# Rated pairwise preference ballot 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 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 the same preference information captured by all of those ballot types, and certain preferences which aren't. Note that with rated pairwise, a voter may indicate they maximally prefer their 1st choice over their 2nd choice, and that they maximally prefer their 2nd choice over their 3rd choice (for this reason, Condorcet cycles can occur based off of the runoff data generated).

Because this ballot type can be more onerous to fill out and count the votes for, it is strongly suggested to use one of the simpler, less expressive implementations discussed below instead. These generally work by allowing the voter to fill out a rated ballot, and then generating some form of rated pairwise preferences from the rated preferences based on the voter's input.

## Example

With some simplification, this can be visualized as (example using pairwise counting for a single voter, with 6 candidates A through F):
Voter's ballot Ranking:
1st choice(s) 2nd choice(s) 3rd choice(s)
A B, C D
Rated pairwise matchups (below):
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
(Partial) Interpretation of ballot
 1st (choice) 2nd 3rd Last 1st --- 80% or 0.8 votes (20% or 0.2 margin) >=80% >=80% 2nd (choice) 60% or 0.6 --- 80% or 0.8 (50% or 0.5) >=80% 3rd <=30% 30% or 0.3 --- 90% or 0.9 (40% or 0.4) Last <=50% <=50% 50% or 0.5 ---
(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 ).

• 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  for discussion on transitivity.

## Vote-counting

If using the negative vote-counting approach, the precinct vote-counters would mark the following (the maximum support a candidate gets in any matchup can be put as the amount of support they get in every matchup, with fractional negative votes in certain matchups to yield the correct values):

 A B C D A 0.9 (-0.1) 0 0 B (-0.4) 1 (-0.2) 0 C (-0.65) (-0.65) 0.95 0 D (-0.40) (-0.15) 0 0.55

or

 A B C D A 0.75 (-0.55) (-0.55) (-0.15) B (-0.6) 0.6 (-0.6) (-0.1) C (-0.6) (-0.6) 0.6 (-0.1) D (-0.4) (-0.4) (-0.4) 0.4

which would then become the above tables after the math had been applied.

## Connection to other ballot types

This approach is a generalization of various other ballot types in the sense that if every voter expresses the same margins-based or winning votes-based preference for each candidate in each head-to-head matchup as they would if they were rating them on a scale with all other candidates (i.e. a voter who would give a candidate 80% support on a rated ballot's scale would give that candidate a 30% margin in a head-to-head matchup against a candidate they'd rate a 50% on the same scale), then it reduces to a rated ballot (with the same logic following for an Approval ballot, since an Approval ballot is a restricted form of a rated ballot), and if every voter expresses a maximal preference for their preferred candidate in each matchup, then it reduces to a ranked ballot (at least, if using the Condorcet interpretation). Here are examples:

### Approval ballot

Ballot: AB (CD disapproved)

This translates into (with margins expressed in parentheses where necessary) a rated pairwise ballot of:

A B C D
A (1) 1

(margin of 0)

1 1
B 1

(margin of 0)

(1) 1 1
C 0 0 (0) 0
D 0 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)

A B C D
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
D 0 0 0 (0)

### Ranked ballot

Ballot: A>B=C>D

A B C D
A --- 1 1 1
B 0 --- 0 (or 1) 1
C 0 0 (or 1) --- 1
D 0 0 0 ---

(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 methods 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:

A B C D
A --- 1 1 1
B 0.7 --- 0.7 0.7
C 0.3 0.3 --- 0.3
D 0 0 0 ---

As can be seen, the margin is the same in the winning votes-based table and the margin-based table (i.e. in the A vs B matchup, the voter contributed only 0.3 points more to A than to B), but some different information is collected; in other words, both approaches will give an accurate final margin in each head-to-head matchup, but can lead to more or less votes for both candidates in each matchup depending on how voters scored them. In addition, this explains why Score voting is precinct-summable to a much easier degree than Category:Pairwise counting-based voting methods; because the voter is assumed to express the same score in every runoff, the score itself can be used to represent their support for the candidate in all of their head-to-head matchups.

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:

Scores for candidates 0 (points/stars) 1 2 3 4 5
A
B
C
D
1st (choice) vs 2nd (choice)
2nd vs 3rd
3rd vs 4th
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

These are ways to use the rated pairwise ballot that limit expressiveness, but still collect more pairwise information than other ballot types. They are mostly listed in order of simplicity for vote-counting and level of expressiveness.

