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

[[File:Adding ballot matrices in negative pairwise counting approach.png|thumb|1088x1088px|Note in "Step 1: Combination" that the two ballots' negative pairwise matrices are added up.[[File:Pairwise counting negative counting with ranked ballot GIF.gif|thumb|454x454px|GIF for negative counting. Click on the image and then the thumbnail of the image to see the animation.]]]]

The negative counting approach is an alternative method of doing [[pairwise counting]]. Depending on implementation, it is faster (i.e. requires less marks and tallying) than regular pairwise counting when voters rank multiple candidates last, and otherwise equally fast. The idea of negative pairwise counting is that, whereas 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, depending on how that is 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", 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.

 

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.

* A [[User:BetterVotingAdvocacy/Negative vote-counting approach for pairwise counting#Regular pairwise counting but done by counting first choices separately|simple variation]] on semi-negative counting is to do regular pairwise counting except that each voter's 1st-choice candidate(s) are simply counted as having been marked on those voters' ballots, with no need to do any further vote-counting work for those candidates on those ballots.

 

 

A number of election examples are provided [[User:BetterVotingAdvocacy/Negative vote-counting approach for pairwise counting#Election example comparisons|below]], with analysis of how the various pairwise counting methods would perform when counting the ballots (the analysis being based off of various [[User:BetterVotingAdvocacy/Negative vote-counting approach for pairwise counting#Formula for counting the required number of marks to be made|formulas]] also provided below.)

 

It is possible to make a special marking for a last-choice candidate that indicates they are not preferred over any of the on-ballot (regular) candidates, but that they are preferred over all write-in candidates. It would then only be necessary to record negative votes for matchups involving write-in candidates who are ranked above the last-choice candidate on some ballots. This would mean making at most two additional marks for every last-ranked candidate on a ballot, because in practice, in most elections, voters are only allowed to write in at most one candidate.<ref name=":0">{{Cite web|url=https://forum.electionscience.org/t/negative-vote-counting-approach-for-pairwise-counting/644|title=Negative vote-counting approach for pairwise counting|date=2020-04-27|website=The Center for Election Science|language=en-US|access-date=2020-09-15}}</ref> This can be compared to the regular approach to dealing with write-ins at [[Pairwise counting#Dealing with write-in candidates]].

 

=== Regular pairwise counting but done by counting first choices separately ===

*** If 4 candidates run, then in both approaches, '''3''' marks are made (2nd choice is ranked above 2 candidates and below 1 candidate).

 

Here are some examples for the first numbers in each series (with the upper bound bolded):

{| class="wikitable"

 

===Alternative ways to frame negative pairwise counting===

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).

 

===Inspiration===

 

 

This has the advantage of, when every voter does [[bullet voting]], being counted exactly like an [[FPTP]] election (one mark per ballot for the candidate it marked), which also shows that FPTP can be thought of as a constrained form of Approval.