Difference between revisions of "Ballot"

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[[Pairwise counting]] can be done on all four ballot types, though ranked and rated ballots offer the most information for this purpose.
 
Note that ranked, rated, and Approval ballots can be generalized into one ballot type: allowing the voters to express their rated preferences in every [[head-to-head matchup]]. With some simplification, this can be visualized as (example for a single voter, with 6 candidates A through F):<blockquote>A>B=C>D
{| class="wikitable"
|+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%
| ---
|}
</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 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.
 
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.
 
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