4,193
edits
Pairwise preference: Difference between revisions
Using the table that I created on English Wikipedia and moved to a template ("Template:Burlington 2009 pairwise table") here on electowiki. It would seem that VisualEditor made a few other changes,too.
(Using the table that I created on English Wikipedia and moved to a template ("Template:Burlington 2009 pairwise table") here on electowiki. It would seem that VisualEditor made a few other changes,too.) |
|||
| (3 intermediate revisions by one other user not shown) | |||
Line 100:
|}
===Election examples===
Here is an example of a pairwise victory table for the [
{{Burlington 2009 pairwise table}}
|-▼
▲|}To read this, take for example the cell where BK is compared to AM (the cell with BK on the left and AM on the top); "4067 (AM)" means that 4067 voters preferred AM (Andy Montroll) over BK (Bob Kiss), and "3477 (BK)" means that 3477 voters preferred BK over AM. Because AM got more votes than BK in that matchup, AM won that matchup.
▲== 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).
Line 194 ⟶ 114:
In the context of [[Condorcet methods]]:
*
*
*
*
* 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.
==
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.
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).
===
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 [[Rated pairwise preference ballot#Margins and winning votes approaches]]).
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
*
*
For ranked and choose-one ballots, both margins and winning votes approaches yield the same numbers, since these approaches assume a voter gives either:
* Maximal support/margin (1 vote) to their preferred candidate in the matchup, when they have one (which they can only achieve by giving maximal support to the preferred candidate and no support to the other candidate in the winning votes approach, so as to create maximal distance).
*
===
If every voter indicates the same rated preference for each pair of candidates, then the Smith set is 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>
==
Pairwise preferences can be used to find the [[order of finish]] of any [[Condorcet method]] when there is a [[Condorcet ranking]], and can always be used to calculate a complete result in [[:Category:Defeat-dropping Condorcet methods]]. They can sometimes be used in [[:Category:Runoff-based voting methods]] to avoid having to do additional rounds of counting (i.e. because no matter which candidates enter the runoff, the result is already known).
When combined with rated information, it is possible to surmise some additional information about how voters scored the candidates. For example, if 2/3rds of the voters prefer A>B, yet B has a higher points total than A, then the A>B voters must have received less than half the [[utility]] gain of going from their less-preferred candidate in the matchup to their more-preferred candidate as the B>A voters.
<br />
== 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.
===Incomparability of separate pairwise data sets===
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.
==
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.
===
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; for each pair of candidates, the voters can be asked to raise their hands for the one they prefer, with the pairwise loser being eliminated, and this repeating until only one candidate remains. This rivals the simplicity of [[Approval voting]] for casual usage purposes, since a relatively similar amount of work is done both ways, though it can create failures of [[transitivity]]. See [[:Category:Sequential comparison Condorcet methods]] for more information.
===
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]]. 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]].
=== Number of allowed transitive pairwise preferences
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]] 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 to limit the amount of vote-counting work necessary, since generally matchups between candidates at the same rank do not need to be counted), 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).
Line 254 ⟶ 180:
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 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).
===
When looking at two candidates, a quick way to figure out the number of votes for the first candidate>second candidate and vice versa is to first locate the cell for "first candidate>second candidate", count the minimum number of cells diagonally one must go to be adjacent to the middle dividing line of the matrix (where there is a --- cell), and then going one cell further diagonally (meaning you'll be starting from the closest cell on the opposite side of that dividing line), go that number of cells further diagonally to reach the other cell. For example:
{| class="wikitable"
Line 269 ⟶ 195:
|A
| ---
|
|2
|'''2'''
Line 275 ⟶ 201:
|-
|B
|
| ---
|2
|2
|
|-
| C
|0
| ---
|
|2
|-
| D
|''<u>0</u>''
| 0
|0
| ---
| 0
|-
|
|0
|0
|0
|0
|
|}Try locating A>D (the fifth cell in the second row). To find the reverse, D>A, first you check and see that you have to go one cell down and to the left to be adjacent to the middle dividing line. Then, starting from the cell one cell down and to the left of the middle dividing line, go one cell further down and to the left to reach D>A. In doing this, you would start at A>D, go down to B>C, then jumping over the middle dividing line to C>B, go down to D>A.<references />
[[Category:Condorcet-related concepts]]
| |||