Rated pairwise preference ballot: Difference between revisions
→Preference threshold
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</blockquote>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):<blockquote>
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which would then become the above tables after the math had been applied.
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==Connection to other ballot types==
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 ===
AB (CD disapproved)
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=== Rated ballot ===
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:
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=== Ranked ballot ===
Finally, a ranked ballot of A>B=C>D is:
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<br />
== 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:
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==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===
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===Preference threshold===
<blockquote>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).</blockquote>Example of this "preference threshold" idea with a single voter, using a rated ballot scale of 0 to 5 (threshold indicated with a "'''|'''"):<blockquote>A:5 B:4 | C:2 D:1</blockquote>This is converted into a pairwise table of:
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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
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.
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