Pairwise preference: Difference between revisions

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 tie''': Occurs when two candidates receive the same number of votes in their pairwise matchup. (Note that sometimes it is also called a tie when there is pairwise cycling, though this is different; see the definition two spots below.) Note that some cycles can be symmetrical ties i.e. you can swap the candidates' names without changing the result. (See the [[Condorcet paradox]] article for an example, and the [[neutrality criterion]] and [[tie]] for more information).

'''Pairwise cycling:''' Also known as a [[Condorcet cycle]], it is when within a set of candidates, each candidate has at least one pairwise defeat (when looking only at the matchups between the candidates in the set). Note that some cycles can be symmetrical i.e. you can swap the candidates' names without changing the result. (See the [[Condorcet paradox]] article for an example, and the [[neutrality criterion]] for more information). Such cycles are sometimes called ties.

Note that pairwise counting can be done either by looking at the margins expressed on a voter's ballot, or the "winning votes"-relevant information (see [[Defeat strength]]). For example, a voter who scores one candidate a 5 and the other a 3 on a rated ballot can either be thought of as giving those scores to both candidates in the matchup (winning votes-relevant information) or as giving 2 points to the first candidate and 0 to the second (only the margins). For ranked and choose-one ballots, both margins and winning votes approaches yield the same numbers, since a voter can only give maximal support to at most one candidate in the matchup.

The Smith set is then always full of candidates who are at least weak Condorcet winners i.e. tied for having the most points/approvals. Note that this is not the case if voters are allowed to have preferences that wouldn't be writable on a cardinal ballot i.e. if the max score is 5, and a voter indicates their 1st choice is 5 points better than their 2nd choice, and that their 2nd choice is 5 points better than their 3rd choice, then this would not be an allowed preference in cardinal methods, and thus it would be possible for a Condorcet cycle to occur. Also, if a voter indicates their 1st choice is 2 points better than their 2nd choice, that this likely automatically implies their 1st choice must be at least 2 points better than their 3rd choice, etc. So there seems to be a [[transitivity]] of strength of preference, just as there is a transitivity of preference for rankings.<ref>{{Cite web|url=https://www.reddit.com/r/EndFPTP/comments/fcexg4/score_but_for_every_pairwise_matchup/|title=r/EndFPTP - Score but for every pairwise matchup|website=reddit|language=en-US|access-date=2020-04-24}}</ref>

The [[rated pairwise preference ballot]] collectsallows the voter to express the most nuanced pairwise information of all [[ballot typetypes]].

Pairwise preferences can be used to understand [[Weighted positional method]]<nowiki/>s and their generalizations (such as [[Choose-one voting]], [[Approval voting]], and [[Score voting]]), and [[:Category:Pairwise counting-based voting methods|Category:Pairwise counting-based voting methods]]. In the first 3 methods, a voter is interpreted as giving a degree of support to each candidate in a matchup. Even [[IRV]] can be understood in this way to some extent when observing its compliance with the [[dominant mutual third]] property.

It may help to putinterpret pairwise data by putting the % of the votes a candidate got in the pairwise matchup. So, for example:

Multi-winner methods that use pairwise counting, such as [[CPO-STV]] and [[Schulze STV]], instead of doing pairwise matchups between individual candidates, do pairwise matchups between sets of candidates (called [[winner setsset]]<nowiki/>s).