Rated pairwise preference ballot

Not to be confused with Cardinal pairwise.

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 in every head-to-head matchup (pairwise matchup) between the candidates. It is a generalization of Choose-one ballot, Approval ballot, rated ballot, and ranked ballot in the sense that it is possible to submit preferences mirroring all of those ballot types, but also possible to submit preferences which can't be written in any of those ballot types. With some simplification, this can be visualized as (example using pairwise counting for a single voter, with 6 candidates A through F):

A>B=C>D

Margins-based rated matchups table

	1st	2nd	3rd	Last
1st	---	20%	60%	75%
2nd	0%	---	50%	60%
3rd	0%	0%	---	40%
Last	0%	0%	0%	---

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. "1st" here refers to "1st choice", and "20%" here can be read as "20% of a vote" or "20% support", equivalent to 0.2 votes (or a 2 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 prefer less than D. This table captures the 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.

Certain minimum requirements for transitivity are apparent simply from looking at this table; for example, since the voter expressed a 50% difference 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 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.

This approach is a generalization of the above 3 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. See Pairwise counting#Cardinal methods and Order theory#Strength of preference for more information on this ballot type.

Implementations

One particular, easier approach to implementing this generalized ballot type is to allow the voters to score the candidates on a scale, and then 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. For example:

	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). 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	---

If using Category:Condorcet-cardinal hybrid methods, 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).

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. 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. It is also possible for voters to indicate a preference threshold, meaning that for all preferred candidates, 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).

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.)

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, and more specifically, Condorcet methods.