Editing Condorcet method (section)

===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.