Rated pairwise preference ballot: Difference between revisions
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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.
AB (CD disapproved)
This translates into a (margin-based) rated pairwise ballot of:
{| class="wikitable"
|+
!
!A
!B
!C
!D
|-
|A
| ---
|0
|1
|1
|-
|B
|0
| ---
|1
|1
|-
|C
|0
|0
| ---
|0
|-
|D
|0
|0
|0
| ---
|}
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:
{| class="wikitable"
|+
!
!A
!B
!C
!D
|-
|A
| ---
|0.3
|0.7
|1
|-
|B
|0
| ---
|0.4
|0.7
|-
|C
|0
|0
| ---
|0.3
|-
|D
|0
|0
|0
| ---
|}
Finally, a ranked ballot of A>B=C>D is:
{| class="wikitable"
|+
!
!A
!B
!C
!D
|-
|A
| ---
|1
|1
|1
|-
|B
|0
| ---
|0
|1
|-
|C
|0
|0
| ---
|1
|-
|D
|0
|0
|0
| ---
|}
See [[Pairwise counting#Cardinal methods]] and [[Order theory#Strength of preference]] for more information on this ballot type.
== Implementations ==
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