Pairwise counting

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Pairwise counting is the process of considering a set of items, comparing one pair of items at a time, and for each pair counting the comparison results.

Most, but not all, election methods that meet the Condorcet criterion or the Condorcet loser criterion use pairwise counting.[nb 1]

Example

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:

  • Number of voters who prefer A over B
  • Number of voters who prefer B over A
  • Number of voters who have no preference for A versus B
  • Number of voters who prefer A over C
  • Number of voters who prefer C over A
  • Number of voters who have no preference for A versus C
  • Number of voters who prefer B over C
  • Number of voters who prefer C over B
  • Number of voters who have no preference for B versus C

Often these counts are arranged in a pairwise comparison matrix[1] or outranking matrix[2] table such as below.

Pairwise counts
A B C
A A > B A > C
B B > A B > C
C C > A C > B

In cases where only some pairwise counts are of interest, those pairwise counts can be displayed in a table with fewer table cells.

Notes

References

  1. Mackie, Gerry. (2003). Democracy defended. Cambridge, UK: Cambridge University Press. p. 6. ISBN 0511062648. OCLC 252507400.
  2. Nurmi, Hannu (2012). Felsenthal, Dan S.; Machover, Moshé (eds.). "On the Relevance of Theoretical Results to Voting System Choice". Electoral Systems. Springer Berlin Heidelberg: 255–274. doi:10.1007/978-3-642-20441-8_10. ISBN 978-3-642-20440-1. Retrieved 2020-01-16.