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Summability criterion: Difference between revisions
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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. [[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.
In [[Schulze method|Cloneproof Schwartz Sequential Dropping]], 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.
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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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.
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|Cloneproof Schwartz Sequential Dropping]], a 10×10 matrix is needed. In an [[IRV]] system, however, the number of possible unique votes is over ten factorial--a very large number. The larger the number of candidates, the more error-prone and less practical it becomes to maintain counts of each possible unique vote. Under IRV, therefore, every individual vote (rank list) must be available at a central location to determine the winner. In a major public election, that could be millions or even tens of millions of votes. The votes cannot be compressed by summing as in other election methods because votes may need to be transferred according to which candidates are eliminated in each round.
IRV therefore requires far more data transfer and storage than the other methods. Modern networking and computer technology can handle it, but that is beside the point. The biggest challenge in using computers for public elections will always be security and integrity. If many thousands of times more data needs to be transferred and stored, verification becomes more difficult and the potential for fraudulent tampering becomes substantially greater.
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