Pairwise counting: Difference between revisions

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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.<ref group=nb>[[Nanson's method|Nanson]] meets the [[Condorcet criterion]] and [[Instant-runoff voting]] meets the [[Condorcet loser criterion]].</ref>

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> See the [[Pairwise counting#Condorcet|Condorcet section]] for more information on the use of pairwise counting in [[Condorcet methods]].

== Example without numbers ==

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

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.

== Example with numbers ==

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'''Pairwise tie''': Occurs when two candidates receive the same number of votes in their pairwise matchup.

'''Pairwise order/ranking''': Also known as a [[Condorcet ranking]], is the ranking of candidates such that each candidate is ranked above all candidates they pairwise beat. Sometimes such a ranking does not exist due to the [[Condorcet paradox]]. As a related concept, there is always a [[Smith set ranking|Smith ranking]] that applies to groups of candidates.

'''Pairwise order/ranking''': Also known as a [[Condorcet ranking]], is the ranking of candidates such that each candidate is ranked above all candidates they pairwise beat. Sometimes such a ranking does not exist due to the [[Condorcet paradox]]. As a related concept, there is always a [[Smith set ranking|Smith ranking]] that applies to groups of candidates, and which reduces to the Condorcet ranking when one exists.

== Condorcet ==

In a pairwise comparison matrix/table, often the color green is used to shade cells where more voters prefer the former candidate over the latter candidate than the other way around, the color red is used to shade cells where more voters prefer the latter candidate over the former candidate than the other way around, and some other color (often gray, yellow, or uncolored) is used to shade cells where as many voters prefer one candidate over the other as the other way around (pairwise ties).

In the context of [[Condorcet methods]]:

- A [[Condorcet winner]] is a candidate for whom all their cells are shaded green.

- The [[Smith set]] is the smallest group of candidates such that all of their cells are shaded green except some of the cells comparing each of the candidates in the group to each other.

- The [[Schwartz set]] is the same as the Smith set except some of their cells may be shaded the color for pairwise ties.

- A [[Condorcet loser criterion|Condorcet loser]] is a candidate for whom all their cells are shaded red.

- The '''weak Condorcet winners''' and '''weak Condorcet losers''' are candidates for whom all of their cells are shaded either green (for the weak Condorcet winners) or red (for the weak Condorcet losers) or the color for pairwise ties.

==Notes==