Rated pairwise preference ballot: Difference between revisions
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{{Distinguish|Cardinal pairwise}}[[File:Pairwise relations Score.png|thumb|Pairwise matchups done using a rated ballot to indicate margin-based strength of preference in each matchup.]]A rated or cardinal pairwise preference ballot allows voters to submit their [[Rated ballot|rated]] preferences (i.e. the strength of their preferences) in every [[head-to-head matchup]] ([[pairwise]] matchup) between the candidates. It is a generalization of most other ballot types, such as [[Choose-one ballot]], [[Approval ballot]], [[rated ballot]], and [[ranked ballot]], in the sense that it is possible to submit
Because this ballot type
== Example ==
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This table captures the [[Margins|margin]] in strength of preference; it is instead possible to capture the strength of preference in a way that captures both margins and "[[winning votes]]"-relevant information (i.e. the voter's rated preference for both candidates in the matchup) by, instead of writing 20% for the more-preferred candidate and 0% for the less-preferred candidate, writing, say, 80% and 60% respectively, if that's what the voter's actual preference was.
== Transitivity ==
Certain minimum requirements for [[transitivity]] are apparent simply from looking at this table; for example, since the voter expressed a 50% difference (margin) in support for their 2nd choice>3rd choice, it wouldn't have made sense for them to express less than 50% support for their 1st choice>3rd choice. Another example is that, because they expressed 20% support for 1st>2nd, they must have had at least 20% support for 1st>3rd as well. To put it succinctly, for whatever degree of margin-based support a voter indicates in a given pairwise matchup cell, they must indicate at least that much support in all cells above, to the right, or to the upper-right of this cell. Thus, one way of collecting this pairwise information in a digital interface is to ask voters to start out by filling out the pairwise comparison between "Last choice>1st choice" (which is in the very bottom-left), and then accordingly allow the voter to fill out match-ups going up and/or right while imposing the required transitivity constraints. See [[Order theory#Strength of preference]] for further notes on transitivity in this framework.
Note that it doesn't make sense to allow a voter to indicate no preference between a higher-ranked candidate and a lower-ranked candidate, because then they'd essentially be putting them at the same rank. Thus, for ballot implementation purposes, a voter need only be given the ability to express some sort of positive preference in each matchup. Further, this only need start from the second-lowest allowed positive value, rather than the lowest; for example, if the voter is allowed to give support in increments of 10 (10% support, 20%, etc.), then because it must be assumed the voter gives at least the lowest positive value in a matchup (10%), only 20% and higher increments need to be offered as writable options for the voter.
There are two main ways to think of transitivity for rated pairwise, which are both based on the idea that when looking at the strength of the voter's preference for one candidate over another, the lower bound on the strength of this preference must be based on the strength of the matchups that come in between (i.e. if looking at the voter's ranked preference of A>B>C>D, the strength of, say, B>D, should depend on the strength of B>C and C>D). When a voter indicates they have a 30% preference for A>B, and 40% preference for B>C:
* A>C must be at least 40% (the highest of A>B and B>C)
* A>C must be at least 70% (A>B+B>C)
The second type of transitivity is based on [[Score voting]] and the idea that a voter's preferences should fit in a scale. Note however that with rated pairwise, a cap must be artificially imposed such that a voter's preference can't exceed 100%; this cap is not needed in Score, because in order for the voter to indicate a 100% marginal preference in any pairwise matchup, they must put their preferred candidate at the max score, and their less-preferred candidate at the min score; this inherently prevents them from further increasing their marginal preference by shifting either candidate up or down in terms of score.
Both of these transitivity requirements are automatically fulfilled in standard Condorcet using ranked ballots, because if the voter indicates any preference for A>B and B>C, then this will count as 1 vote (100% support) for A>B and B>C each, and because ranked transitivity ensures that this voter must indicate an A>C preference, that will also be counted as a 100% strong preference.
Also see <ref>https://forum.electionscience.org/t/how-should-transitivity-be-handled-with-rated-pairwise-preferences/693</ref>.
== Vote-counting ==
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See [[Pairwise counting#Cardinal methods]] and [[Order theory#Strength of preference]] for more information on this ballot type.
== Ballot design ==
As seen in the image at the top of the article, it is possible to allow a voter to show their rated preference between every pair of candidates. It is possible to allow the voter to indicate their score for both candidates in the matchup by filling out two scores (if they have a preference), or only one score (if they have no preference, in which case they have to select both candidates in the matchup). So for example, if the voter wished to score both A and B a 3 out of 5 in the A vs B matchup, they'd have to mark that they prefer both A and B, and then bubble in 3 out of 5. If they wanted to indicate A:5 B:3, they'd have to select A, and then bubble in both 5 and 3 as their scores in the matchup.
