Rated pairwise preference ballot

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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.

Because this ballot type requires significantly more information to be collected from the voters and processed by the vote-counters, and has some difficult rationality/transitivity requirements which voters may struggle to follow while filling out their ballots, it is strongly suggested to use one of the simpler, less expressive implementations discussed below instead.

Example[edit | edit source]

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


Winning votes-based rated matchups table
1st 2nd 3rd Last
1st --- 80% 90% 90%
2nd 60% --- 80% 100%
3rd 30% 30% --- 95%
Last 15% 40% 55% ---

This is equivalent to the following table in terms of margins:

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 would then be converted by the vote-counters into (adding a candidate E into the election, who is assumed to be ranked last by the voter):


Winning votes-based table
A --- 0.8 0.8 0.9 0.9
B 0.6 --- 0 0.8 1
C 0.6 0 --- 0.8 1
D 0.3 0.3 0.3 --- 0.95
E 0.15 0.4 0.4 0.55 ---


Margins-based table
A --- 0.2 0.2 0.6 0.75
B 0 --- 0 0.5 0.6
C 0 0 --- 0.5 0.6
D 0 0 0 --- 0.4
E 0 0 0 0 ---

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 (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.

Vote-counting[edit | edit source]

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):

Winning votes-based table
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


Margins-based table
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[edit | edit source]

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. Here are examples:

Approval ballot[edit | edit source]

AB (CD disapproved)

This translates into a (margin-based) rated pairwise ballot of:

A --- 0 1 1
B 0 --- 1 1
C 0 0 --- 0
D 0 0 0 ---

Rated ballot[edit | edit source]

A rated ballot of, with min score 0 and max score 10, A:10 B:7 C:3 (D:0) is a rated pairwise ballot of:

A --- 0.3 0.7 1
B 0 --- 0.4 0.7
C 0 0 --- 0.3
D 0 0 0 ---

Ranked ballot[edit | edit source]

Finally, a ranked ballot of A>B=C>D is:

A --- 1 1 1
B 0 --- 0 1
C 0 0 --- 1
D 0 0 0 ---

Margins and winning votes approaches[edit | edit source]

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 --- 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.[1] 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.

Implementations[edit | edit source]

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[edit | edit source]

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. For example, suppose the following information is collected:

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 --- 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 Smith set.

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[edit | edit source]

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 --- 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[edit | edit source]

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: [2] and some discussion: [3]

Notes[edit | edit source]

Allowing unlimited maximal pairwise preferences[edit | edit source]

The idea of the rated pairwise ballot is to allow voters to indicate their strength of preference, while not being limited to expressing only 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. However, it can be argued to be illogical or undesirable to allow a voter to express several transitive maximally strong pairwise preferences.[4] 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.

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 sets 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.

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.

Condorcet criterion[edit | edit source]

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.

References[edit | edit source]

  1. "r/EndFPTP - Comment by u/curiouslefty on "Score but for every pairwise matchup"". reddit. Retrieved 2020-05-12.
  2. "r/EndFPTP - Poll for 2020 Dem primary using Scored Pairwise Matchups". reddit. Retrieved 2020-04-28.
  3. "r/EndFPTP - Comment by u/Chackoony on "Adjusting pairwise matchup margins to favor higher-utility candidates"". reddit. Retrieved 2020-04-28.
  4. "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?"". reddit. Retrieved 2020-04-27.