Pairwise counting: Difference between revisions
Reintroduced Condorcet loser as e.g. STAR passes it and uses pairwise counting. Summable contingent vote would also use it.
(Reintroduced Condorcet loser as e.g. STAR passes it and uses pairwise counting. Summable contingent vote would also use it.) |
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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">
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==== Example without numbers ====
As an example, if pairwise counting is used in an election that has three candidates named A, B, and C, the following pairwise counts are produced:
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In general, for N candidates, there are 0.5*N*(N-1) pairwise matchups. For example, for 2 candidates there is one matchup, for 3 candidates there are 3 matchups, for 4 candidates there are 6 matchups, for 5 candidates there are 10 matchups, for 6 candidates there are 15 matchups, and for 7 candidates there are 21 matchups.
==== Example with numbers ====
{{Tenn_voting_example}}
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==== Example using various ballot types ====
[See [[:File:Pairwise counting procedure.png|File:Pairwise_counting_procedure.png]] for an image explaining all of this).
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Suppose there are five candidates A, B, C, D and E.
===== Sufficiently expressive ballot types =====
====== Ranked ballots ======
Using ranked ballots, suppose two voters submit the ranked ballots A>B>C, which means they prefer A over B, B over C, and A over C, with all three of these ranked candidates being preferred over either D or E. This assumes that unranked candidates are ranked equally last.
====== Rated ballots ======
Now suppose the same two voters submit [[Rated voting|rated ballots]] of A:5 B:4 C:3, which means A is given a score of 5, B a score of 4, and C a score of 3, with D and E left blank. Pairwise preferences can be inferred from these ballots. Specifically A is scored higher than B, and B is scored higher than C. It is known that these ballots indicate that A is preferred over B, B over C, and A over C. If blank scores are assumed to mean the lowest score, which is usually a 0, then A and B and C are preferred over D and E.
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([https://star.vote star.vote] offers the ability to see the pairwise matrix based off of rated ballots.)
===== Inexpressive ballot types =====
====== Choose-one and Approval ballots ======
Pairwise counting also can technically be done using [[Choose-one voting]] ballots and [[Approval voting]] ballots (by giving one vote to the marked candidate in a matchup where only one of the two candidates was marked), but such ballots do not supply information to indicate that the voter prefers their 1st choice over their 2nd choice, that the voter prefers their 2nd choice over their 3rd choice, and so on.
===== Dealing with unmarked/last-place candidates =====
Note that when a candidate is unmarked it is generally treated as if the voter has no preference between the unmarked candidates (a candidate who is marked on the ballot is considered '''explicitly''' supported, and a candidate who is unmarked is '''implicitly''' unsupported). When the voter has no preference between certain candidates, which can also be seen by checking if the voter ranks/scores/marks multiple candidates in the same way (i.e. they say two candidates are both their 1st choice, or are both scored a 4 out of 5), then it is treated as if the voter wouldn't give a vote to any of those candidates in their matchups against each other.
=== Dealing with write-in candidates ===
[[File:Approaches for handling write-in candidates in pairwise counting.png|thumb|837x837px]]
The difficulty of handling [[Write-in candidate|write-in candidat]]<nowiki/>es depends on how a voter's preference between ranked and unranked candidates is counted.
# If the voter is treated as preferring ranked candidates over unranked candidates (which is the near-universal approach), then write-ins can be difficult to count using pairwise counting, because the vote-counters don't know who they are and thus can't directly record voter preferences in matchups between on-ballot mainstream candidates and write-in candidates.
A comprehensive approach is to, in each [[precinct]], count the number of ballots that explicitly rank each (non-write-in) candidate. When a write-in candidate is found on a ballot, then before that ballot is counted, the number of votes each non-write-in candidate gets against the write-in candidate is the number of ballots they were so far ranked on. The ballot is then counted, and the write-in candidate is treated as a non-write-in candidate from that point onwards (from the perspective of this algorithm). When the pairwise vote totals are summed up from each precinct, then if in one precinct a write-in candidate wasn't marked by any voters but in another they were, then similarly the number of votes each candidate in the first precinct is treated as getting against the write-in candidate are the number of ballots that ranked them in the first precinct. <ref>{{Cite web|url=https://electowiki.org/wiki/Talk:Condorcet_method|title=Condorcet method|date=2020-05-14|website=Electowiki|language=en|access-date=2020-05-14}}</ref><ref>{{Cite web|url=https://www.reddit.com/r/EndFPTP/comments/fsa4np/possible_solution_to_the_condorcet_writein_problem/fm7bgpd|title=r/EndFPTP - Comment by u/ASetOfCondors on ”Possible solution to the Condorcet write-in problem”|website=reddit|language=en-US|access-date=2020-05-14}}</ref> ▼
# If the voter is treated as having no preference between ranked and unranked candidates, then there are no issues to consider with counting write-in candidates under the regular approach.
Below are some ways of dealing with write-ins if unranked candidates are treated in the first way described above.
