User:BetterVotingAdvocacy/Negative vote-counting approach for pairwise counting: Difference between revisions

The negative counting approach is an alternative method of doing [[pairwise counting]]. The idea of negative pairwise counting is that, whereasWhereas regular pairwise counting operates from the perspective that a candidate ("Candidate A") ranked by a voter is ''not'' preferred over any other candidates <u>except</u> any candidates ranked below Candidate A by that voter, negative pairwise counting operates from the opposite perspective, which is that Candidate A is ''preferred'' over every other candidate <u>except</u> any candidates ranked above (and optionally, [[User:BetterVotingAdvocacy/Negative vote-counting approach for pairwise counting#Dealing with equal-ranking|equally to]]) Candidate A by that voter. Both approaches give the same final pairwise vote totals for an election (except in elections allowing equal-ranking and/or write-in candidates, depending on how those are counted). An example of how the information garnered by negative pairwise counting is translated into a final pairwise total: if it is known that a) 5 voters ranked a "Candidate B" on their ballots, and b) only 3 of these voters ranked Candidate B below or equal to "Candidate C", then logically, the other 2 voters <u>must</u> have ranked Candidate B above Candidate C.

Depending on implementation, negative pairwise counting is faster (i.e. requires less marks and tallying) than regular pairwise counting when voters rank multiple candidates last, and otherwise equally fast. [[User:BetterVotingAdvocacy/Negative vote-counting approach for pairwise counting#Semi-negative pairwise counting|Semi-negative pairwise counting]], which is theoretically even faster than negative pairwise counting, is based on using both of the regular and negative pairwise counting techniques for each voter's ballot, depending on which is faster at each step of counting each ballot. There is also a [[User:BetterVotingAdvocacy/Negative vote-counting approach for pairwise counting#Regular pairwise counting but done by counting first choices separately|simple variation]] of semi-negative pairwise counting which does not involve using any negative numbers, but which only sometimes outperforms negative pairwise counting.

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When a voter only ranks candidates using the first two ranks on their ballot (i.e. either 1st or last rank), then negative pairwise counting becomes essentially equivalent to [[Approval voting]]'s vote-counting procedure for that voter's ballot (this insight is part of the [[#Inspiration]] for the idea).

There are also vote-counting techniques based on similar principles to negative pairwise counting that [[User:BetterVotingAdvocacy/Negative vote-counting approach for pairwise counting#Negative counting used for non-pairwise methods|can be used in non-pairwise contexts]], such as for [[Score voting]] and various [[Proportional Representation]] methods.

When a voter ranks more than half of the candidates, it is possible to further reduce the number of marks counted compared to negative pairwise counting or regular pairwise counting. This is because when a voter ranks one candidate below more than half of the candidates, instead of counting negative marks in that candidate's matchups against all of those candidates, it would be faster to count positive (regular) marks for that candidate against the fewer than the half of the candidates that they're ranked above, instead of counting negative marks in that candidate's matchups against all of the candidates that they're ranked below. Example of this approach with a voter ranking all of 5 candidates:

Note that regular pairwise counting can have its required number of marks reduced, without using any negative numbers, by counting 1st choices separately from all other ranks; see the section above [[Pairwise counting#Uses for first choice information]]. This is essentially equivalent to doing semi-negative pairwise counting with negative counting only applied to 1st choices and regular pairwise counting applied to all other candidates. Once all ballots have been counted, the same final "calculation" step from negative pairwise counting applies, wherein all the ballots that were counted as marking a candidate in previous steps of vote-counting (in this approach, only those ballots that were counted as ranking a candidate 1st) are counted as giving that candidate a vote in any pairwise matchups against other candidates.