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. In the context of voting theory, it involves comparing pairs of candidates or winner sets (usually using majority rule) to determine the winner and loser of the [[Pairwise matchup|pairwise matchup]].
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]].
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The following terms are often used when discussing pairwise counting:
'''Pairwise matchup''': Also known as a head-to-head matchup, it is when voters are asked to indicate their preference between two candidates or winner sets, with the one that voters prefer winning. It is usually done on the basis of majority rule (i.e. if more voters prefer one candidate over the other than the number of voters who have the opposing preference, then the candidate preferred by more voters wins the matchup) using [[Choose-one voting|choose-one voting]], though see the [[Pairwise counting#Cardinal methods|Cardinal methods]] section for alternative ways. Pairwise matchups can be simulated from ranked or rated ballots and then assembled into a table to show all of the matchups simultaneously; see above.
'''Pairwise win/beat''' and '''pairwise lose/defeated''': When one candidate receives more votes in a pairwise matchup/comparison against another candidate, the former candidate "pairwise beats" the latter candidate, and the latter candidate "pairwise loses." Often this is represented by writing "Pairwise winner>Pairwise loser"; this can be extended to show a [[beatpath]] by showing, for example, "A>B>C>D", which means A pairwise beats B, B pairwise beats C, and C pairwise beats D (though it may or may not be the case, depending on the context, that, for example, A pairwise beats C).
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==Notes==
[[File:Pairwise counting procedure.png|thumb|The procedure for pairwise counting with various ballot formats and examples.]]
One of the notable aspects of pairwise counting is that it can be used to find a Condorcet winner or member of the Smith set in a simple manner without needing to be done with written ballots; see [[:Category:Sequential comparison Condorcet methods]] for more information.{{reflist|group=nb}}
== References ==
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