The below-discussed negative counting approach automatically handles write-ins, and requires less markings than the above-mentioned approach when equal-rankings are counted as a vote for both candidates in a matchup. ▼
==== Non-comprehensive approaches ====
==Notes==▼
* Write-in candidates can be banned. This is the usual approach.
** Write-in candidates can be allowed to run, but with the caveat that only the pairwise preferences of ballots that rank them contribute votes in pairwise matchups featuring them.
*** A slight modification is to comprehensively count only those write-in candidates who are ranked on a significant number of ballots i.e. two rounds of counting may be necessary in each precinct sometimes, one to determine how many ballots write-ins are ranked on, and a second for the major write-ins.
==== Comprehensive approaches ====
These approaches collect all of the pairwise information for write-in candidates i.e. there would be no change in vote totals if the write-in candidate suddenly became one of the on-ballot candidates.
* In each [[precinct]], count the number of ballots that explicitly rank each (non-write-in) candidate. When a write-in candidate is found on a ballot, then before that ballot is counted, give each non-write-in candidate a number of votes against the write-in equal to the number of ballots where that non-write-in candidate was explicitly ranked. Then count the ballot and treat the write-in candidate as a non-write-in candidate from that point onwards (from the perspective of this algorithm).
**When creating a precinct subtotal, also record, for each candidate, how many ballots that candidate was explicitly ranked on.
▲
▲* The
==Count complexity==
==== Sequentially examining each rank on a voter's ballot ====
[[File:Pairwise counting with ranked ballot GIF.gif|thumb|576x576px|A GIF for pairwise counting with a [[ranked ballot]], which shows how to sequentially count it one rank at a time. Click on the image and then the thumbnail of the image to see the animation.]]The naive way of counting pairwise preferences implies determining, for each pair of candidates, and for each voter, if that voter prefers the first candidate of the pair to the second or vice versa. This requires looking at ballots <math>O(Vc^2)</math> times.
If reading a ballot takes a lot of time, it's possible to reduce the number of times a ballot has to be consulted by noting that:
* if a voter ranks X first, he prefers X to everybody else
* if he ranks Y second, he prefers Y to everybody but X
and so on. In other words, a ballot can be more quickly counted by examining candidates in each of its ranks sequentially from the highest rank on downward. The pairwise matrix still has to be updated <math>O(Vc^2)</math> times, but a ballot only has to be consulted <math>Vc</math> times at most. If the voters only rank a few preferences, that further reduces the counting time.
A special case of this speedup is to separately record the first preferences of each ballot, as in a [[First_past_the_post]] count. A voter who ranks a candidate X uniquely first must rank X above every other candidate and no other candidate above X, so there's no need to look at Y>X preferences at all.
===== Uses for first choice information =====
(This actually collects more information than the usual pairwise approach; specifically, if no voters equally rank candidates 1st, then it is possible to determine who the [[FPTP]] winner is, and further, if it can be determined that there is only one candidate in the [[Dominant mutual third set]], then that candidate is the [[IRV]] winner.)
If using pairwise counting for a [[rated method]], one helpful trick is to put the rated information for each candidate in the cell where each candidate is compared to themselves. For example, if A has 50 points (based on a [[Score voting]] ballot), B has 35 points, and C has 20, then this can be represented as:
{| class="wikitable"
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Pairwise counting can be used to tally the results of [[Choose-one voting]], [[Approval voting]], and [[Score voting]]; in these methods, a voter is interpreted as giving a degree of support to each candidate in a matchup, which can be reflected either using margins or (in the case of Score) the voter's support for both candidates in the matchup. See [[rated pairwise preference ballot#Margins and winning votes approaches]] for an example.
==Negative vote-counting approach==▼
=== Defunct sections ===▼
See [[Negative vote-counting approach for pairwise counting]] for an alternative way to do pairwise counting. The negative counting approach can be faster than the approach outlined in this article in some cases; for example, a voter who votes A>B when there are 10 candidates requires 9+8=17 markings to be made to count their ballot under the usual approach, but only 3 in the negative counting approach.▼
These sections were at one time part of this particle, but have been shifted to the [[Pairwise preference]] article. They are kept here only to avoid breaking any links pointing to them.
See [[Pairwise preference#Election examples]]
See [[Pairwise preference#Definitions]].
See [[Pairwise preference#Condorcet]].
See [[Pairwise preference#Strength of preference]] and [[rated pairwise preference ballot]].
===Notes===
▲==Negative vote-counting approach==
Image to right shows interpretation of ranked ballot.
▲See [[Negative vote-counting approach for pairwise counting]]. The negative counting approach can be faster than the approach outlined in this article in some cases; for example, a voter who votes A>B when there are 10 candidates requires 9+8=17 markings to be made to count their ballot under the usual approach, but only 3 in the negative counting approach.
==References==
<references />
▲==Notes==
[[Category:Majority-related concepts]]▼
<references group="nb" />
[[Category:Majority–minority relations]]
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