# Summability criterion (Wikipedia version)

The summability criterion is a voting system criterion, used to objectively compare voting systems. The criterion states:

Each vote should be able to be mapped onto a summable array, such that its size at most grows polynomially with respect to the amount of candidates, the summation operation is associative and commutative and the winner could be determined from the array sum for all votes cast alone.

## Complying methods

Majority Choice Approval, Schulze method, Approval voting, Range voting, Borda count, Nanson's method, and Plurality voting all comply. Instant-Runoff Voting does not comply.

## Commentary

Summability is the only publicly expressed criterion that addresses implementation logistics. Election methods that comply with the summability criterion are easier to implement than those that do not. Those who support the summability criterion say that it is essential for ensuring the integrity of an election.

Under methods that do not comply with the summability criterion, usually every individual vote (rank list) or at least every unique vote and its number of occurrences, must be available at a central location to determine the winner. The votes cannot feasibly be compressed by summing, as in other election methods.

## Summability of various methods

In plurality voting, the number of ballots for each candidate may be counted, and these totals reported from each precinct.

In Approval voting, Borda count, and Range voting, each ballot contains votes for more than one candidate, and, with the last two, these votes may have different values. However, the sum of all values for each candidate may be found at each precinct and reported.

With Bucklin voting, the precinct totals for each candidate at each rank may be summed and reported.

In many Condorcet methods, each ballot can be represented as a two-dimensional square array referred to as a pairwise matrix. The sum of these matrices may be reported from each precinct.

Instant-runoff voting does not comply with the summability criterion. [1]

## References

1. Gaming the Vote, Why Elections Aren't Fair (and What We Can Do About It), William Poundstone, New York: Hill and Wang, 2008, p. 170.