Summability criterion: Difference between revisions
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== Examples == |
== Examples == |
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=== Summable methods === |
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==== Points-scoring methods ==== |
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===== Positional methods ===== |
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In [[plurality voting]], each vote is equivalent to a one-dimensional array with a 1 in the element for the selected candidate, and a 0 for each of the other candidates. The sum of the arrays for all the votes cast is simply a list of vote counts for each candidate. |
In [[plurality voting]], each vote is equivalent to a one-dimensional array with a 1 in the element for the selected candidate, and a 0 for each of the other candidates. The sum of the arrays for all the votes cast is simply a list of vote counts for each candidate. |
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<math>c</math> possible ranks. This ''positional matrix'' can then be used to compute the result for any weighted positional method after the fact, or for [[Bucklin voting]]. |
<math>c</math> possible ranks. This ''positional matrix'' can then be used to compute the result for any weighted positional method after the fact, or for [[Bucklin voting]]. |
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==== Cardinal methods ==== |
===== Cardinal methods ===== |
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[[Approval voting]] is the same as plurality voting except that more than one candidate can get a 1 in the array for each vote. Each of the selected or "approved" candidates gets a 1, and the others get a 0. |
[[Approval voting]] is the same as plurality voting except that more than one candidate can get a 1 in the array for each vote. Each of the selected or "approved" candidates gets a 1, and the others get a 0. |
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For example, with [[Score voting]], a voter who votes A:10 B:6 C:3 D:1 is treated as giving a 10 to A, a 6 to B, etc. Comparisons across different score scales can be made by dividing the score by the max score (i.e. instead of a 6, treat it as a 6/10=0.6, etc.) so that a voter who scores a candidate a 3 out of 5 and a voter who scores a candidate a 6 out of 10 can have their scores treated and counted the same without any issues. |
For example, with [[Score voting]], a voter who votes A:10 B:6 C:3 D:1 is treated as giving a 10 to A, a 6 to B, etc. Comparisons across different score scales can be made by dividing the score by the max score (i.e. instead of a 6, treat it as a 6/10=0.6, etc.) so that a voter who scores a candidate a 3 out of 5 and a voter who scores a candidate a 6 out of 10 can have their scores treated and counted the same without any issues. |
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=== Pairwise methods === |
==== Pairwise methods ==== |
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Some voting methods, such as [[STAR voting]] can be made precinct-summable using pairwise information alongside other pieces of information. |
Some voting methods, such as [[STAR voting]] can be made precinct-summable using pairwise information alongside other pieces of information. |
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==== Condorcet methods ==== |
===== Condorcet methods ===== |
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See [[pairwise counting]]. |
See [[pairwise counting]]. |
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[[:Category:Condorcet-cardinal hybrid methods]] require one additional piece of information per candidate: the score for the candidate. This can be stored in the cell comparing the candidate to themselves (i.e. A>A would have candidate A's score). |
[[:Category:Condorcet-cardinal hybrid methods]] require one additional piece of information per candidate: the score for the candidate. This can be stored in the cell comparing the candidate to themselves (i.e. A>A would have candidate A's score). |
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=== Median methods === |
==== Median methods ==== |
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The [[Majority Judgement]] and [[Bucklin]] methods can be made summable by counting the number of voters who gave a candidate each specific grade or rank. This is in contrast to other rated methods, where the grade level doesn't matter, only the strength/quality/degree of the grade (i.e. in other methods, two 1/5s are equivalent to one 2/5). |
The [[Majority Judgement]] and [[Bucklin]] methods can be made summable by counting the number of voters who gave a candidate each specific grade or rank. This is in contrast to other rated methods, where the grade level doesn't matter, only the strength/quality/degree of the grade (i.e. in other methods, two 1/5s are equivalent to one 2/5). |
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=== Non-summable methods === |
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===== Instant-runoff voting ===== |
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IRV does not comply with the summability criterion. In the IRV system, a count can be maintained of identical votes, but votes do not correspond to a summable array. The total possible number of unique votes grows factorially with the number of candidates. |
IRV does not comply with the summability criterion. In the IRV system, a count can be maintained of identical votes, but votes do not correspond to a summable array. The total possible number of unique votes grows factorially with the number of candidates. |
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