Pairwise preference: Difference between revisions
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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. A major argument in favor of analyzing pairwise preferences is that it minimizes the ability of [[strategic nomination]] to affect the race, that is, [[Independence of irrelevant alternatives]] is maximally satisfied (though not completely, if using [[majority rule]]) by ensuring candidates who enter or drop out of the race play less of a role in deciding which of the remaining candidates wins.
=== Ways of collecting pairwise information ===
==== Manually doing each pairwise matchup ====
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).
==== Collecting pairwise information from ranked ballots ====
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 [[rated ballot]]<nowiki/>s are also sometimes used; 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.
=== Identifying the winner of the matchup ===
To identify which candidate wins a specific pairwise matchup, such as between candidates A and B, subtract the value of B>A (the number of voters who prefer B to A) from A>B. If the resulting value is positive, then candidate A won the matchup. If it is zero, then there is a pairwise tie. If the result is negative, then candidate B won the matchup. See [[margins]].
== Definitions ==
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'''Pairwise order/ranking''': Also known as a [[Condorcet ranking]], it is a 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.
==Election examples==▼
== Presentation ==
This section covers how to demonstrate pairwise preferences.
Sometimes only the "dominance relation" (wins, losses, and ties) is shown, rather than the exact numbers. So for example, if A beat B in their pairwise matchup, it'd be possible to write "Win" (or a green checkmark) in the A>B cell and "Loss" (or a red X) in the B>A cell.
=== Percentages ===
It may help to interpret pairwise data by putting the % of the votes a candidate got in the pairwise matchup. So, for example:▼
{| class="wikitable"▼
|+▼
!▼
!A▼
!B▼
|-▼
| A▼
| ---▼
|'''56%'''▼
|-▼
| B▼
|'''44%'''▼
| ---▼
|}▼
▲===Election examples===
Here is an example of a pairwise victory table for the [https://en.wikipedia.org/wiki/2009_Burlington_mayoral_election Burlington 2009] election:
{| class="wikitable"
Line 129 ⟶ 158:
== 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.▼
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).
=== Margins and winning votes approaches ===
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>▼
▲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 [[
=== Transitivity requirements ===
▲
== Criticism ==
One major criticism of pairwise preferences is that they are harder to understand and think about because a candidate's quality can't be completely summed up into one number, like in [[cardinal method]]<nowiki/>s. ▼
Another criticism is that it can be harder to do [[pairwise counting]] than it is to count the vote in other methods, such as [[Approval voting]]. The [[Rated pairwise preference ballot#Rated or ranked preference]] implementation can potentially mitigate this criticism, because for every voter who indicates a rated preference, at most only one piece of information need be collected from their ballot for every candidate they marked (their score for the candidate), rather than several pairwise preferences. ▼
The nature of pairwise preferences prevents direct comparisons of candidates from two separate elections, unlike with [[rated method]]<nowiki/>s or other methods. For example, it is possible to compare Reagan's [[approval rating]] in polls from the 1980s to Obama's in the 2010s without having to ask voters about both in the same election/poll, but their pairwise matchup against each other can't be evaluated like that. ▼
== Notes ==
The [[rated pairwise preference ballot]] allows the voter to express the most nuanced pairwise information of all [[ballot types]].
The interpretation of pairwise ties can conceptually link different concepts together sometimes. For example, the [[Smith set]] and [[Schwartz set]] are identical except that one treats a tie as counting against both tied candidates (i.e. it's as bad as a defeat) in terms of their deservingness to be in the set or not, while the other treats a tie as having no relevance to the quality of either of the tied candidates. ▼
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.▼
=== Casual usage ===
One of the notable aspects of pairwise preferences is that a Condorcet winner or member of the Smith set can be found in a simple manner without needing to be done with written ballots;
=== Understanding non-pairwise methods using pairwise preferences ===
▲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 [[:
=== Required amount of information to collect ===
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]].
