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B is the CW, so they'd win the first seat. If their supporters' ballots are reweighted by half, then C pairwise beats A 48.5 to 34 and wins, despite A being bullet voted by a Droop quota. One complicated way of possibly fixing this is to, after electing B, say that if B hadn't been in the election, C would have been the winner, and therefore both B and C voters' ballots should be reweighted by half since they both rank C above all candidates other than B (A), thus allowing A to beat C 34 to 33.
B is the CW, so they'd win the first seat. If their supporters' ballots are reweighted by half, then C pairwise beats A 48.5 to 34 and wins, despite A being bullet voted by a Droop quota. One complicated way of possibly fixing this is to, after electing B, say that if B hadn't been in the election, C would have been the winner, and therefore both B and C voters' ballots should be reweighted by half since they both rank C above all candidates other than B (A), thus allowing A to beat C 34 to 33.


== Negative pairwise counting approach ==
== Vote-counting ==

=== Sequential Monroe Voting ===
[[SMV]] can be done by creating a weighted positional matrix (similar to that for [[Bucklin voting]]; see [[Summability criterion]]) which shows how many voters max-scored a candidate, one-point-below-max, and so on. This section will assume the use of a score scale of 0 to 5. A candidate's quota score can be reconstructed by looking at the number of voters who gave them a 5, and adding in all of the voters who scored them a 4 if there isn't a quota scoring them a 5, and repeating until a quota of voters has been found. The total points given by this group of voters is the candidate's quota score. Once the candidate with the highest quota score has been elected, reweighting occurs; if fractional reweighting is used, then at most there are 5 additional possibilities for every additional round for how voters can score the remaining candidates. For example, in the second round, either voters give a candidate an integer score between 0 and 5, or they give the candidate a fractional score whose strength depends on which of the 5 positive scores between 0 and 5 they gave to the winner of the first round.

This idea only requires two passes of all ballots for each winner elected; one to compute the round's matrix, and the second to isolate the ballots that are in a candidate's quota. It is possible to use the below-discussed idea for SPAV in SMV by, between rounds, only keeping track of the changes in scores for ballots that were reweighted. For example, it is possible to find the second round's matrix by noting that 5 voters are no longer giving a candidate a 4, with 3 of them now giving that candidate a 2.8 and 2 of them giving the candidate a 0. Thus, only changes would have to be made to the first round's matrix.

=== Negative pairwise counting approach ===
It's possible to fit both [[Negative vote-counting approach for pairwise counting]] info and rated info in a pairwise matrix by either creating a cell for each candidate that holds both pieces of info, or creating a separate column to contain one of the pieces of info. Example:
It's possible to fit both [[Negative vote-counting approach for pairwise counting]] info and rated info in a pairwise matrix by either creating a cell for each candidate that holds both pieces of info, or creating a separate column to contain one of the pieces of info. Example:
{| class="wikitable"
{| class="wikitable"
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When doing the "semi-negative" counting procedure mentioned in the previous paragraph, some voters will be able to contribute votes to both candidates in a matchup, while other voters won't, purely based on how highly or lowly they ranked them. If this creates legal or procedural issues, it is possible to have each precinct only submit the [[Margins|margin]] they found in every pairwise matchup, rather than the votes on both sides as well. In other words, if, for the A vs B matchup, in Precinct 1 A has 15 votes and B 10, while in P2 A has 7 and B 8, then it is possible for P1 to submit that A has 5 votes more than B, and P2 to submit that B has 1 vote more than A. This can be used to find that A has 4 votes more than B in the combined electorate of the two precincts.
When doing the "semi-negative" counting procedure mentioned in the previous paragraph, some voters will be able to contribute votes to both candidates in a matchup, while other voters won't, purely based on how highly or lowly they ranked them. If this creates legal or procedural issues, it is possible to have each precinct only submit the [[Margins|margin]] they found in every pairwise matchup, rather than the votes on both sides as well. In other words, if, for the A vs B matchup, in Precinct 1 A has 15 votes and B 10, while in P2 A has 7 and B 8, then it is possible for P1 to submit that A has 5 votes more than B, and P2 to submit that B has 1 vote more than A. This can be used to find that A has 4 votes more than B in the combined electorate of the two precincts.