### 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.

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:

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:

A B C
A 15 3 4
B 2 12 5
C 1 7 13
This can be interpreted as a regular Pairwise comparison matrix except that the score/points total for each candidate is recorded in their cell (i.e. A>A shows A's score). So, for example, 3 voters indicated a ranked preference for A>B, and some number of other voters indicated a rated preference such that their combined support for A added up to 15 points for A. Supposing the max score is 5 and min score is 0, in this example 15/5=3 votes would be added to every Head-to-head matchup in favor of A, 12/5=2.4 votes in favor of B's matchups, and 7/5=1.4 votes in favor of C's matchups. So the final table would be:
A B C
A --- 6 7
B 4.4 --- 7.4
C 2.4 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).

### Preference threshold

It's possible to modify Score to be more like a traditional Condorcet method by allowing voters to write the scores they would give to every possible pair of candidates in a Score runoff, and then using a Condorcet method to process this, treating a score of, say, A5 B3 (where the max score is 5) as 0.4 votes for A>B. As this would be utterly infeasible with just a few candidates running however, one way to accomplish most of the same objective is to allow voters to mark on their ballots that they want their vote strategically optimized, meaning that if their cardinally expressed preferences are A5 B3 Z2, instead of having their vote considered as B3 Z2 in an B vs. Z runoff, it would be considered as B5 Z0 (if the max score is 5), which is functionally equivalent to the Plurality voting runoffs that are used for the traditional Condorcet winner definition. It is also possible for voters to indicate a preference threshold, meaning that for all preferred candidates (candidates above or at the threshold), no strategic optimization is applied to pairwise matchups between them, but all other matchups are strategically optimized. With this modification, if all voters use strategic optimization, Score becomes a traditional Condorcet method (which will need a cycle resolution method to be applied at times), but if no voters strategically optimize, it remains Score (which never needs cycle resolution methods to be applied).
Example of this "preference threshold" idea with a single voter, using a rated ballot scale of 0 to 5 (threshold indicated with a "|"):
A:5 B:4 | C:2 D:1
This is converted into a pairwise table of:
A B C D
A --- 1 1 1
B 0.8 --- 1 1
C 0 0 --- 1
D 0 0 0 ---

The matchup between A and B is treated as weak because both candidates come before the threshold (i.e. the voter only gives 0.2 more votes to A than B, which is their scored preference of (5-4)/5=1/5th or 0.2 votes; keep in mind that when changing the scale from 0 to 5 to 0 to 1, the scores of 5 and 4 become 1 and 0.8 respectively, which is what you see in the pairwise table. It is also possible to put 0.2 and 0 instead, which captures only the margin and not the winning votes for the matchup), while all other matchups are treated as maximal (despite, for example, A>C only having a scored preference of (5-2)/5=3/5th or 0.6 votes, it is instead treated as a maximal preference of 1 vote).

A voter who sets their preference threshold at the same score they gave their favorite candidate or higher is essentially casting a ranked-preference ballot, while if they set it at the lowest score, they are casting a rated ballot. It is possible to only treat voters' preferences as maximal in matchups between preferred candidates and dispreferred candidates, but this would make it no longer possible to effectively cast a ranked ballot using this approach.

It is possible to allow for multiple preference thresholds on a single ballot, such that the matchups between candidates in between thresholds aren't maximized, but all other matchups are. For example, a voter voting A:5 B:4 | C:3 D:2 | E:1 | could have the A vs B and C vs D matchups treated as weak, but the A>C and D>E preferences, for example, treated as strong. Fractional preference thresholds can even be applied; see fractional optimization below.

### Fractional optimization

This strategic optimization can be done fractionally to allow a voter to customize how much optimization they want to be done with their scores in each runoff.

An example of fractional optimization on a scale of 0 to 5:

A:5 B:3 C:2 at 60% optimization

This would become an optimized preference of 76% or 0.76 votes for A>B, 68% for B>C, and 84% for A>C. This is derived by looking at how far apart the rated preference and maximal preference values are, applying the % of optimization to this difference, and adding the resulting value to the rated value. So for example, A>B is a rated preference of (5-3)/5=2/5ths or 40% strength. That is 60% shy of 100%, and 60% optimization multiplied by this 60% difference is 36%, which added back to the rated preference of 40% yields 76%.

0% optimization is equivalent to a rated ballot, while 100% is equivalent to a ranked preference.
Note that when designing a ballot to allow voters to indicate strength of preference in pairwise matchups, it could be done by allowing the voters to rank or score the candidates themselves, and then indicate "between your 1st choice(s) and 2nd choice(s), what scores would you give to each in a pairwise matchup?" or "between the candidates you scored (max score) and the candidates you scored (max score - 1), what scores would you give in their pairwise matchups?", etc. Here is an example of one such setup:  and some discussion: