Mutual majority criterion: Difference between revisions
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== Notes == |
== Notes == |
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The smallest mutual majority set can be found by looking for the [[Smith 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: |
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35 A>B |
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35 B>A |
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30 C>B |
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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. |
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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. |
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. |
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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. |
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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 |
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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