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[[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]]. ItDepending on implementation, it is faster (i.e. requires less marks and tallying), dependingthan onregular implementation,pairwise counting when voters rank multiple candidates last, and otherwise equally fast. RatherThe thanidea of negative pairwise counting ais voter'sthat, preferencewhereas forregular apairwise candidatecounting theyoperates rankedfrom (i.e.the overperspective lower-rankedthat candidatesa candidate ("Candidate A"), itranked countsby thata voter's is '''lack'not'' ofpreferred preferenceover forany thatother candidatecandidates (i.e.<u>except</u> overany thecandidates candidateranked themselvesbelow andCandidate higher-rankedA candidates).by Inthat other wordsvoter, negative pairwise counting a)operates treatsfrom athe voteropposite asperspective, havingwhich "approved"is allthat ofCandidate theA candidatesis they''preferred'' over every other candidate <u>except</u> any candidates ranked above last-place(and optionally, b[[User:BetterVotingAdvocacy/Negative vote-counting approach for pairwise counting#Dealing with equal-ranking|equally to]]) countsCandidate theA by that voter's. Both approaches give the same final pairwise preferencesvote onlytotals amongfor thean candidateselection they(except "approved"in elections allowing equal-ranking, anddepending c)on aggregateshow thosethat twois piecescounted). An example of how the information forgarnered allby votersnegative (suchpairwise thatcounting foris eachtranslated candidate,into youa havefinal thepairwise total: numberif ofit is known that a) 5 voters whoranked a "approvedCandidate B", themand alongb) withonly the3 relevantof pairwisethese preferences),voters andranked Candidate B below or equal to "Candidate C", then calculateslogically, the finalother pairwise2 totalsvoters using<u>must</u> thehave aggregatedranked informationCandidate B above Candidate C.
 
Another way of looking at negative pairwise counting is that it a) treats each voter as having "approved" all of the candidates they ranked above last-place, b) counts each voter's pairwise preferences only among the candidates they "approved", c) aggregates those two pieces of information for all voters into the form of a pairwise table (such that for each candidate, you have the total number of voters who "approved" that candidate, along with the incomplete information about the candidate's performance in pairwise matchups against all other candidates), and then d) calculates the final pairwise totals for the election using that pairwise table.
An example of negative pairwise counting: if 5 voters ranked a candidate (A), and 3 of them didn't rank A above some other candidate (B), then 2 voters must have ranked A above or equal to B. When a voter only ranks candidates as their 1st choice(s) or last choices (i.e. uses equal-ranking with only two ranks), then negative pairwise counting becomes essentially equivalent to [[Approval voting]]'s vote-counting procedure for that voter's ballot.
 
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.
Semi-negative pairwise counting, which is theoretically even faster, is based on using both of the regular and negative pairwise counting techniques. 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 being marked by those voters, with no need to do pairwise counting for those specific candidates. 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.) There are also vote-counting techniques based on similar principles that can be used in non-pairwise contexts, such as for [[Score voting]] and various [[PR]] methods.
 
* 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.
 
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 (see [[#Inspiration]] for more information).
 
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.)
 
There are also vote-counting techniques based on similar principles that can be used in non-pairwise contexts, such as for [[Score voting]] and various [[Proportional Representation]] methods.
 
== Description ==
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=== Dealing with equal-ranking ===
 
The negative counting approach, depending on implementation, can require even more markings when a) equal-ranking is allowed and b) it is desired to have traditional pairwise vote totals. AnyHowever, any implementation of negative counting will give accurate information about whowhich candidate won, tied, or lost in each pairwise matchup, (i.e.along with the pairwise [[margins]]), howeverof victory/defeat. This is because if there are 2 candidates A and B, with the (explicit) votes being:<blockquote>2 A>B
 
1 B>A
Line 593 ⟶ 601:
 
===Inspiration===
[[Approval voting]] can be thought of as a [[Smith-efficient]] [[Condorcet method]] (i.e. one type of pairwise voting method) where, when a voter approves a candidate, they are assumed to vote for them in every head-to-head matchup against other candidates(see [[Self-referential Smith-efficient Condorcet method]]). Further, approving a candidate can be thought of as ranking them 1st, while disapproving a candidate can be thought of as ranking them last. Given that connection, and that in Approval voting, the vote-counting is done by counting the number of ballots that approve/mark a candidate, rather than by harder [[pairwise counting]], it is clear that a similar counting procedure could be applied to pairwise counting itself. The only complexity is that when voters rank candidates, they are allowed to express that they don't prefer certain candidates in certain matchups; thus, negative votes are necessary to communicate that lack of preference for specific matchups.
 
Given that connection, and that in Approval voting, the vote-counting is done by simply counting the number of ballots that approve/mark a candidate (rather than by using any complex [[pairwise counting]]), it is clear that an Approval-voting-style vote-counting procedure could be applied in order to obtain pairwise information from non-Approval-Voting ballots; however, the only complexity is that when voters are allowed to rank/score candidates, they are allowed to express more than two levels of preference, which allows them to indicate that they don't prefer certain candidates in certain matchups, while still preferring those candidates in other matchups (i.e. a voter who ranks "A>B>C" can indicate that B is superior to C but inferior to A; this can't be done on an Approval voting-style ballot); thus, counting "negative votes" is necessary during the vote-counting procedure in order to record that lack of preference/dispreference for those middle-ranked candidates in specific matchups.
 
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.
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