User:BetterVotingAdvocacy/Negative vote-counting approach for pairwise counting: Difference between revisions
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== Notes ==
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.
=== Comprehensive example ===
Here is a more comprehensive picture of how the negative approach works with an example: for a voter who voted A>B=C>D with 2 unranked candidates E and F, their ballot would be processed into this matrix:
{| class="wikitable"
|+
!
!A
!B
!C
!D
!E
!F
|-
|A
|''(Ranked on) 1 ballot''
|
|
|
|
|
|-
|B
| -1 vote
|''1 ballot''
|
|
|
|
|-
|C
| -1 vote
|
|''1 ballot''
|
|
|
|-
|D
| -1 vote
| -1 vote
| -1 vote
|''1 ballot''
|
|
|-
|E
|
|
|
|
|
|
|-
|F
|
|
|
|
|
|
|}
A second voter who voted C>D=E>A would have their ballot processed into a matrix like this:
{| class="wikitable"
!
!A
!B
!C
!D
!E
!F
|-
|A
|''(Ranked on) 1 ballot''
|
| -1 vote
| -1 vote
| -1 vote
|
|-
|B
|
|
|
|
|
|
|-
|C
|
|
|''1 ballot''
|
|
|
|-
|D
|
|
| -1 vote
|''1 ballot''
|
|
|-
|E
|
|
| -1 vote
|
|''1 ballot''
|
|-
|F
|
|
|
|
|
|
|}
The two ballots can be combined into this matrix:
{| class="wikitable"
!
!A
!B
!C
!D
!E
!F
|-
|A
|''(Ranked on) 2 ballots''
|
| -1 vote
| -1 vote
| -1 vote
|
|-
|B
| -1 vote
|''1 ballot''
|
|
|
|
|-
|C
| -1 vote
|
|''2 ballots''
|
|
|
|-
|D
| -1 vote
| -1 vote
| -2 votes
|''2 ballots''
|
|
|-
|E
|
|
| -1 vote
|
|''1 ballot''
|
|-
|F
|
|
|
|
|
|
|}
Supposing that these two voters were the only voters in the precinct, this matrix would be the matrix containing all of the necessary information to get the final vote totals for this precinct, so the precinct could send it along. Each different precinct's matrix could be summed in the way shown above for the two voters. Once all precinct matrices have been summed, the following is the math done to get the final vote totals:
{| class="wikitable"
!
!A
!B
!C
!D
!E
!F
|-
|A
|''2 ballots''
|2 (votes)
|1
|1
|1
|2
|-
|B
|0
|''1 ballot''
|1
|1
|1
|1
|-
|C
|1
|2
|''2 ballots''
|2
|2
|2
|-
|D
|1
|1
|0
|''2 ballots''
|2
|2
|-
|E
|1
|1
|0
|1
|''1 ballot''
|1
|-
|F
|0
|0
|0
|0
|0
|0
|}
For example, look at the row for candidate A. Because A was ranked on 2 ballots, they received 2 votes in every head-to-head matchup against another candidate. But because they were ranked below C, D, and E on 1 ballot, their votes against C, D, and E were reduced by 1 vote, so that they only had 1 vote in favor of them against those 3 candidates in the respective head-to-head matchups.
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.
=== Dealing with truncation ===
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