Mutual majority criterion: Difference between revisions
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The '''Mutual majority criterion''' is a criterion for evaluating [[voting system]]s. 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|majority criterion]] for sets of candidates. Thus, it is often called the '''Majority criterion for [[Solid coalition|solid coalitions]].'''
The mutual majority criterion is implied by the [[Smith criterion]].
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18 C:10 A:9 B:8 (C>A>B >D=E)
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.
; Systems which pass
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10 C(>A=B) </blockquote>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 counting|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 counting|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: <blockquote>
▲ 30: 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:<blockquote>35: A2>B
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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.
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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