Pairwise counting: Difference between revisions
→Negative vote-counting approach
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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.
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.
Writeup on solving the write-in issue for pairwise counting: <blockquote> Bonus: The votes for each candidate can be placed in the blank cell comparing themselves to themselves in the pairwise matrix i.e. A>A would contain A's votes [number of voters ranking them].<ref>{{Cite web|url=https://www.reddit.com/r/EndFPTP/comments/fsa4np/possible_solution_to_the_condorcet_writein_problem/|title=Possible solution to the Condorcet write-in problem|last=|first=|date=|website=|url-status=live|archive-url=|archive-date=|access-date=}}</ref></blockquote>▼
The negative counting approach requires even more markings when it is desired to have comprehensive vote totals, rather than only information about who won, tied, or lost each matchup. This is because if there are 2 candidates A and B, with 2 voters preferring A>B, 1 preferring B>A, and 5 voting A=B, 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 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). ▼
▲Writeup on solving the write-in issue for pairwise counting:<blockquote> Bonus: The votes for each candidate can be placed in the blank cell comparing themselves to themselves in the pairwise matrix i.e. A>A would contain A's votes [number of voters ranking them].<ref>{{Cite web|url=https://www.reddit.com/r/EndFPTP/comments/fsa4np/possible_solution_to_the_condorcet_writein_problem/|title=Possible solution to the Condorcet write-in problem|last=|first=|date=|website=|url-status=live|archive-url=|archive-date=|access-date=}}</ref></blockquote>
It is not necessary to mark that a voter ranked a candidate if they ranked that candidate as their last choice, because this means they wouldn't vote for that candidate in any matchups. This advice is less relevant when write-ins are allowed, however, because even if a voter ranks a candidate last among the candidates named on their ballot, they are still implicitly ranking that candidate above all write-in candidates they didn't rank on their ballot.
=== Dealing with equal-ranking ===
The negative counting approach requires even more markings when equal-ranking is allowed, and it is desired to have comprehensive vote totals, rather than only information about who won, tied, or lost each matchup. This is because if there are 2 candidates A and B, with: <blockquote>2 A>B
1 B>A
▲
=== 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.
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