also encompasses rated voting methods:
If a majority of voters unanimously vote a given set of candidates above a given rating or ranking, and all other candidates below that rating or ranking, then the winner must be from that set.
Note that the logical implication of the mutual majority criterion is that a candidate from the smallest set of candidates preferred by the same absolute majority of voters over all others must win; this is because if, for example, 51 voters prefer A over B, and B over C, with the other 49 voters preferring C, then not only is (A, B) a set of candidates preferred by an absolute majority over all others (C), but candidate A is also a candidate preferred by an absolute majority over all others (B and C), and therefore A must win in order to satisfy the criterion.
It is sometimes simply (and confusingly) called the Majority criterion.
Notes
Voting methods which pass the majority criterion but not the mutual majority criterion (some ranked methods fall under this category, notably FPTP) possess a spoiler effect, since if all but one candidate in the mutual majority drops out, the remaining candidate in the mutual majority is guaranteed to win, whereas if nobody had dropped out, a candidate not in the mutual majority might have won.
The mutual majority criterion doesn't apply to situations where there are large "sides" if enough voters are indifferent to the large sides. Example:
51 A>C
49 B
10 C(>A=B)
The last line "10 C(>A=B)" should be read as "these 10 voters prefer C as their 1st choice and are indifferent between A and B." Even though candidate A is preferred by the (same) majority of voters in pairwise matchups against B (51 vs. 49) and C (51 vs. 10), candidate A technically is not preferred by an absolute majority (i.e. over half of all voters), and C would beat A in some mutual majority-passing methods, such as Bucklin. A "mutual plurality" criterion might make sense for these types of situations where a plurality of voters prefer a set of candidates above all others, and everyone in that set pairwise beats everyone outside of the set; this mutual plurality criterion implies the mutual majority criterion (because a majority is a plurality, and anyone who is preferred by an absolute majority over another candidate is guaranteed to pairwise beat that candidate, thus all candidates in the mutual majority set pairwise beat all other candidates). The Smith criterion implies this mutual plurality criterion (because the Smith criterion implies that someone from the smallest set of candidates that can pairwise beat all others must win, and this smallest set must be a subset of any set of candidates that can pairwise beat all candidates not in the set). IRV doesn't pass the mutual plurality criterion; example:
15: A1>A2>B
20: A2>B
30: B
20: C1>B
15: C2>C1>B
B is ranked above all other candidates by 30 voters, whereas no other set of candidates is ranked above all others by more than 20 voters. Yet after a few eliminations, this becomes:
35: A2>B
30: B
35: C2>B
and B is eliminated first, despite pairwise dominating everyone else (i.e. being the Condorcet winner). This is an example of the Center squeeze effect. If there are some losing candidates ranked above the mutual majority set of candidates by some voters in the majority, this voids the criterion guarantee. Example:
26 A>B
25 B
49 C
Despite B being preferred by an absolute majority over C, and the only candidate preferred by any voters in that absolute majority over or equally to B being A (with no voters in the majority preferring anyone over A), the mutual majority criterion doesn't guarantee that either A or B must win. It has been argued that to avoid the Chicken dilemma, C must win here (and C would win in some mutual majority-passing methods, such as IRV, which is often claimed to resist the chicken dilemma), but methods that do so have a spoiler effect, since if A drops out, B must win by the majority (and thus mutual majority) criterion.
By analogy to the majority criterion for rated ballots, one could design a mutual majority criterion for rated ballots, which would be the mutual majority criterion with the requirement that each voter in the majority give at least one candidate in the mutual majority-preferred set of candidates a perfect (maximal) score.