Pairwise preference: Difference between revisions
→Condorcet
(→Notes) |
|||
Line 29:
== Condorcet ==
In a pairwise comparison matrix/table, often the color green is used to shade cells with pairwise victories (where more voters prefer the former candidate over the latter candidate than the other way around), the color red is used to shade cells with pairwise defeats (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 with pairwise ties (where as many voters prefer one candidate over the other as the other way around
Pairwise comparison tables are often ordered to create a [[Smith set ranking|Smith ranking]] of candidates, such that whenever possible, the candidates
▲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).
▲Pairwise comparison tables are often ordered to create a [[Smith set ranking|Smith ranking]] of candidates, such that the candidates at the top [[Pairwise counting#Terminology|pairwise beat]] all candidates further down the table, all candidates directly below the candidates at the top pairwise beat all candidates further down the table, etc.
In the context of [[Condorcet methods]]:
Line 42 ⟶ 41:
* 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.
<br />
== Strength of preference ==
Cardinal methods can be counted using pairwise counting by comparing the difference in scores (strength of preference) between the candidates, rather than only the number of voters who prefer one candidate over the other. See the [[rated pairwise preference ballot]] article for a way to do this on a per-matchup basis.
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 support to at most one candidate in the matchup.
Line 49:
Essentially, instead of doing a pairwise matchup on the basis that a voter must give one vote to either candidate in the matchup or none whatsoever, a voter could be allowed to give something in between (a partial vote) or even one vote to both candidates in the matchup (which has the same effect on deciding which of them wins the matchup as giving neither of them a vote, as it does not help one of them get more votes than the other).
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>
== Notes ==
Line 56:
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.
Pairwise preferences require (N^2 - N) pieces of information for N candidates. This is because each candidate can get a different number of votes in favor of then in each of their matchups against other candidate, resulting in 0.5*(N^2 - N) matchups. See also [[Precinct summability]].
The interpretation of pairwise ties can conceptually link different concepts together sometimes. For example, the [[Smith set]] and [[Schwartz set]] are identical except that, essentially, the Smith set treats a tie as counting against both tied candidates (i.e. it's as bad as a defeat), while the Schwartz set treats a tie as having no relevance to the quality of either of the tied candidates. ▼
▲The interpretation of pairwise ties can conceptually link different concepts together sometimes. For example, the [[Smith set]] and [[Schwartz set]] are identical except that
It may help to interpret pairwise data by putting the % of the votes a candidate got in the pairwise matchup. So, for example:
| |||