Pairwise preference: Difference between revisions
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[[File:Pairwise counting table with links between matchups.png|thumb|444x444px|Green arrows point from the loser of the matchup to the winner (based on who got more votes). Yellow arrows indicate a tie. Red arrows (not shown here) indicate the opposite of green arrows (i.e. who lost the matchup).For example, the B>A matchup points to A>B with a green arrow because A pairwise beats B (wins head-to-head).]]Pairwise preferences are voters' preferences between pairs of candidates (their preferences in head-to-head matchups between candidates). [[Pairwise counting]] is used to extract this information from voters' [[ballot]]<nowiki/>s, and it is then used in all [[:Category:Pairwise counting-based voting methods|Category:Pairwise counting-based voting methods]], which are mostly just the [[Condorcet methods]], to help determine the winner of the election.
== Pairwise matchups ==
A pairwise matchup is when voters choose between two candidates, with there being a winner and loser, or a tie (the possibility of which will only intermittently be discussed throughout this article). The idea is that when there are only two options to choose from, it's always possible to get a [[majority]] in favor of one of them, because any votes that don't go to one must have gone to the other.
The most direct way to conduct a pairwise comparison is to ask voters "Who do you prefer between these two candidates" for every pair of candidates. However, this would be rather onerous when there are more candidates running, and could even result in violations of [[transitivity]]: a voter could say they prefer A>B (A over B in the A vs B matchup), B>C, and C>A, which means that if these were the only 3 candidates in the election, and the voter had total power to decide which of them won, then they'd be unable to make up their mind, since for whichever one they choose, they'd want to pick someone else (in fact, when voters express these cyclical preferences on their ballots, the common approaches to making their preferences "rational"/acyclical are to either ignore the last or lowest part of the cycle, such that A>B>C>A becomes A>B>C, or to treat all candidates as equally preferred, i.e. A=B=C, though the noncyclical preferences the voter expressed in regard to these candidates versus other candidates are still respected).
Thus, in the context of voting, it's more common to ask voters to indicate their preference using a [[Ballot types|ballot type]] that automatically imposes transitivity, usually with a [[ranked ballot]] (though theoretically any ballot type that allows a voter to indicate at least one pairwise preference works for this purpose). The use of a transitive ballot type has the further advantage that, because it is usually assumed a voter prefers every candidate they mark a preference for over every unmarked candidate, voters don't have to explicitly mark all of their preferences.
== Definitions ==
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== Notes ==
The [[rated pairwise preference ballot]] allows the voter to express the most nuanced pairwise information of all [[ballot types]].
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!B
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| A
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|'''56%'''
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| B
|'''44%'''
| ---
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