Weighted positional method: Difference between revisions
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==Criterion compliances==
Every weighted positional method passes the [[participation criterion]], the [[consistency criterion]], and mono-add-top; and is [[summability criterion|summable]] with order k=1.<ref name="Votedesc">{{cite web |url=http://www.9mail
Every weighted positional method except for [[First Past the Post electoral system|First past the post]] fails the [[later-no-harm criterion]]<ref name="Votedesc"/> and the [[majority criterion]]. Since [[First Past the Post electoral system|First past the post]] fails the [[Condorcet criterion]], and Condorcet implies majority, every weighted positional method fails Condorcet.
The [[Borda count]] is the only weighted positional method that never ranks the Condorcet winner last.<ref>{{Cite journal|last=Smith|first=John H.|date=1973|title=Aggregation of Preferences with Variable Electorate|url=http://www.jstor.org/stable/1914033|journal=Econometrica|volume=41|issue=6|pages=1027–1041|doi=10.2307/1914033|issn=0012-9682}}</ref> It follows that the only Condorcet-compliant runoff method that eliminates one loser at a time and is based on a weighted positional method is [[w:Nanson's_method#Baldwin_method|Baldwin]] (Borda-elimination).
Every [[resolvability criterion|resolvable]] weighted positional method fails [[clone independence]]: Plurality fails to vote-splitting, and every other method can be made to fail the majority criterion even for clones, hence turning a majority loser into a winner.<ref>{{cite web|url=http://lists.electorama.com/pipermail/election-methods-electorama.com/2017-December/001642.html|title=Resolvable weighted positional systems all fail independence of clones|website=Election-methods mailing list archives|date=2017-12-03|last=Munsterhjelm|first=K.}}</ref>
===Majority criterion===
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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.
==
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>.
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.
==Some
* 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|Votedesc}}{{rp|5}}{{r|rdrr_votesys_dowdall_method}}
* [[Borda Count]]
* 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|Votedesc}}{{rp|22}}
* The Eurovision Song Contest uses a method where each country's panel of judges rates the top 10 songs. The first pick gets 12 points, the second 10, the third 8, and the rest get <math>m - i</math> (i.e. 7, 6, ..., 1) with all the rest of the songs getting 0. This can be represented by the vector <math>a_\mathrm{Eurovision} = (12, 10, 8, 7, ..., max(m - i, 0), ..., 0)</math>
==Notes==
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).
==References==
{{reflist|refs=
<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>
}}
[[Category:Single-winner voting methods]]
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