User:BetterVotingAdvocacy/Negative vote-counting approach for pairwise counting

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).

GIF for negative counting. Click on the image and then the thumbnail of the image to see the animation.

There are two steps to the negative counting approach: the information captured by the vote-counters, and the math done to find the final result.

1. 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 also marked both candidates on their ballot).
2. Math: The final number of votes for the first candidate against the second candidate in each head-to-head matchup is then found by subtracting the second value for the first candidate from the first value.

The 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.

Note: When using this approach, there is an important caveat when dealing with voters who explicitly rank two candidates equally; see the #Dealing with equal-ranking section below.

Example

If 10 voters vote A>B and 5 voters vote B, then A is explicitly marked on 10 ballots and B on 15, with B being explicitly ranked below A on 10 ballots, and A being explicitly ranked below B on 0 ballots. The number of votes in favor of each candidate is then:

• A>B is (10-0)=10 votes.
• 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 comprehensive vote totals, rather than only information about who won, tied, or lost each matchup (i.e. the pairwise margins). This is because if there are 2 candidates A and B, with the votes being:

2 A>B

1 B>A

5 A=B

then either it can be marked that A wins against B by:

• 2 votes to 1
• 7 votes to 6

This is because the voters who equally ranked A and B can be considered to, in the A vs B matchup, either be voting for:

• Neither of them (similar to choose-one FPTP voting; this is the traditional pairwise counting approach).
• Both of them (similar to Approval voting).

This is related 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 using the winning-votes version of a Category:Defeat-dropping Condorcet method, because not only does it matter who wins the matchup, but also exactly how many voters genuinely preferred the winner to the loser.

Example of the two approaches to equal-ranking

If a voter had ranked 9 of 10 candidates as their 1st choices, and the 10th candidate was unranked (i.e. implicitly ranked last), then at least 9 marks must be made in both approaches, to indicate that 1 voter has ranked each of the 9 candidates who are the voter's 1st choice. 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:

• 2 markings can be made (1 negative vote for A>B and 1 for B>A). (Note that in essence, this approach involves counting the number of voters who explicitly ranked a given candidate below or equal to the other candidate, rather than only below.)
• 1 negative marking can be made for the A vs B matchup in general, which is later interpreted as a negative vote for both candidates.

In this example, there are 0.5*(9*8)=0.5*72=36 matchups to count between equally-ranked candidates. Accordingly, either 36*2=72 or 36 markings can be made.

Dealing with last-place candidates

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.

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.

Comparison to the regular approach

Comparison between the regular approach and negative counting:

• The regular approach: The precinct vote-counters manually count all of the voter's preferences in each head-to-head matchup; in other words, a candidate is assumed to be preferred only in the matchups where the vote-counters mark them as being so.
  • This can be slow, and also can make it difficult to accommodate write-in candidates (see the above section), 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.
• Negative counting approach: The vote-counters mark a candidate as being ranked on a ballot, assume the voter who marked them prefers that candidate in every matchup, and then show which matchups this is not true for.

Note that this negative counting is faster when voters rank only a few of all candidates, and potentially 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.

Connection to cardinal methods

This approach can be considered an Approval voting-based or cardinal approach, because when explicit equal-rankings are counted as a vote for both candidates in the matchup, then each voter that votes Approval-style (i.e. explicitly ranks some candidates 1st and all others implicitly last) will have their ballot counted like an Approval ballot (i.e. all approved candidates receive one mark, and all disapproved candidates receive no marks).

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, rather than 1 vote; 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 treated as if only a partial voter supported a candidate (see KP transform).