Weighted positional method: Difference between revisions
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With more work, the examples can be generalized to any number of candidates greater then 3 by assuming every voter ranks all the other candidates in the same order. |
With more work, the examples can be generalized to any number of candidates greater then 3 by assuming every voter ranks all the other candidates in the same order. |
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==Generalizations== |
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It is possible to view [[Approval voting]] and [[Score voting]] as a more general weighted positional method, where each voter has some freedom in what <math>a</math> vector to choose. For Approval, the voter's <math>a</math> vector has value 1 for every approved candidate and 0 otherwise - i.e. 1 down to the voter's [[approval cutoff]] and then 0 below - while for Score voting, the voter directly specifies <math>a</math>. |
It is possible to view [[Approval voting]] and [[Score voting]] as a more general weighted positional method, where each voter has some freedom in what <math>a</math> vector to choose. For Approval, the voter's <math>a</math> vector has value 1 for every approved candidate and 0 otherwise - i.e. 1 down to the voter's [[approval cutoff]] and then 0 below - while for Score voting, the voter directly specifies <math>a</math>. |
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As a result, [[Score voting]] fails every criterion that does not involve removing or adding candidates, and that least one weighted positional method fails. The voters could just happen to rate the candidates the same way a weighted positional method would score them, and then the failure example for that method would also apply to Score. |
As a result, [[Score voting]] fails every criterion that does not involve removing or adding candidates, and that least one weighted positional method fails. The voters could just happen to rate the candidates the same way a weighted positional method would score them, and then the failure example for that method would also apply to Score. |
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==Some Methods== |
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* Nauru island adopted a positional method with harmonic weights where the k<sup>th</sup>-ranked candidate gets a score of 1/k. It can be represented by the <math>a</math> vector: <math>a_\mathrm{Nauru} = (1, 1/2, ..., 1/k, ..., 1/m)</math>. This is also sometimes called the Dowdall method.{{r|smith2006}}{{rp|5}}{{r|rdrr_votesys_dowdall_method}} |
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* [[Borda Count]] |
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* As shown above, many systems can be represented as weighted positional methods including Plurality, Antiplurality, [[Vote For and Against]], and Dabagh's "vote and a half" method where a voter assigns 1 point to their favorite and half a point to their second favorite.{{r|smith2006}}{{rp|22}} |
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==Notes== |
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All weighted positional methods can be understood in a [[pairwise counting]] context. For example, in Borda, if a voter gives every candidate the same number of points in a matchup as they give them overall, then the winner of all matchups is the Borda winner. The connection can be further understood by dividing the total number of points a voter gave a candidate by the maximum number of points they could have given any candidate i.e. a voter who gave one candidate 7 points out of a max of 7 and another 6 out of 7 contributed a pairwise margin of 1 point, or 1/7th of a vote, to the former candidate in the matchup between the two). |
All weighted positional methods can be understood in a [[pairwise counting]] context. For example, in Borda, if a voter gives every candidate the same number of points in a matchup as they give them overall, then the winner of all matchups is the Borda winner. The connection can be further understood by dividing the total number of points a voter gave a candidate by the maximum number of points they could have given any candidate i.e. a voter who gave one candidate 7 points out of a max of 7 and another 6 out of 7 contributed a pairwise margin of 1 point, or 1/7th of a vote, to the former candidate in the matchup between the two). |
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==References== |
==References== |
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{{reflist|refs= |
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<ref name=smith2006>{{cite journal |url=http://www.9mail.de/m-schulze/votedesc.pdf |access-date=2022-01-29 |title=Descriptions of single-winner voting systems |date=July 12, 2006 |last1=Smith }}</ref> |
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<ref name=rdrr_votesys_dowdall_method>{{cite web |url=https://rdrr.io/cran/votesys/man/dowdall_method.html |title=dowdall_method: Dowdall Method|access-date=2022-01-29 }}</ref> |
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}} |
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[[Category:Single-winner voting methods]] |
[[Category:Single-winner voting methods]] |
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