Pairwise counting: Difference between revisions
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== Procedure ==
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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 ===
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.
# 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.
==== Non-comprehensive approaches ====
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* 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.
*
**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>
* The [[Negative vote-counting approach for pairwise counting]] automatically handles write-ins, and requires less markings than the above-mentioned approach when explicit equal-rankings are counted as a vote for both candidates in a matchup.
==Notes==
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