Summability criterion: Difference between revisions
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== Examples ==
=== Positional methods ===
In [[
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]].
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. ▼
=== Condorcet methods ===
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.
=== Instant-runoff voting ===
▲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.
== Importance of summability ==
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In first-order summable election systems, adding new ballots to the count (say, ballots that were found after the initial count, or late absentee ballots, or ballots that were initially ruled invalid) is as simple as "summing" the original result with the newly-found ballots. Under non-summable systems, though, finding new ballots means all ballots might possibly need to be recounted. This is not a big problem for computer recounts, but manual recounts can be extremely time-consuming and expensive.
== References ==▼
[[Category:Voting system criteria]]▼
<references />
== See also ==
*[[Voting system]]
*[[Monotonicity criterion]]
*[[Condorcet Criterion]]
*[[Generalized Condorcet criterion]]
*[[Favorite betrayal criterion]]
*[[Participation criterion]]
''Some parts of this article are derived with permission from text at http://electionmethods.org''
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{{fromelectorama|Summability_criterion}}
▲[[Category:Voting system criteria]]
▲== References ==
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