Summability criterion

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The summability criterion is a criterion about the counting process of voting systems. 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 cnk. If there is no value of k for which the method is kth-order summable, the method is non-summable.

Summable Methods

Methods and their summability levels.
k=1 k=2 k=3 non-summable


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, 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.

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.

Suppose, for example, that the number of candidates is ten. Under first-order summable methods like plurality or Approval voting, the votes at any level (precinct, ward, county, etc.) can be compressed into a list of ten numbers. For Schulze, 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.

To illustrate this point, consider the verification of a vote tally for a national office. In a plurality election, each precinct verifies its vote count. This can be an open process where The counts for each precinct in a county can then be added to determine the county totals, and anyone with a calculator or computer can verify that the totals are correct. The same process is then repeated at the state level and the national level. If the votes are verified at the lowest (precinct) level, the numbers are available to anyone for independent verification, and election officials could never get away with "fudging" the numbers.


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 must be recounted. This is not a big problem for computer recounts, but manual recounts can be extremely time-consuming and expensive.

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