Pairwise counting: Difference between revisions

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(Cleanup (remove duplicate references). Also clarify write-in handling.)
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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, the number of votesgive each non-write-in candidate getsa number of votes against the write-in candidateequal isto the number of ballots theywhere werethat sonon-write-in farcandidate rankedwas onexplicitly ranked. TheThen ballotcount isthe then counted,ballot and treat the write-in candidate is treated 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.
**When combining 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 treat all the numberballots offrom votesthe eachfirst precinct to rank every explicitly ranked candidate inabove the firstwrite-in: precinctfor iseach treatedcandidate asin gettingthe first precinct, against the write-in candidatein the second, areadd the number of ballots that ranked themvoters in the first precinct who explicitly ranked that non-write-in candidate. <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 [[Pairwise counting#Negative vote-counting approach|negative for pairwise vote-counting approach]] 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. However, it requires a post-processing stage to convert the Condorcet matrix into the more familiar form before usage by Condorcet methods.
 
==Count complexity==
==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).]]
 
=== Quicker ways to do pairwise counting ===
Also see the [[Negative vote-counting approach for pairwise counting]].
 
==== Sequentially examining each rank on a voter's ballot ====
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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.
==== Counting first choices separately ====
For every voter who ranks a candidate as their only 1st choice, because it is known they prefer this candidate over all others, it is possible to speed up the usual pairwise counting procedure by modifying it in the following way:
 
* Information collected by vote-counters:
*# Count the '''['''number of voters who rank a candidate as their only 1st choice''']'''
*# For those voters' ballots, don't count any preferences in head-to-head matchups involving their unique 1st choice candidate
* Math: Each candidate has as many votes added in favor of them as the '''['''number of ballots that ranked them uniquely 1st''']'''
 
This can be further generalized to handle ballots that equally rank multiple candidates 1st, but with a caveat; see [[Negative vote-counting approach for pairwise counting#Dealing with equal-ranking]].
 
===== 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.)
 
=== Techniques for when one is collecting both rated and pairwise information ===
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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==References==
<references />
 
▲==Notes==
<references group="nb" />▼
 
[[Category:Majority-related concepts]]
[[Category:Condorcet-related concepts]]
 
▲<references group="nb" />
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