=== Examples ===
==== Examples ====
Here's a theoretical example to compare negative pairwise counting with the regular approach (taken from <ref>https://www.reddit.com/r/EndFPTP/comments/fylh2p/how_are_elections_run_under_condorcet_reported/fn1vztw/</ref>): a 2018 election in Washington state involved 28 candidates <ref>https://en.wikipedia.org/wiki/2018_United_States_Senate_election_in_Washington</ref>. For simplicity's sake, suppose there had only been 100 voters in that election, with 80% of them ranking 5 candidates and 20% ranking all 28 candidates. (See the "Formula for number of marks that need to be made" section in the negative counting article for info on the following calculations.)
Here's a theoretical example to compare negative pairwise counting with the regular approach (taken from <ref>https://www.reddit.com/r/EndFPTP/comments/fylh2p/how_are_elections_run_under_condorcet_reported/fn1vztw/</ref>): a 2018 election in Washington state involved 28 candidates <ref>https://en.wikipedia.org/wiki/2018_United_States_Senate_election_in_Washington</ref>. For simplicity's sake, suppose there had only been 100 voters in that election, with 80% of them ranking 5 candidates and 20% ranking all 28 candidates. (See the "Formula for number of marks that need to be made" section in the negative counting article for info on the following calculations.)


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Some caveats are that the work in the regular approach could have been reduced significantly because many of these candidates could have potentially been left off of the ballot if there had been tighter ballot access requirements, and the number of rankings could have been limited (i.e. voters could have been given only 5 ranks to put the candidates into), which would make some voters have to equally rank more candidates. These factors also reduce the amount of work for the negative counting approach to some extent.
Some caveats are that the work in the regular approach could have been reduced significantly because many of these candidates could have potentially been left off of the ballot if there had been tighter ballot access requirements, and the number of rankings could have been limited (i.e. voters could have been given only 5 ranks to put the candidates into), which would make some voters have to equally rank more candidates. These factors also reduce the amount of work for the negative counting approach to some extent.


=== Negative counting in non-pairwise methods ===
==== Negative counting in non-pairwise methods ====
Negative counting approaches can be applied to various voting methods. For example, it's possible to reserve a special mark in Score voting that indicates that a voter gave every non-write-in candidate the max score, and then also count negative scores for the voter in such a way as to reproduce their actual scores. The practicality of this would likely be limited to ballots that max-score nearly all of the candidates, though. This type of special mark actually changes the worst-case number of marks needed to count Approval; if every voter approves every candidate (which is unlikely, since the voter is indicating no preference between any of the non-write-in candidates), then only 1 mark needs to be made per ballot, rather than [number of candidates] marks. In fact, such a mark reduces the worst-case number of marks from [number of candidates] down to roughly [0.5*(number of candidates)], because when, say, a voter approves one more than half of the candidates, that can be counted with the special mark along with negative marks for the one less than half of the candidates disapproved by the voter, for a total of [0.5*(number of candidates)] marks, rather than this same number plus one. Of course, reducing the number of marks that need to be made doesn't always result in less work (manual or cognitive) overall.
Negative counting approaches can be applied to various voting methods. For example, it's possible to reserve a special mark in Score voting that indicates that a voter gave every non-write-in candidate the max score, and then also count negative scores for the voter in such a way as to reproduce their actual scores. The practicality of this would likely be limited to ballots that max-score nearly all of the candidates, though. This type of special mark actually changes the worst-case number of marks needed to count Approval; if every voter approves every candidate (which is unlikely, since the voter is indicating no preference between any of the non-write-in candidates), then only 1 mark needs to be made per ballot, rather than [number of candidates] marks. In fact, such a mark reduces the worst-case number of marks from [number of candidates] down to roughly [0.5*(number of candidates)], because when, say, a voter approves one more than half of the candidates, that can be counted with the special mark along with negative marks for the one less than half of the candidates disapproved by the voter, for a total of [0.5*(number of candidates)] marks, rather than this same number plus one. Of course, reducing the number of marks that need to be made doesn't always result in less work (manual or cognitive) overall.


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