Mutual majority criterion: Difference between revisions
→Finding the mutual majority set
No edit summary |
|||
Line 102:
=== 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: ▼
==== Pairwise counting ====
▲Note that the mutual majority set is a pairwise-dominating set (every candidate in it [[pairwise]] beats every candidate not in it). So one way to find it would be to find the [[Smith set ranking]], and then look for the smallest group of candidates highest in the Smith ranking who are preferred by a mutual majority, if there is one. 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
Line 111 ⟶ 113:
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.
=== Bucklin approach ===
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
| |||