Mutual majority criterion: Difference between revisions
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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>
== Notes ==
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. ▼
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: <blockquote>17 A>B>C ▼
17 A=B>C ▼
17 C>A>B ▼
49 D>E>F </blockquote>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. ▼
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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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.
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]]
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
32 B>A
33 C>B
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.
▲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
▲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.
▲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:
▲49 D>E>F
==References==
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