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Mutual majority criterion: Difference between revisions
Renamed Category:Majority-related concepts to Category:Majority–minority relations
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{{wikipedia}}
The '''
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]].'''
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; Systems which pass
:[[Baldwin's method|Borda-Elimination]], [[Bucklin voting|Bucklin]], [[Coombs]], [[IRV]], [[Kemeny-Young]], [[Nanson's
; Systems which fail
: most [[Cardinal voting|rated methods]] (such as [[Approval voting]], [[Score|Score voting]], and [[STAR|STAR voting]]), [[Black]], [[Borda]], [[Dodgson]], [[Minmax]], [[Sum of Defeats]]
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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. This is also why [[:Category:Sequential loser-elimination methods|Sequential loser-elimination methods]] whose base methods pass the majority criterion pass the mutual majority criterion.
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.
In contrast to the [[dominant mutual third]] set, a mutual majority set is always also a dominant mutual majority set. Every coalition that has majority support also pairwise beats the rest of the candidates, but that is not true of all coalitions supported by more than 1/3 of the voters.
=== Dominant mutual plurality criterion ===
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=== 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:
{{ballots|
4: L>C>R
3: R>C>L
2: C>L>R}}
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.
=== Finding the mutual majority set ===
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[[Category:Voting system criteria]]
[[Category:
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