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 ==
With some simplification, this can be visualized as (example using [[pairwise counting]] for a single voter, with 6 candidates A through F):<blockquote>'''<u>Voter's ballot</u>'''
Ranking: </blockquote>
{| class="wikitable"
!1st choice(s)
!2nd choice(s)
!3rd choice(s)
|-
|A
|B, C
|D
|}
<blockquote>Rated pairwise matchups (below):</blockquote>
{| class="wikitable"
|+
!
!0% (support)
!10%
!20%
!30%
!40%
!50%
!60%
!70%
!80%
!90%
!100%
|-
|1st choice>2nd choice
|
|
|
|
|
|
|X
|
|X
|
|
|-
|2nd choice>3rd choice
|
|
|
|X
|
|
|
|
|X
|
|
|-
|3rd choice>last choice
|
|
|
|
|
|X
|
|
|
|X
|
|}
<blockquote>'''<u>(Partial) Interpretation of ballot</u>'''
{| class="wikitable"
|+
!
|1st (choice)
|2nd
|3rd
Line 52 ⟶ 82:
|-
|1st
| ---
|80% or 0.8 votes
(20% or 0.2
margin)
|''<small>>=80%</small>''
|''<small>>=80%</small>''
|-
|2nd
(choice)
|60% or 0.6
| ---
|80% or 0.8
(50% or 0.5)
|''<small>>=80%</small>''
|-
|3rd
|''<small><=30%</small>''
|30% or 0.3
| ---
|90% or 0.9
(40% or 0.4)
|-
|Last
|''<small><=50%</small>''
|''<small><=50%</small>''
|50% or 0.5
| ---
|}</blockquote>(Note: The italicized cells are explained below in the "Transitivity" section).
So this voter expressed a ranked preference, and also expressed, in the head-to-head matchup table, their strength of preference in every head-to-head matchup between each of the candidates in each rank, by marking their support for both their more-supported and less-supported candidate in each matchup.
"80%" here can be read as "80% of a vote" or "80% support", equivalent to 0.8 votes (or an 8 out of 10 on a rated ballot). This can be read as, for example, "1st>3rd" referring to the voter's support for A>D, and "2nd>last" referring to the voter's support for either B or C over all candidates they didn't rank (i.e. that they prefer less than D).
== 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% marginal support for their 1st choice>3rd choice. Another example is that, because they expressed 20% marginal support for 1st>2nd, they must have had at least 20% marginal 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.
=== Margins-based transitivity ===
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>C) >= ((A>B)+(B>C)) )
**A>C must ''equal'' 70%. (A>C = A>B + A>C)
The second type of transitivity is based on [[Score voting]] and the idea that a voter's preferences should fit in a scale (see <ref name=":0">https://imgur.com/a/ssXCQIE</ref>).
* Note however that with rated pairwise, a cap must be artificially imposed such that a voter's preference can't exceed 100% in any matchup.
** 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.
This form of ballot may be cast by first requesting a full ranking, followed by pairwise margins between neighboring candidates. Margins of 0 would indicate equal rankings. Visually, this can be presented as a cardinal evaluation for the ">" operation itself (bounded by 0.0 and 1.0 in each pairwise matchup in this case):
[0.5] [0.3] [0.0] [1.0]
A > B > C = D > E
We then see that A > C,D must be at least 80% of a margin. Traditional ranked ballots simply assume a 100% margin for all ">".
Thus, 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.
=== Transitivity for support on both sides of a matchup ===
Not only is it possible to consider transitivity for what the margins should look like, but if a voter is allowed to express support on both sides of a matchup (i.e. they can say A is 80% supported and B is 50% supported, rather than saying they 30% prefer A over B), then it's also necessary to consider how transitivity should work for support on both sides of the matchup.
* When considering which form of transitivity to use here, a basic test is that that form of transitivity should work properly with a Score voting ballot. For example, if a voter scores A:4 B:2 and B:2 C:1, then in the A vs C matchup, they must give A:4 B:2.
Here are some ideas, using the matchup between 1st choice and 3rd choice as an example:
* Take the greatest score given to any candidate in the 1st>2nd or 2nd>3rd matchup, and then subtract the necessary margin from this score. If the resulting number is less than the minimum score, then increase it enough that it becomes the minimum score, and increase the "greatest score" by the same amount, capping that at the maximum score. The two scores are the scores used for both candidates in the 1st>3rd matchup.
Also see <ref>https://forum.electionscience.org/t/how-should-transitivity-be-handled-with-rated-pairwise-preferences/693</ref> for discussion on transitivity.