=== Number of allowed transitive pairwise preferences ===
▲The interpretation of pairwise ties can conceptually link different concepts together sometimes. For example, the [[Smith set]] and [[Schwartz set]] are identical except that one treats a tie as counting against both tied candidates (i.e. it's as bad as a defeat) in terms of their deservingness to be in the set or not, while the other treats a tie as having no relevance to the quality of either of the tied candidates.
Most pairwise criteria ([[Condorcet criterion]], [[Smith]], etc.) assume a voter may indicate as many transitive pairwise preferences as desired i.e. they may place each candidate in a separate rank. Some [[:
▲Most pairwise criteria ([[Condorcet criterion]], [[Smith]], etc.) assume a voter may indicate as many transitive pairwise preferences as desired i.e. they may place each candidate in a separate rank. Some [[:Category:Pairwise counting-based voting methods|Category:Pairwise counting-based voting methods]] actually violate this by limiting the number of [[slot]]<nowiki/>s voters have, such as common implementations of [[Smith//Score]]. This can be done for practical reasons (to keep the ballot smaller, potentially), or for more philosophical reasons; some object to the idea that a voter should be able to put a full vote "between" every transitive pair of candidates (because it may be unlikely for voters to honestly feel such maximally strong preferences), and so wish to limit the number of available ranks. Indeed, when a voter can only indicate two ranks (or also give candidates partial support between these two ranks), then you get [[Score voting]], because if you give 1 vote to help A beat B, then you must give 0 votes for B>C (or if you give 0.6 votes A>B, then you can't give 0.5 votes B>C). The [[Rated pairwise preference ballot]] can be implemented with fewer ranks than candidates in this manner, which then forces [[preference compression]] (or, more complexly, no, or a less strict, limitation on ranks might be imposed, but the voter might be required to indicate a weak preference between at least some of the ranks).
Because of [[preference compression]], which can happen also for [[strategic voting]] purposes i.e. [[Min-max voting]], it's not always possible to get accurate pairwise data from [[rated ballot]]<nowiki/>s. Thus, it is often useful to differentiate between a candidate who gets at least half of all voters to prefer them over their opponents in head-to-head matchups, rather than only at least half of all voters ''with preferences in the relevant matchups'' (i.e. they tie or [[Majority-beat]] their opponents), since no matter what preferences preference-compressing voters have in those matchups, the candidate in question will at least tie or win the matchup no matter what. Example:<blockquote>25 A:5 B:4
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49 C:5 D:5 </blockquote>Here, the [[CW]] based on honest pairwise preferences is B, but because of min-max voting (potentially done to ensure at least one of A or B enter the runoff rather than both C and D), it looks like A is the CW.
=== Multi-winner pairwise methods ===
▲It may help to interpret pairwise data by putting the % of the votes a candidate got in the pairwise matchup. So, for example:
▲{| class="wikitable"
▲|+
▲!
▲!A
▲!B
▲|-
▲| A
▲| ---
▲|'''56%'''
▲|-
▲| B
▲|'''44%'''
▲| ---
▲|}
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 set]]<nowiki/>s).
▲The nature of pairwise preferences prevents direct comparisons of candidates from two separate elections, unlike with [[rated method]]<nowiki/>s or other methods. For example, it is possible to compare Reagan's [[approval rating]] in polls from the 1980s to Obama's in the 2010s without having to ask voters about both in the same election/poll, but their pairwise matchup against each other can't be evaluated like that.
▲One major criticism of pairwise preferences is that they are harder to understand and think about because a candidate's quality can't be completely summed up into one number.
▲Another criticism is that it can be harder to do [[pairwise counting]] than it is to count the vote in other methods, such as [[Approval voting]]. The [[Rated pairwise preference ballot#Rated or ranked preference]] implementation can potentially mitigate this criticism, because for every voter who indicates a rated preference, at most only one piece of information need be collected from their ballot for every candidate they marked (their score for the candidate), rather than several pairwise preferences.
▲One of the notable aspects of pairwise preferences is that a Condorcet winner or member of the Smith set can be found in a simple manner without needing to be done with written ballots; see [[:Category:Sequential comparison Condorcet methods]] for more information.
=== Reading the pairwise table ===
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