However, this can be difficult to fill out, and it also can make it possible for the voter to indicate an intransitive (cyclical) preference i.e. they vote that they prefer A>B, B>C, and C>A, which creates an A>B>C>A cycle. Even if the voter votes in a manner that is consistent with a ranking, it is possible they might indicate a preference that doesn't satisfy "rated pairwise" transitivity (see the [[#Transitivity]] section). For these reasons, here is a ballot type where it is impossible for the voter to indicate any inconsistent preferences:
{| class="wikitable"
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!Scores for candidates
!0 (points/stars)
!1
!2
!3
!4
!5
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|A
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|B
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|C
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|D
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|1st (choice) vs 2nd (choice)
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|2nd vs 3rd
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|3rd vs 4th
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|4th vs last
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* So here, the voter indicates their score for each candidate individually, and this is used to generate their ranked preference (it is possible to make that portion of the ballot a ranked ballot rather than a rated ballot). Then, their scores in the matchups between candidates at each rank is used to generate their rated pairwise preferences i.e. if they ranked D>B>C>A and said they have a "1st choice:5 and 2nd choice:4" preference (i.e. they'd score their 1st choice candidate(s) a 5 and 2nd choices a 4 in a matchup between one candidate from each rank), then this would be considered as them giving D a 5 and B a 4 in the D vs B matchup. To calculate the voter's preference for D>C (which is 1st choice vs 3rd choice in the example), for example, there are various ways to do so, but the one most reminiscent of Score voting is to add up the margin expressed in the 1st vs 2nd matchup, and the 2nd vs 3rd matchup. So if the voter expressed 2nd:5 3rd:2, that is a margin of 3 points in favor of 2nd, so adding that to the (5-4)=1 point margin in favor of 1st choice in the 1st vs 2nd matchup, that is a 4 point margin in favor of 1st in the 1st vs 3rd matchup. This is based on the second type of transitivity described in the [[#Transitivity]] section.
* In order to determine the actual scores in the 1st vs 3rd matchup, there are two main ways:
** either only the margin itself could be used (so 1st:4 3rd:0)
** or the score for the higher-preferred candidate and lower-preferred candidate in the transitive matchups can be added up separately, and then both are moved downwards if necessary until the score for the higher-preferred candidate is at or below the max score (with the less-preferred candidate having to get at least the min score). So here, that would mean adding '''1st:5''' ''2nd:4'' and '''2nd:5''' ''3rd:2'' to get a score for the higher-preferred candidate (the 1st choice, in the 1st vs 3rd matchup) of 5+5='''10''' and a score for the less-preferred candidate of 4+2=''6''. Because the voter can't be allowed to give a candidate more than the max score in any matchup, the 10 points for the more-preferred candidate has to be subtracted by 5 points to yield 5 points, the max score. Subtracting the same 5 points from the score for the less-preferred candidate yields 6-5=1 point. So the final result is that the voter would be treated as scoring 1st:5 3rd:1 in the 1st vs 3rd matchup.
If the voter only partially filled out their pairwise preferences, but filled out their scored preferences, then the scored preferences could be used in various ways to "auto-complete" (infer) the rated pairwise preferences.
==Implementations ==
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===Rated or ranked preference===
One particular, easier approach to implementing this generalized ballot type is to allow the voters to score the candidates on a scale, and also allow them to check a box indicating whether they have rated or ranked preferences. If using [[pairwise counting]], this can be counted by, for voters who indicate rated preferences, collecting their scores directly, and for those with ranked preferences, doing regular pairwise counting.
The ballot might look like this:
{| class="wikitable"
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!
!0
!1
!2
!3
!4
!5
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|A
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|X
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|B
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|X
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|C
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|X
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|Maximize preference?
|(Checkbox)
|Yes
|(Checkbox)
|No
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|}
Here is how this vote would be counted. If the voter checked the box for "Yes" (i.e. they indicate they do want to maximize their preference), then the value not in parentheses would be counted, but if they checked the box for "No", then the value in parentheses is counted. Note that it is possible to convert between points and votes by rescaling the points to a scale of 0 to 1 (i.e. a 4 out of 5 becomes a 0.8 out of 1) and vice versa with votes:
{| class="wikitable"
|+
!
!A
!B
!C
|-
|A
| ---
|1 vote (or 2 points)
|0 votes (or 2 points)
|-
|B
|0 votes (or 1 point)
| ---
|0 votes (or 1 point)
|-
|C
|1 vote (or 5 points)
|1 vote (or 5 points)
| ---
|}
Because there is rated information collected here, it is possible, even if the voter indicates a desire to maximize their pairwise power, to observe whether the voter did [[normalization]] or not by checking if they put their highest-scored candidate at the max score and their lowest-scored candidate at the min score. If they did not, then the rated margin between these two candidates, divided by the [max score - min score], could be used as the voter's pairwise power in every matchup i.e. if the voter's highest-scored and lowest-scored candidates are scored a 4 out of 5 and 2 out of 5 respectively, then that is a vote that is only 2/5ths as powerful as it could be (via normalization), so it could be justified to allow the voter to only cast up to 2/5ths of a vote in each pairwise matchup where they have a preference.
For example, suppose the following information is collected:
{| class="wikitable"
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=== Condorcet criterion ===
The [[Condorcet criterion]] is defined based on electing a candidate who would win a pairwise matchup against every other candidate. This is generally done based on [[majority rule]]. If a rated pairwise ballot is used, then it can be thought of as allowing each matchup to be done on the basis of [[utilitarianism]] instead, though the utility expressed in each matchup is less connected than it is in Score voting (see [[#Transitivity]]).
==== Rated pairwise Condorcet winner ====
In the same way that rated ballots offer an intuitive justification for the Score voting winner, and ranked ballots (to some extent) likewise for Condorcet winners, the rated pairwise ballot intuitively justifies a third type of result: electing a candidate (the "RPCW", who is potentially one among a group of candidates, if there is a [[Condorcet cycle]], since there will be a multi-member [[Smith set]] then) who can pairwise beat every other candidate when voters are allowed to submit fractional votes in Condorcet matchups.
==References==
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