Pairwise counting: Difference between revisions

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[[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).]]

[[File:Pairwise counting negative counting with ranked ballot GIF.gif|thumb|454x454px|GIF for negative counting. Click on the image and then the thumbnail of the image to see the animation.]]

The usual approach to pairwise counting is for the precinct vote-counters to mark all of the voter's preferences in each head-to-head matchup. This can be slow, and also can make it difficult to accommodate write-in candidates, since the vote-counters won't know ahead of time who those candidates are, and thus won't be able to indicate preferences in those matchups. An alternative method of pairwise counting is the "negative votes/counting" approach: the precinct vote-counters simply indicate how many voters ranked/rated/marked a candidate on their ballot, and which candidates the voter ranked above (and equal to, depending on implementation; see below) the candidates they marked. In other words, instead of a candidate being assumed to be preferred only in the matchups where the vote-counters mark them as being so, the vote-counters assume a voter prefers a candidate they marked in all matchups against other candidates, and then work to indicate which matchups this is not true for.

The usual approach to pairwise counting is for the precinct vote-counters to mark all of the voter's preferences in each head-to-head matchup. This can be slow, and also can make it difficult to accommodate write-in candidates, since the vote-counters won't know ahead of time who those candidates are, and thus won't be able to indicate preferences in those matchups. An alternative method of pairwise counting is the "negative votes/counting" approach: the precinct vote-counters simply indicate how many voters ranked/rated/marked a candidate on their ballot, and which candidates the voter ranked above (and equal to, depending on implementation; see below) the candidates they marked. In other words, instead of a candidate being assumed to be preferred only in the matchups where the vote-counters mark them as being so, the vote-counters assume a voter prefers a candidate they marked in all matchups against other candidates, and then work to indicate which matchups this is not true for. This approach can also be considered an Approval voting-based or [[cardinal]] approach, because when voters vote Approval-style (rank some candidates 1st and all others last) their ballots are counted like Approval ballots.

Note that this is faster when voters rank only a few of all candidates, and slower otherwise. For example, a voter who votes A>B when there are 10 candidates can be assumed to vote for A and B in every matchup, except they don't prefer B>A. Usually, this would require manually marking those positive preferences, resulting in 9 marks to show A being preferred to all other candidates, and 8 marks to show B preferred to all candidates except A. But negative counting only requires 3 marks: 1 each for A and B to indicate they are preferred in every matchup, and 1 to indicate that this isn't the case for B>A.

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1 B>A

5 A=B </blockquote>then either it can be marked that A wins against B 2 to 1, or 7 to 6. This is because the voters who equally ranked A and B can be considered to either be voting for both of them, or neither of them, in their matchup. This is similar to how, in [[Approval voting]], if A has 30 approvals and B 20, and no other information is supplied, then it is impossible to know whether the 20 voters who approved B also approved A or not. This ~~issue~~ is ~~most~~ ~~relevant~~ ~~when trying to get accurate [[winning votes]] totals. To do so~~, ~~either two markings can be made (1 negative vote~~ for ~~A>B and 1 for B>A) or one (1 negative marking for the A vs B matchup in general~~, ~~which is later interpreted as a negative vote for both candidates). This can make negative counting significantly harder than~~ the ~~regular~~ ~~approach~~ ~~for some ballots: if a voter had ranked 9~~ of 10 candidates 1st and the 10th candidate last, there would only be 9 marks to make in the regular approach, but at least 10 + 9 + 36 or 72 = 55 or 91 markings (depending on whether 1 mark or 2 marks are made for equal-ranked matchups) with the negative approach.

5 A=B </blockquote>then either it can be marked that A wins against B 2 to 1, or 7 to 6. This is because the voters who equally ranked A and B can be considered to either be voting for both of them, or neither of them, in their matchup. This is similar to how, in [[Approval voting]], if A has 30 approvals and B 20, and no other information is supplied, then it is impossible to know whether the 20 voters who approved B also approved A or not. This can change who wins if, for example, using the winning votes version of [[Schulze]].

This issue is most relevant when trying to get accurate [[winning votes]] totals. To do so, either two markings can be made (1 negative vote for A>B and 1 for B>A) or one (1 negative marking for the A vs B matchup in general, which is later interpreted as a negative vote for both candidates). This can make negative counting significantly harder than the regular approach for some ballots: if a voter had ranked 9 of 10 candidates 1st and the 10th candidate implicitly last, there would only be 9 marks to make in the regular approach, but at least 9 + 36 or 72 = 45 or 81 markings (depending on whether 1 mark or 2 marks are made for equal-ranked matchups) with the negative approach. If no marks are made for equal-ranked matchups, then the same number of only 9 marks would have to be made under the negative approach.

=== Using with strength of preference ===

Negative vote-counting can be used to count weak pairwise preferences (i.e. if a voter only wants to give 0.4 votes in a matchup; see [[Rated pairwise preference ballot#Implementations]]) by counting only a "partial ballot" marking a candidate, and partial (i.e. weighted or fractional) negative votes in certain matchups.

Negative vote-counting can be used to count weak pairwise preferences (i.e. if a voter only wants to give 0.4 votes in a matchup; see [[Rated pairwise preference ballot#Implementations]]) by counting only a "partial ballot" marking a candidate, and partial (i.e. weighted or fractional) negative votes in certain matchups. In other words, it is as if a voter only partially approved (see [[KP transform]]) a candidate.

==References==