== Vote-counting ==
Line 250 ⟶ 226:
==Connection to other ballot types==
This approach is a generalization of
=== Approval ballot ===
Ballot: AB (CD disapproved)
This translates into
{| class="wikitable"
|+
Line 265 ⟶ 241:
|-
|A
|''(1)''
|
(margin of 0)
|1
|1
|-
|B
|
(margin of 0)
|''(1)''
|1
|1
Line 279 ⟶ 257:
|0
|0
|''(0)''
|0
|-
Line 286 ⟶ 264:
|0
|0
|''(0)''
|}
(Note that you can put the number of approvals a candidate has in every pairwise matchup in their self-comparison cell i.e. the value of "1" in A>A can show that candidate A has 1 approval in every matchup).
=== Rated ballot ===
Scale: 0 to 10
Ballot: A:10 B:7 C:3 (since D wasn't scored, we can assume D:0)
{| class="wikitable"
|+
Line 300 ⟶ 281:
|-
|A
|''(1)''
|1 (0.3)
|1 (0.7)
|1
|-
|B
|0.7 (0)
|''(0.7)''
|0.7 (0.4)
|0.7
|-
|C
|0.3 (0)
|0.3 (0)
|''(0.3)''
|0.3
|-
Line 321 ⟶ 302:
|0
| 0
|''(0)''
|}
=== Ranked ballot ===
{| class="wikitable"
|+
Line 343 ⟶ 324:
|0
| ---
|0 (or 1)
|1
|-
|C
|0
|0 (or 1)
| ---
|1
Line 358 ⟶ 339:
| ---
|}
(Note that many ranked voting methods can be seen as, at different points in time, assuming the voter indicates some form of rated pairwise preference that diverges from this. For example, IRV assumes a voter only wishes to support their 1st choice in all pairwise matchups (though their 1st choice can change at different points in time), [[Borda]] and all [[Weighted positional method]]<nowiki/>s assume the voter wants to vote in a way more constrained than Score voting, etc. But pretty much all of them tend to interpret a ranked preference A>B as either A>B or A=B, but never B>A).
== Margins and winning votes approaches ==
To show "winning votes"-relevant information, take the above rated ballot of A:10 B:7 C:3 (D:0), and portray it instead as:
Line 395 ⟶ 378:
See [[Pairwise counting#Cardinal methods]] and [[Order theory#Strength of preference]] for more information on this ballot type.
== Ballot design ==
=== Rated pairwise preferences in all runoffs ===
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).
=== Ranked ballot with rated pairwise preferences between candidates one (or fewer) ranks apart ===
For these reasons, here is a ballot type where it is impossible for the voter to indicate any inconsistent preferences:
{| class="wikitable"
|+
!Scores for candidates
!0 (points/stars)
!1
!2
!3
!4
!5
|-
|A
|
|
|
|
|
|
|-
|B
|
|
|
|
|
|
|-
|C
|
|
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|-
|D
|
|
|
|
|
|
|-
|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
|
|
|
|
|
|
|}
* 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 ==
Line 400 ⟶ 476:
===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.
{| class="wikitable"
|+
Voter's ballot
!
!0
!1
!2
!3
!4
!5
|-
|A
|
|
|X
|
|
|
|-
|B
|
|X
|
|
|
|
|-
|C
|
|
|
|
|
|X
|-
|Maximize preference?
|(Checkbox)
|Yes
|(Checkbox)
|No
|
|
|}
Here is how this vote would be counted, as seen below. 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 points out of 5 becomes 0.8 votes 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"
|+
Line 444 ⟶ 588:
|8.4
| ---
|}Note that if every voter indicates rated preferences, the [[Smith set]] of the collected pairwise preferences will be the Score winner (or the candidates tied with the most points), while if every voter indicates ranked preferences, it will be the regular (Condorcet/majority rule-based) Smith set.
==== Vote-counting ====
If using [[:Category:Condorcet-cardinal hybrid methods]], or any voting method where you want to store both the candidate's actual score and their support in head-to-head matchups (both rated and pairwise preference), it is likely best to store the scores of each voter in one of two separate data values in each candidate's cells i.e. if a voter expressed a rated preference, put their score for a candidate only in the "rated preference" value, but if they expressed a ranked preference, put the score only in the "score for candidate" value. So, for example, a voter expressing a ranked preference who scored candidate A a 5 would be treated as giving A 0 points in the "rated preference" data value but 5 points in the "score for candidate" data value (which could be read as "0, 5" in the A>A cell). This would then be tabulated by giving each candidate as many points as they have in the rated preference data value i.e. a candidate with 51 points in the rated preference value and 37 in the score value would have those values treated such that, supposing a max score of 5, 51/5=10.2 votes would be added to all of their pairwise matchups in favor of them, and 51 points would be added to their score value to find that they have 88 points overall. This actually is easier to count than having to do pairwise counting with only ranked ballots, because for each voter who expresses a rated rather than a ranked preference, their support for a candidate in a head-to-head matchup can be summarized as one data value (the score for the candidate) rather than up to (number of candidates - 1) data values (i.e. the fact that they give that candidate 1 vote in each head-to-head matchup against a lower-ranked candidate).
