Talk:Dominant mutual third set: Difference between revisions
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[[User:Kristomun]], I think this may have some kind of STV-PR generalization. In a 2-seat election, we know that anyone who has over 1/4th of the active votes at any point in STV is guaranteed to be one of the final 3 remaining candidates, since it's impossible for 3 other candidates to each have more votes than this candidate (since they'd each have to have over 1/4th of the active votes, resulting in more than 100% of votes total being allocated to different candidates), which is what would enable them to survive elimination longer. So, we can say that when over 1/3rds of the voters prefer someone from the "dominant mutual quarter" set (DMT but for 1/4th of the electorate) over anyone else who survives until the final round, then the dominant mutual quarter candidate must win. In general, someone who is preferred by a solid coalition of 1/(k+2)th of the voters (k being the number of seats) and preferred by 1/(k+1)th of all voters over any other given rival must win. I'm not sure if there's a way to extract more from this insight, though. [[User:BetterVotingAdvocacy|BetterVotingAdvocacy]] ([[User talk:BetterVotingAdvocacy|talk]]) 03:48, 27 March 2020 (UTC) |
[[User:Kristomun]], I think this may have some kind of STV-PR generalization. In a 2-seat election, we know that anyone who has over 1/4th of the active votes at any point in STV is guaranteed to be one of the final 3 remaining candidates, since it's impossible for 3 other candidates to each have more votes than this candidate (since they'd each have to have over 1/4th of the active votes, resulting in more than 100% of votes total being allocated to different candidates), which is what would enable them to survive elimination longer. So, we can say that when over 1/3rds of the voters prefer someone from the "dominant mutual quarter" set (DMT but for 1/4th of the electorate) over anyone else who survives until the final round, then the dominant mutual quarter candidate must win. In general, someone who is preferred by a solid coalition of 1/(k+2)th of the voters (k being the number of seats) and preferred by 1/(k+1)th of all voters over any other given rival must win. I'm not sure if there's a way to extract more from this insight, though. [[User:BetterVotingAdvocacy|BetterVotingAdvocacy]] ([[User talk:BetterVotingAdvocacy|talk]]) 03:48, 27 March 2020 (UTC) |
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== Smith-efficiency == |
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[[User:BetterVotingAdvocacy]], I don't think what you said is true: that electing from the DMT set implies Smith when the Smith set is a subset of the DMT set is. Consider e.g. an election where more than a third of the voters vote ABCD in some order above everybody else, and that there are say, 26 candidates. Suppose furthermore that each voter in a majority votes (some random permutation of a random subset of candidates E..Z) > (some permutation of A, B, and C) > D > everybody else. Now {A,B,C,D} is the smallest DMT set, but D is beaten pairwise by A, B, and C, and thus D is not in the Smith set. So the Smith set is {A,B,C} which is a subset of the smallest DMT set {A,B,C,D}. Then our contrived DMT-passing method could elect D, which would be in the DMT set but not the Smith set. |
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E.g. |
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{{ballots| |
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12: D>B>C>A>E>F>G>H>I |
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11: A>B>C>D>E>F>G>H>I |
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11: C>A>B>D>E>F>G>H>I |
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20: E>A>B>C>D |
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20: F>B>C>A>D |
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20: G>C>A>B>D}} |
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Or the obligatory anti-IRV example: |
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{{ballots| |
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12:D>A>B>C>E>F>G |
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11:D>B>A>C>E>F>G |
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11:C>D>A>B>E>F>G |
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19:E>A>B>C>D>F>G |
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21:F>B>C>A>D>E>G |
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20:G>C>A>B>D>E>F}} |
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Here the smallest DMT set is {ABCD}. IRV elects D. The Smith set is {ABC}. [[User:Kristomun|Kristomun]] ([[User talk:Kristomun|talk]]) 22:10, 8 May 2020 (UTC) |
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Revision as of 22:10, 8 May 2020
Possible multi-winner generalizations
User:Kristomun, I think this may have some kind of STV-PR generalization. In a 2-seat election, we know that anyone who has over 1/4th of the active votes at any point in STV is guaranteed to be one of the final 3 remaining candidates, since it's impossible for 3 other candidates to each have more votes than this candidate (since they'd each have to have over 1/4th of the active votes, resulting in more than 100% of votes total being allocated to different candidates), which is what would enable them to survive elimination longer. So, we can say that when over 1/3rds of the voters prefer someone from the "dominant mutual quarter" set (DMT but for 1/4th of the electorate) over anyone else who survives until the final round, then the dominant mutual quarter candidate must win. In general, someone who is preferred by a solid coalition of 1/(k+2)th of the voters (k being the number of seats) and preferred by 1/(k+1)th of all voters over any other given rival must win. I'm not sure if there's a way to extract more from this insight, though. BetterVotingAdvocacy (talk) 03:48, 27 March 2020 (UTC)
Smith-efficiency
User:BetterVotingAdvocacy, I don't think what you said is true: that electing from the DMT set implies Smith when the Smith set is a subset of the DMT set is. Consider e.g. an election where more than a third of the voters vote ABCD in some order above everybody else, and that there are say, 26 candidates. Suppose furthermore that each voter in a majority votes (some random permutation of a random subset of candidates E..Z) > (some permutation of A, B, and C) > D > everybody else. Now {A,B,C,D} is the smallest DMT set, but D is beaten pairwise by A, B, and C, and thus D is not in the Smith set. So the Smith set is {A,B,C} which is a subset of the smallest DMT set {A,B,C,D}. Then our contrived DMT-passing method could elect D, which would be in the DMT set but not the Smith set.
E.g.
12: D>B>C>A>E>F>G>H>I 11: A>B>C>D>E>F>G>H>I 11: C>A>B>D>E>F>G>H>I 20: E>A>B>C>D 20: F>B>C>A>D 20: G>C>A>B>D
Or the obligatory anti-IRV example:
12:D>A>B>C>E>F>G 11:D>B>A>C>E>F>G 11:C>D>A>B>E>F>G 19:E>A>B>C>D>F>G 21:F>B>C>A>D>E>G 20:G>C>A>B>D>E>F
Here the smallest DMT set is {ABCD}. IRV elects D. The Smith set is {ABC}. Kristomun (talk) 22:10, 8 May 2020 (UTC)