Statement of Criterion
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.
Complying Methods
All of the methods in the compliance table above comply with the summability criterion except Instant Runoff Voting (IRV).
Commentary
The summability criterion is the only criteria discussed on this webpage that addresses implementation logistics. Election methods that comply with the summability criterion are substantially easier to implement with integrity than those that do not. All the election methods listed in Table 1 comply except Instant Runoff Voting (IRV).
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 Condorcet voting, 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. The larger the number of candidates, the more error-prone and less practical it becomes to maintain counts of each possible unique vote. It becomes impractical with more than about six candidates.
Suppose, for example, that the number of candidates is ten. In our current plurality system, the votes at any level (precinct, county, state, or national) can be compressed into a list of ten numbers. The same is true for an Approval system. For a Condorcet system, a 10x10 matrix is needed. In an IRV system, however, the number of possible unique votes is over ten factorial -- a huge number.
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 our current plurality system, each precinct verifies its vote count. 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.
The point is that once 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. At the lowest level, ballot problems such as "hanging chads" could be a problem, but adding the vote counts will certainly not be a problem. And this applies not only to conventional plurality elections, it applies also to Condorcet, Approval, and even Borda -- but not IRV
In an IRV election, the voting data cannot be "compressed" by adding the vote totals together at each level, so verification of the tally results becomes nearly impossible. The final result depends on all the votes, but even if the individual votes are all counted correctly, nobody can verify that the total pool of votes has not been tampered with at some level of the tallying process. And with IRV's erratic properties, someone could lower the rankings of a candidate to make him win or raise the rankings of a candidate to make him lose. It's a prescription for disaster and voter cynicism.
If IRV were superior otherwise, then its failure to comply with the summability criterion might be excusable. But IRV has been shown to fail with respect to every one of the criteria listed, including such basic criteria as monotonicity. To accept the additional security risks that IRV poses would therefore be the epitome of folly.