User:BetterVotingAdvocacy/Negative vote-counting approach for pairwise counting: Difference between revisions
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*[[RCV]]: ''354'' marks (249 voters' 1st choices + 105 votes transferred throughout<ref>https://i0.wp.com/evanstondems.com/wp-content/uploads/2020/02/RCVPrez-Results.png?fit=1024%2C341&ssl=1</ref>)
== Using negative pairwise counting while collecting additional non-pairwise information ==
=== Cardinal information ===
In some contexts (i.e. when precinct-summably counting [[STAR voting]]) it may be useful to collect both pairwise counts and scores for candidates. In such circumstances, one can take advantage of the fact that if:
* 3 voters scored Candidate A a 3/5
** 2 of those 3 voters scored A below B
then this can be recorded by, for Candidate A, tallying not just the number of voters who scored them, but specifically the number of voters who scored them at a specific score (except optionally the minimum score, which is usually 0). So if you know that:
* 5 score A a 5/5
* 2 score A a 4/5
* 3 voters score A a 3/5
* 0 voters scored A any other score
then 10 voters total (explicitly) scored A. This allows you to figure out not only the number of voters scoring a candidate above-last, but also the score of the candidate (since you can multiply the number of voters giving the candidate a particular score by that score, and add this up for all possible scores).
* Note that the amount of cardinal information collected here allows you to also compute the results of cardinal methods like [[Majority Judgement]].
* This requires just as many marks as negative pairwise counting done without collecting scores, because in either vote-counting scheme, one mark is made to count that a voter scored a candidate.
** Data-wise, this requires collecting [[number of candidates]*[number of possible scores]] data values rather than only [number of candidates] values to know how many voters scored a candidate (i.e. because rather than all of those voters being considered to be one group, they are separated into [number of possible scores] groups).
=== Bucklin (number of voters giving a particular ranking to a candidate) ===
For [[Bucklin voting]] or any median-based ranked voting method, instead of counting the number of voters who ranked a candidate, you can count specifically the number of voters who ranked that candidate 1st, 2nd, etc. To tally this, count only the number of voters who ranked each candidate 1st, and if no candidate has a majority, add in the number of voters who ranked them 2nd, and repeat for 3rd, 4th, etc.
== Connection to cardinal methods ==
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== Notes ==
[[File:Negative vote-counting approach to pairwise counting.png|thumb|1114x1114px|Negative vote-counting approach for pairwise counting (Note: Regular approach may be better in some use cases; see cited discussions in text to the left).]]
An alternative way to do the negative approach, which is more similar to the regular approach, is to, when candidate B is explicitly ranked below A on a ballot, instead of counting -1 votes for B>A, count 1 vote for A>B, and later on, when the math is done, the number of votes for B>A is the number of ballots ranking B minus the number of votes for A>B. In other words, a part of the regular pairwise counting approach is used, but only in matchups where both candidates are explicitly ranked by the voter (i.e. a voter who voted A>B and left C unranked would have their vote for A>B counted, but not their vote for A>C, because later on it will be inferred that they must have preferred A to C by virtue of having ranked A but not C).▼
In practice, to make pairwise counting easier, voters could be provided with two or fewer ranks than the number of candidates, with equal-ranking being allowed so that voters could do [[preference compression]]. This way, a voter who would usually indicate a preference that would have to be counted between two candidates would have to indicate no preference between them instead.
=== Alternative ways to frame negative pairwise counting ===
▲An alternative way to do the negative approach, which is more similar to the regular approach, is to, when candidate B is explicitly ranked below A on a ballot, instead of counting -1 votes for B>A, count 1 vote for A>B, and later on, when the math is done, the number of votes for B>A is the number of ballots ranking B minus the number of votes for A>B. In other words, a part of the regular pairwise counting approach is used, but only in matchups where both candidates are explicitly ranked by the voter (i.e. a voter who voted A>B and left C unranked would have their vote for A>B counted, but not their vote for A>C, because later on it will be inferred that they must have preferred A to C by virtue of having ranked A but not C).
=== Inspiration ===
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Note: In the B vs C matchup, B has 1 vote and C has 2 votes; yet in reality, 1 of the 2 voters equally ranked B and C (giving 0 votes to either of them), and the other ranked C over B (giving 1 vote to C and 0 to B); this discrepancy can be explained as being because the equal-ranking voter was treated as voting for both B and C, rather than neither of them.
=== Negative counting used for non-pairwise methods ===▼
Negative numbers can be used to count Approval voting, and can also be used in the context of most PR methods. ▼
* For Approval, when a voter
** Note that the same trick works for Score, but doesn't always speed the counting up there. For example, a voter who scores a candidate a 3 out of 5 can be thought of as giving them 5 positive points and -2 negative points as well.
* For sequential PR methods, some selection algorithm generally picks the a winner in a given round, then reweighting is applied to the votes, and this repeats. Because reweighting generally only alters some votes, it's often possible to treat the votes in one round as being the votes from a previous round, except some candidates have lost some support. [[SPAV]] is one such example. <ref>{{Cite web|url=https://forum.electionscience.org/t/possible-trick-for-counting-spav-and-cardinal-pr-faster/657|title=Possible trick for counting SPAV (and cardinal PR) faster|date=2020-05-26|website=The Center for Election Science|language=en-US|access-date=2020-07-21}}</ref>▼
=== Dealing with truncation ===
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=== Independence of unranked candidates ===
The negative approach doesn't require additional marks to be made for a given ballot when candidates are added to the election that that ballot doesn't vote for. For example, at most 3 marks need be made for a voter whose ballot is A>B, regardless of whether there are 2 candidates in the election or 100.
▲=== Negative counting used for non-pairwise methods ===
▲Negative numbers can be used to count Approval voting, and can also be used in the context of most PR methods.
▲* For Approval, when a voter approved more than half of the candidates, they can be considered to disapprove fewer than half the candidates. That is, a ballot approving A, B, and C, but not D, can be counted as a ballot that approves everyone except D.
▲** Note that the same trick works for Score, but doesn't always speed the counting up there. For example, a voter who scores a candidate a 3 out of 5 can be thought of as giving them 5 positive points and -2 negative points as well.
▲* For sequential PR methods, some selection algorithm generally picks the a winner in a given round, then reweighting is applied to the votes, and this repeats. Because reweighting generally only alters some votes, it's often possible to treat the votes in one round as being the votes from a previous round, except some candidates have lost some support. [[SPAV]] is one such example. <ref>{{Cite web|url=https://forum.electionscience.org/t/possible-trick-for-counting-spav-and-cardinal-pr-faster/657|title=Possible trick for counting SPAV (and cardinal PR) faster|date=2020-05-26|website=The Center for Election Science|language=en-US|access-date=2020-07-21}}</ref>
== References ==
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