Line 500 ⟶ 645:
==Notes ==
There is most likely no simple way to create a PR method using rated pairwise ballots, partially because there are no good summary statistics to describe voters' preferences with these ballots (i.e. one voter's 1st choice may be given a different strength of preference in some matchups than another voter's, etc.) Possibly such a thing could be aided by, when some number of winners are desired, allowing voters to express preferences between [[winner set]]<nowiki/>s rather than only pairs of candidates, though this is likely much less practical.
The main voting methods with which this ballot type can be used in the single-winner case are the [[:Category:Pairwise counting-based voting methods]]. However, it can also be used with many [[:Category:Runoff-based voting methods|Category:Runoff-based voting methods]] by only using the rated pairwise information in the runoff i.e. [[IRV]] can be done until all but two candidates are eliminated, at which point the rated pairwise-preferred wins.
If using an implementation of this ballot type involving a single rated ballot, with the additional goal of using the rated information in its direct form as well (i.e. you're using the pairwise information to find the Smith set and the rated information to find the Score winner in [[Smith//Score]]), then it may be useful to include an [[approval threshold]] so that a voter trying to express a pairwise preference without expressing rated support for a given candidate can do so.
=== Notation ===
==== Ranked preferences with rated pairwise information ====
The following notation can be used to describe at least some of a voter's rated pairwise preferences:
A 20% (or 0.2)>B 30%>C 40%>D 50%>E 60%(>last-ranked candidates)
This means the voter indicated a 20% marginal preference in the A vs B matchup (i.e. gave A 0.2 more votes than B), B 30% support against C, etc. It is possible to instead write, say, A 80% to 60%>B 80% to 50%>C (or perhaps it would look better as A 80%>60% B 80%>50% C), which would show not only the marginal preference, but the actual support given to both candidates in the matchup (i.e. instead of only saying the voter gives 0.2 votes more to A than B, you can say the voter gave A 0.8 votes and B 0.6 votes in the A vs B matchup, which conveys more information).
Depending on which [[Rated pairwise preference ballot#Transitivity|transitivity requirements]] are used, this can give varying levels of information about the voter's preference in matchups between candidates that are more than one rank apart. For example, when looking at the A>C matchup:
* If using the first type of transitivity (the weaker one), A>C need only be at least 30% strong.
* With the second type of transitivity, A>C needs to be at least or exactly 50% strong (20%+30%).
If using the rated pairwise ballot design which [[Rated pairwise preference ballot#Ballot design|prevents the voter]] from indicating inconsistent preferences, with the assumption that A>C always equals A>B + B>C, then the above notation gives complete information on all of the voter's rated pairwise preferences. This is because it is possible to figure out the strength of their preference in any given matchup by adding up the margins of each intervening transitive matchup (capping it at 100% support if necessary).
==== Rated pairwise preferences in all runoffs ====
To express that A has 50% support in the A vs B matchup while B has 30%, and that B has 60% support in the B vs C matchup while C has 20%:
A:50% B:30%, B:60% C:20
=== Allowing unlimited maximal pairwise preferences ===
The idea of the rated pairwise ballot is to allow voters to indicate their strength of preference, while not being forced to express at most one maximally strong transitive pairwise preference. For example, on a rated ballot, if a voter expresses that A is maximally better than B (by putting A at the max score and B at the lowest score), then B is automatically treated as being no better than any other candidate i.e. because there is no further room on the scale to down-score another candidate (this image shows this: <ref name=":0" />). However, it can be argued to be illogical or undesirable to allow a voter to express several transitive maximally strong pairwise preferences.<ref>{{Cite web|url=https://www.reddit.com/r/EndFPTP/comments/fylh2p/how_are_elections_run_under_condorcet_reported/fnjdafj|title=r/EndFPTP - Comment by u/MuaddibMcFly on ”How are Elections Run under Condorcet reported with [typical races are normally reported by Points (%)] and - which form of Condorcet Voting would be easiest to implement?”|website=reddit|language=en-US|access-date=2020-04-27}}</ref> A way to partially address this concern is to limit the number of ranks a voter may use, or put limits on the total amount of allowed differentiation between each consecutive rank.
=== 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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