Pairwise counting: Difference between revisions

Note that more generally, for example, the pairwise count "Number of voters who prefer B over C" can be thought of as "the number of votes helping B beat or tie C in the B vs. C [[Pairwise matchup|pairwise matchup]]". Also, in order to know the number of voters who have no preference between two candidates, the only values that need be known are the number of voters who prefer the first over the second, the number of voters that prefer the second over the first, and the number of total voters in the election. This is done by subtracting the first two categories (which together are the number of voters who have any preference between the two candidates) from the number of total voters.

In general, for N candidates, there are 0.5*N*(N-1) pairwise matchups to consider. So for 1 candidate, there are 0 matchups, 2 candidates, 1 matchup, 3 candidates, 3 matchups, 4 candidates, 6 matchups, 5 candidates, 10 matchups, 6 candidates, 15 matchups, 7 candidates, 21 matchups, etc.

Often these counts are arranged in a ''pairwise comparison matrix''<ref name=":0">{{Cite book|url=https://books.google.com/?id=q2U8jd2AJkEC&lpg=PA6&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>{{Cite journal|title=On the Relevance of Theoretical Results to Voting System Choice|url=http://link.springer.com/10.1007/978-3-642-20441-8_10|publisher=Springer Berlin Heidelberg|work=Electoral Systems|date=2012|access-date=2020-01-16|isbn=978-3-642-20440-1|pages=255–274|doi=10.1007/978-3-642-20441-8_10|first=Hannu|last=Nurmi|editor-first=Dan S.|editor-last=Felsenthal|editor2-first=Moshé|editor2-last=Machover}}</ref>'' table such as below.

Note that since a candidate can't be pairwise compared to themselves (i.e. candidate B can't be compared to candidate B, since there's only one candidate in the comparison), the cell that does so is always empty.

When it is only desired to know which candidate wins the pairwise matchup, say, between candidates and B, one can subtract the value of B>A from A>B. If the resulting value is positive, it means A won the matchup, and if it is 0, it means there is a pairwise tie; otherwise it means B won the matchup. (See the [[Pairwise counting#Terminology|Terminology]] section).