Pairwise counting: Difference between revisions
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'''Pairwise counting''' is the process of considering a set of items, comparing one pair of items at a time, and for each pair counting the comparison results. In the context of voting theory, it involves comparing pairs of candidates or winner sets (usually using majority rule) to determine the winner and loser of the [[Pairwise matchup|pairwise matchup]]. This is done by looking at voters' (usually [[Ranked ballot|ranked]] or [[Rated ballot|rated]]) ballots to count, for each pair of candidates, which one they indicated a preference for, if they did. The [[pairwise preference]] article discusses how pairwise comparison information can be used.
Most, but not all, election methods that meet the [[Condorcet criterion]] or the [[Condorcet loser criterion]] use pairwise counting.<ref group=nb>[[Nanson's method|Nanson]] meets the [[Condorcet criterion]] and [[Instant-runoff voting]] meets the [[Condorcet loser criterion]].</ref> See the [[Pairwise counting#Condorcet|Condorcet section]] for more information on the use of pairwise counting in [[Condorcet methods]].
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|6658 (AM) –
104 (wi)
|}To read this, take for example the cell where BK is compared to AM (the cell with BK on the left and AM on the top); "4067 (AM)" means that 4067 voters preferred AM (Andy Montroll) over BK (Bob Kiss), and "3477 (BK)" means that 3477 voters preferred BK over AM. Because AM got more votes than BK in that matchup, AM won that matchup.
== Terminology ==▼
See [[Pairwise preference#Definitions]]. ▼
== Condorcet ==▼
See [[Pairwise preference#Condorcet]]. ▼
== Cardinal methods ==▼
See [[Pairwise preference#Strength of preference]] and [[rated pairwise preference ballot]]. ▼
==Notes==
[[File:Pairwise counting table with links between matchups.png|thumb|444x444px|Green arrows point from the loser of the matchup to the winner. Yellow arrows indicate a tie. Red arrows (not shown here) indicate the opposite of green arrows (i.e. who lost the matchup).For example, the B>A matchup points to A>B with a green arrow because A pairwise beats B (head-to-head).]]
[[File:Negative vote-counting approach to pairwise counting.png|thumb|1114x1114px|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).]]▼
[[File:Pairwise counting procedure.png|thumb|The procedure for pairwise counting with various ballot formats and examples.]]
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! A
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|'''50 points'''
| A>B
|A>C
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!A
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!D
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|}Try locating A>D (the fifth cell in the second row). To find the reverse, D>A, first you check and see that you have to go one cell down and to the left to be adjacent to the middle dividing line. Then, starting from the cell one cell down and to the left of the middle dividing line, go one cell further down and to the left to reach D>A. In doing this, you would start at A>D, go down to B>C, then jumping over the middle dividing line to C>B, go down to D>A.
One of the notable aspects of pairwise counting is that it can be used to find a Condorcet winner or member of the Smith set in a simple manner without needing to be done with written ballots; see [[:Category:Sequential comparison Condorcet methods]] for more information.
For equally ranked candidates, such as A=B>C>D, you put a negative vote for both A>B and B>A. Bonus: The votes for each candidate can be placed in the blank cell comparing themselves to themselves in the pairwise matrix i.e. A>A would contain A's votes.<ref>{{Cite web|url=https://www.reddit.com/r/EndFPTP/comments/fsa4np/possible_solution_to_the_condorcet_writein_problem/|title=Possible solution to the Condorcet write-in problem|last=|first=|date=|website=|url-status=live|archive-url=|archive-date=|access-date=}}</ref></blockquote>{{reflist|group=nb}}▼
▲See [[Pairwise preference#Strength of preference]] and [[rated pairwise preference ballot]].
==Negative vote-counting approach==
== References ==▼
▲[[File:Negative vote-counting approach to pairwise counting.png|thumb|1114x1114px|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).]]
The usual approach to pairwise counting is for the precinct vote-counters to mark all of the voter's preferences in each head-to-head matchup. This can be slow, and also can make it difficult to accommodate write-in candidates, 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. An alternative method of pairwise counting is the "negative votes/counting" approach: the precinct vote-counters simply indicate how many voters ranked/rated/marked a candidate on their ballot, and which candidates the voter ranked above (or equal to, depending on implementation) the candidates they marked. In other words, the vote-counters assume a voter prefers a candidate they marked in all matchups against other candidates, and then work to indicate which matchups this is not true for.
Note that this is faster when voters rank only a few of all candidates, and 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 vs A.
The negative counting approach requires even more markings when it is desired to have comprehensive vote totals, rather than only information about who won, tied, or lost each matchup. This is because if there are 2 candidates A and B, with 2 voters preferring A>B, 1 preferring B>A, and 5 voting A=B, then either it can be marked that A wins against B 2 to 1, or 7 to 6. This is because the voters who equally ranked A and B can be considered to be voting for both of them in their matchup. This is similar 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 issue is most relevant when trying to get accurate [[winning votes]] totals. To do so, either two markings can be made (1 negative vote for A>B and 1 for B>A) or one (1 negative marking for the A vs B matchup in general, which is later interpreted as a negative vote for both 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.
<references />
[[Category:Majority-related concepts]]
[[Category:Condorcet-related concepts]]
<references group="nb" />
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