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.

Example
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.

Complying and non-complying methods

 * Systems which pass
 * Borda-Elimination, Bucklin, Coombs, IRV, Kemeny-Young, Nanson's method, 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:

or more simply,

A generalized form that also encompasses rated voting methods:

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. This usage is due to Woodall.

Stronger forms
The mutual majority criterion is implied by the dominant mutual third property, which itself is implied by the Smith criterion.

Weaker forms
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. An even weaker criterion along these lines would be that the mutual majority must give everyone they prefer a perfect score; Majority Judgment passes this.

Dominant mutual plurality criterion
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.

Semi-mutual majority
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. All major defeat-dropping Condorcet methods elect B here, since they have the weakest pairwise defeat.

Independence of mutual majority-dominated alternatives
Similar to Independence of Smith-dominated Alternatives, a "independence of mutual majority-dominated alternatives" criterion could be envisioned.

Both instant-runoff voting and Descending Acquiescing Coalitions fail this criterion, as can be shown by Left, Center, Right scenarios when y+z also constitutes a majority.

For instance:

The smallest mutual majority set is {L, C}, and C beats L pairwise, so in any election where those two candidates are the only one in the running, C wins. However, IRV first eliminates C and then L beats R. DAC first excludes R from the set of viable candidates (because the {L, C} coalition is the largest). Then L has the greatest first preference count of the two and thus wins.

Pairwise counting
Note that the mutual majority set is a pairwise-dominating set (every candidate in it pairwise beats every candidate not in it). So one way to find it would be to find the Smith set ranking, and then look for the smallest group of candidates highest in the Smith ranking who are preferred by a mutual majority, if there is one.

The smallest mutual majority set can be found in part by looking for the Smith set, because the Smith set is always a subset of the mutual majority set when one exists, and then adding in candidates into the mutual majority set who are preferred by enough of the voters who helped the candidates in the Smith set beat other candidates to constitute a mutual majority. Example:

35 A>B

35 B>A

30 C>B

The Smith set is just B here. When looking at the 70 voters who helped B beat C and the 65 for B>A, it's clear that a majority of them prefer A over C, and that an absolute majority of voters prefer either A or B over C. So the smallest mutual majority set is A and B.

Bucklin approach
An alternative way to find the smallest mutual majority set would be to use a modified version of Bucklin voting: for each voter, assume they "approve" all of their 1st choices. Find the ballot which approves the most candidates; for each other ballot, until it approves as many candidates as this "most-approvals" ballot, the most-approvals ballot should be prevented from approving any more candidates. Once a ballot approves as many or more candidates than the most-approvals ballot, it should be considered the most-approvals ballot instead, and likewise, it should stop approving additional candidates. For each ballot that is not a most-approvals ballot, approve all candidates at the next consecutive rank where candidates haven't been approved yet for that ballot. Do this until some candidate(s) are approved by a majority of voters, and then check if all ballots approving each majority-approved candidate do not approve anyone else. If so, then the majority-approved candidates are the smallest mutual majority set, but if not, then there is no smallest mutual majority set. For example: 17 A>B>C

17 A=B>C

17 C>A>B

49 D>E>F 34 voters approve A as their 1st choice, 17 B, 17 C, and 49 D. The 17 A=B voters approve both A and B, two candidates, making them the most-approvals voters currently, so they are not allowed to approve any more candidates for now. Adding in the next rank, 17 voters now approve B as their 2nd choice, 17 A, and 49 E. Now 51 voters approve A, so check whether they are a mutual majority. In this case, the only candidates any of the 51 voters prefer more than or equally to A are B and C; it is seen that all 51 voters prefer any of A, B, or C over all other candidates (D, E, and F), so ABC is the smallest mutual majority set.