Mutual majority criterion: Difference between revisions

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It is an extension of (and also implies) the [[Majority criterion|majority criterion]] for sets of candidates. Thus, it is often called the '''Majority criterion for [[Solid coalition|solid coalitions]].'''

== Example ==

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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):<blockquote>17 A:10 B:9 C:8 (A>B>C >D=E)

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49 D:10 E:10 (D>E >A=B=C)</blockquote>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

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It is sometimes simply (and confusingly) called the '''Majority criterion.''' This usage is due to Woodall.<ref name="Woodall 1994 Properties">{{cite journal | last=Woodall |first=D. |title=Properties of preferential election rules | journal=Voting matters | issue=3 | pages=8–15 | year=1994 | url=http://www.votingmatters.org.uk/ISSUE3/P5.HTM}}</ref>

== Related forms of the criterion ==

=== Stronger forms ===

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

=== Weaker forms ===

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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.

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== Notes ==

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All [[Condorcet methods]] pass mutual majority when there is a [[Condorcet winner]], since if there is a mutual majority set, all candidates in it pairwise beat all candidates not in it by virtue of being preferred by an absolute majority; since the CW isn't pairwise beaten by anyone, they must be in the set. [[Smith-efficient]] [[Condorcet methods]] always pass mutual majority.

=== 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: <blockquote>51 A>C

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30: B

35: C2>B</blockquote>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: <blockquote>26 A>B

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49 C</blockquote>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 [[:Category:Defeat-dropping Condorcet methods|defeat-dropping Condorcet methods]] elect B here, since they have the weakest pairwise defeat.

=== Independence of mutual majority-dominated alternatives ===

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By analogy to the [[~~Majority criterion for rated ballots|~~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.

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Similar to [[Independence of Smith-dominated Alternatives]], a "independence of mutual majority-dominated alternatives" criterion could be envisioned. Example where IRV fails:

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Similar to [[Independence of Smith-dominated Alternatives]], a "independence of mutual majority-dominated alternatives" criterion could be envisioned. Example where IRV fails:

35 A>B

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A and B are a mutual majority, so the criterion would require allowing C to be eliminated, at which point, B would the majority's 1st choice and thus win. But IRV eliminated B first and then elects A.

=== Finding the mutual majority set ===

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: