Rated pairwise preference ballot: Difference between revisions
→Example
Line 76:
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>
Either
Line 83:
|+Winning votes-based table
!
|
|
|
|D
|E
|-
|
| ---
|0.8
|0.8
|0.9
|0.9
|-
|
|0.6
| ---
|0
|0.8
|1
|-
|
|0.6
|0
| ---
|0.8
|1
|-
|D
|0.3
|0.3
|0.3
Line 106 ⟶ 117:
|0.95
|-
|E
|0.15
|0.4
|0.4
|0.55
Line 120 ⟶ 132:
| C
|D
|E
|-
| A
| ---
| 0.2
| 0.
|0.6
|0.75
|-
Line 130 ⟶ 144:
|0
| ---
|0
|0.5
|0.6
Line 137 ⟶ 152:
|0
| ---
|0.
|0.6
|-
|D
Line 143 ⟶ 159:
|0
| 0
| ---
|0.4
|-
|E
|0
|0
|0
|0
| ---
|}
</blockquote>
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):
{| class="wikitable"
|+Winning votes-based table
!
|A
|B
|C
|D
|-
|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''
|}
or
{| class="wikitable"
|+Margins-based table
!
|A
|B
|C
|D
|-
|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.
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.
Line 151 ⟶ 248:
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.
==Connection to other ballot types==
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