User:BetterVotingAdvocacy/Negative vote-counting approach for pairwise counting: Difference between revisions
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# Vote-counting: The precinct vote-counters count the following values for a given candidate:
##The number of voters who ranked/rated/marked a candidate on their ballot.
##In each head-to-head matchup, the number of voters who ''explicitly'' ranked that candidate '''below''' the other candidate ("explicitly" meaning they
##*(This can be considered as, for a given ballot that ranks two candidates A and B as B>A, either:
##**Counting the number of votes for A''<''B. This yields a ''positive'' value.
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# Math: The final number of votes for the first candidate against the second candidate in each head-to-head matchup is then found by, if treating the second value as a positive number, subtracting the second value for the first candidate from the first value (addition is done instead of subtraction if the second value is treated as a negative number).
#* This can be succinctly summarized as, for finding the vote totals in a matchup between candidates A and B:<blockquote>A>=B* = A - A<B</blockquote>* meaning that the number of ballots ranking A over or (explicitly) equal to B, is equal to the number of ballots ranking A, minus the number of ballots ranking A below B (i.e. B over A).
The number of ballots marking each candidate can be placed in the blank cell comparing themselves to themselves in the pairwise matrix i.e. for candidate A, the cell A>A would contain the number of voters ranking A.<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>
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'''<big>Note</big>''': When using this approach, there is an important caveat to consider when dealing with voters who explicitly rank two candidates equally; see the [[#Dealing with equal-ranking]] section below.
=== Example ===
If the votes are:<blockquote>10: A>B
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* B>A is (15-10)=5 votes.
=== Dealing with equal-ranking ===
The negative counting approach, depending on implementation, can require even more markings when equal-ranking is allowed and it is desired to have traditional pairwise vote totals. Any implementation of negative counting will give accurate information about who won, tied, or lost each matchup (i.e. the pairwise [[margins]]), however. This is because if there are 2 candidates A and B, with the votes being: <blockquote>2 A>B
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Note: Voters who don't rank either of the candidates in a matchup are generally considered to equally rank them, but no implementation of negative counting would consider them to vote for both candidates in the matchup. Only voters who have marked both candidates can be counted that way.
==== Example of the two approaches to equal-ranking ====
Suppose a voter had ranked 9 of 10 candidates as their 1st choices, and the 10th candidate was unranked (i.e. implicitly ranked last).
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In addition:
===== Explicitly equal-ranked candidates both get a vote =====
No extra work needs to be done.
===== Equal-ranked candidates don't get votes =====
For each matchup, the following number of markings can be made for two candidates A and B:
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In this example, there are 0.5*(9*8)=0.5*72='''36''' matchups to count between equally-ranked candidates. Accordingly, at least either 36*2='''72''' or '''36''' markings must be made.
=== Dealing with last-place candidates ===
It is not necessary to make any markings for a candidate a voter ranked as their last choice, because this means they wouldn't vote for that candidate in any matchups.
==== Write-in candidates ====
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 of the write-in candidates they didn't rank on their ballot.
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=== Dealing with truncation ===
When voters aren't allowed to do [[truncation]] (and [[Write-in candidate|Write-in candidat]]<nowiki/>es aren't allowed), then it can be useful to skip the part of the negative counting procedure where the vote-counters mark how many ballots a candidate is ranked on, and instead assume a candidate is preferred in every matchup on every voter's ballot. The negative votes would then be applied as usual.
* This works because every candidate will actually be marked on every voter's ballot.
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