# Difference between revisions of "Pairwise counting"

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(Clarify that not all methods that pass CW or CL use pairwise matrices.) |
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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. |
'''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. |
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− | Most election methods that meet the [[Condorcet criterion]] or the [[Condorcet loser criterion]] use pairwise counting |
+ | Most, but not all, election methods that meet the [[Condorcet criterion]] or the [[Condorcet loser criterion]] use pairwise counting.<ref group=nb>[[Nanson's method|Nanson]] meets the [[Condorcet criterion]] and [[Instant-runoff voting]] meets the [[Condorcet loser criterion]].</ref> |

== Example == |
== Example == |

## Revision as of 04:51, 21 January 2020

**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.

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

- ↑ Nanson meets the Condorcet criterion and Instant-runoff voting meets the Condorcet loser criterion.

## References

- ↑ Mackie, Gerry. (2003).
*Democracy defended*. Cambridge, UK: Cambridge University Press. p. 6. ISBN 0511062648. OCLC 252507400. - ↑ 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.