Summability criterion: Difference between revisions
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{{Merge|Summability criterion (Wikipedia version)|date=February 2019}}
The '''summability criterion''' is a criterion about the counting process of voting systems, which describes how precinct-summable a voting method is (i.e. whether there is a way for two areas to transmit their vote totals and add this up to find the combined vote total, and if so, how easy it is, or if all the votes need to be taken to a centralized counting location to find the combined result). Unlike most other voting system criteria, it does not relate to the end result, only to the process.
Each vote should map onto a summable array, where the summation operation is associative and commutative, and the winner should be determined from the array sum for all votes cast. An election method is ''kth-order summable'' if there exists a constant ''c'' such that in any election with ''n'' candidates, the required size of the array is at most ''cn<sup>k</sup>''. If there is no value of ''k'' for which the method is ''k''th-order summable, the method is ''non-summable''.
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Any [[weighted positional method]] can be summed this way, but with different one-dimensional arrays depending on the method. Alternatively, precincts may sum up the number of times each candidate was ranked at each of the <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]].
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.
=== Condorcet methods ===
See [[pairwise counting]].
In [[Schulze method|Schulze]] and many other summable [[Condorcet method|Condorcet methods]], each vote is equivalent to a two-dimensional array referred to as a pairwise matrix. If candidate A is ranked above candidate B, then the element in the A row and B column gets a 1, while the element in the B row and A column gets a 0. The pairwise matrices for all the votes are summed, and the winner is determined from the resulting pairwise matrix sum. The precincts' matrices may be added together to get the matrix for the whole electorate, just like a precinct's voters' matrices may be added together to get the matrix for that precinct.▼
▲In [[Schulze method|Schulze]] and many other summable [[Condorcet method|Condorcet methods]], each vote is equivalent to a two-dimensional array referred to as a pairwise matrix. If candidate A is ranked above candidate B, then the element in the A row and B column gets a 1, while the element in the B row and A column gets a 0. The pairwise matrices for all the votes are summed, and the winner is determined from the resulting pairwise matrix sum. The precincts' matrices may be added together to get the matrix for the whole electorate, just like a precinct's voters' matrices may be added together to get the matrix for that precinct.
For example, a voter who ranks all of the candidates A>B=C>D is treated as, in a matrix, giving:
{| class="wikitable"
|+
!
!A
!B
!C
!D
|-
|A
| ---
|'''(A>B)''' 1
|1
|1
|-
|B
|'''(B>A)''' 0
| ---
|0
|1
|-
|C
|0
|0
| ---
|1
|-
|D
|0
|0
|0
| ---
|}
If some other voter ranked B above A, then that would be added into this matrix by adding a 1 to the B>A cell (i.e. increasing it from 0 to 1), etc. [[:Category:Condorcet-cardinal hybrid methods|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).
=== Instant-runoff voting ===
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== Importance of summability ==
The summability criterion addresses implementation logistics. Election methods with lower summability levels are substantially easier to implement with integrity than methods with higher summability levels or methods that are non-summable. In addition, summability points to the simplicity of understanding how voters' support for candidates influences who wins in the voting method.
Suppose, for example, that the number of candidates is ten. Under first-order summable methods like [[plurality voting|plurality]] or [[Approval voting]], the votes at any level (precinct, ward, county, etc.) can be compressed into a list of ten numbers. For [[Schulze method|Schulze]], a 10×10 matrix is needed (although only 10*9=90 data values are actually kept). In an [[IRV]] system, however, each precinct would need to send a list of ten numbers, the number of first-place votes for each candidate. The central system would then return to each precinct a candidate to eliminate. Each precinct would then return the first-place votes for each of the nine remaining candidates, and receive another candidate to eliminate. This would be repeated at most 9 times. This is more than the others.
IRV therefore requires more data transfer and storage than the other methods. The biggest challenge in using computers for public elections will always be security and integrity. If N-1 times more data needs to be transferred and stored, verification becomes more difficult and the potential for fraudulent tampering becomes slightly greater.
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