Editing Condorcet method (section)

==Basic procedure==
===Voting===

In a Condorcet election the voter ranks the list of candidates in order of preference. If a ranked ballot is used, the voter gives a "1" to their first preference, a "2" to their second preference, and so on. Some Condorcet methods allow voters to rank more than one candidate equally so that the voter might express two first preferences rather than just one.<ref>{{Cite web|title=Condorcet|url=https://www.equal.vote/condorcet|access-date=2021-04-25|website=Equal Vote Coalition}}</ref> If a scored ballot is used, voters rate or score the candidates on a scale, for example as is used in [[Score voting]], with a higher rating indicating a greater preference.<ref>{{Cite journal |last1=Igersheim |first1=Herrade |last2=Durand |first2=François |last3=Hamlin |first3=Aaron |last4=Laslier |first4=Jean-François |date=January 2022 |title=Comparing voting methods: 2016 US presidential election |url=https://halshs.archives-ouvertes.fr/halshs-01972097/document  |journal=European Journal of Political Economy |language=en |volume=71 |pages=102057 |doi=10.1016/j.ejpoleco.2021.102057}}</ref> When a voter does not give a full list of preferences, it is typically assumed that they prefer the candidates that they have ranked over all the candidates that were not ranked, and that there is no preference between candidates that were left unranked. Some Condorcet elections permit [[write-in candidate]]s.

===Finding the winner===

The count is conducted by pitting every candidate against every other candidate in a series of hypothetical one-on-one contests. The winner of each pairing is the candidate preferred by a majority of voters. Unless they tie, there is always a majority when there are only two choices. The candidate preferred by each voter is taken to be the one in the pair that the voter ranks (or rates) higher on their ballot paper. For example, if Alice is paired against Bob it is necessary to count both the number of voters who have ranked Alice higher than Bob, and the number who have ranked Bob higher than Alice. If Alice is preferred by more voters then she is the winner of that pairing. When all possible pairings of candidates have been considered, if one candidate beats every other candidate in these contests then they are declared the Condorcet winner. As noted above, if there is no Condorcet winner a further method must be used to find the winner of the election, and this mechanism varies from one Condorcet consistent method to another.<ref name=":2" /> In any Condorcet method that passes [[Independence of Smith-dominated alternatives]], it can sometimes help to identify the [[Smith set]] from the head-to-head matchups, and eliminate all candidates not in the set before doing the procedure for that Condorcet method.

===Pairwise counting and matrices===

Condorcet methods use pairwise counting. For each possible pair of candidates, one pairwise count indicates how many voters prefer one of the paired candidates over the other candidate, and another pairwise count indicates how many voters have the opposite preference. The counts for all possible pairs of candidates summarize all the pairwise preferences of all the voters.

Pairwise counts are often displayed in a ''pairwise comparison matrix'',<ref name=":0">{{Cite book|url=https://books.google.com/books?id=q2U8jd2AJkEC&pg=PA6|title=Democracy defended|last=Mackie, Gerry.|date=2003|publisher=Cambridge University Press|isbn=0511062648|location=Cambridge, UK|pages=6|oclc=252507400}}</ref> or ''outranking matrix'',<ref>{{Citation|last=Nurmi|first=Hannu|s2cid=12562825|chapter=On the Relevance of Theoretical Results to Voting System Choice|date=2012|pages=255–274|editor-last=Felsenthal|editor-first=Dan S.|publisher=Springer Berlin Heidelberg|doi=10.1007/978-3-642-20441-8_10|isbn=9783642204401|editor2-last=Machover|editor2-first=Moshé|title=Electoral Systems|series=Studies in Choice and Welfare}}</ref> such as those below. In these [[Matrix (mathematics)|matrices]], each row represents each candidate as a 'runner', while each column represents each candidate as an 'opponent'. The cells at the intersection of rows and columns each show the result of a particular pairwise comparison. Cells comparing a candidate to themselves are left blank.<ref name=":1">{{Cite journal |last=Young |first=H. P. |date=1988 |title=Condorcet's Theory of Voting |url=https://www.cs.cmu.edu/~arielpro/15896s15/docs/paper4a.pdf |url-status=live |journal=American Political Science Review |language=en |volume=82 |issue=4 |pages=1231–1244 |doi=10.2307/1961757 |issn=0003-0554 |jstor=1961757 |s2cid=14908863 |archive-url=https://web.archive.org/web/20181222192924/http://www.cs.cmu.edu/~arielpro/15896s15/docs/paper4a.pdf |archive-date=2018-12-22}}</ref><ref>{{Cite journal|last=Hogben|first=G.|date=1913|title=Preferential Voting in Single-member Constituencies, with Special Reference to the Counting of Votes|url=http://rsnz.natlib.govt.nz/volume/rsnz_46/rsnz_46_00_005780.html|journal=Transactions and Proceedings of the Royal Society of New Zealand|volume=46|pages=304–308}}</ref>

Imagine there is an election between four candidates: A, B, C, and D. The first matrix below records the preferences expressed on a single ballot paper, in which the voter's preferences are (B, C, A, D); that is, the voter ranked B first, C second, A third, and D fourth. In the matrix a '1' indicates that the runner is preferred over the 'opponent', while a '0' indicates that the runner is defeated.<ref name=":1" /><ref name=":0" />

{| class="wikitable" style="width:13em;margin:auto;text-align:center"
! {{diagonal split header|Runner|&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Opponent}} !! A !! B !! C !! D
|-
! A
|| — || 0 || 0 || 1
|-
! B
|| 1 || — || 1 || 1
|-
! C
|| 1 || 0 || — || 1
|-
! D
|| 0 || 0 || 0 || —
|-
| colspan=6 style="line-height: 10px;" | <small>A '1' indicates that the runner is preferred over the opponent; a '0' indicates that the runner is defeated.</small>
|}

Using a matrix like the one above, one can find the overall results of an election. Each ballot can be transformed into this style of matrix, and then added to all other ballot matrices using [[matrix addition]]. The sum of all ballots in an election is called the sum matrix. Suppose that in the imaginary election there are two other voters. Their preferences are (D, A, C, B) and (A, C, B, D). Added to the first voter, these ballots would give the following sum matrix:
{| class="wikitable" style="width:13em;margin:auto;text-align:center"
! {{diagonal split header|Runner|&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Opponent}} !! A !! B !! C !! D
|-
! A
|| —
|| 2 || 2 || 2
|-
! B
|| 1 || — || 1 || 2
|-
! C
|| 1 || 2 || — || 2
|-
! D
|| 1 || 1 || 1 || —
|}

When the sum matrix is found, the contest between each pair of candidates is considered. The number of votes for runner over opponent (runner, opponent) is compared with the number of votes for opponent over runner (opponent, runner) to find the Condorcet winner. In the sum matrix above, A is the Condorcet winner because A beats every other candidate. When there is no Condorcet winner Condorcet completion methods, such as Ranked Pairs and the Schulze method, use the information contained in the sum matrix to choose a winner.

Cells marked '—' in the matrices above have a numerical value of '0', but a dash is used since candidates are never preferred to themselves. The first matrix, that represents a single ballot, is inversely symmetric: (runner, opponent) is ¬(opponent, runner). Or (runner, opponent) + (opponent,  runner) = 1. The sum matrix has this property: (runner, opponent) + (opponent, runner) = N for N voters, if all runners were fully ranked by each voter.