Summability criterion

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

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

Majority Choice Approval, Cloneproof Schwartz Sequential Dropping, Approval voting, Cardinal Ratings, Borda count, and Plurality voting all comply. Only Instant-Runoff Voting does not comply.

Commentary

The summability criterion is the only criteria 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 the criterion is essential for ensuring the integrity of an election.

Under methods that do not comply with the summability criterion, every individual vote (rank list) must be available at a central location to determine the winner. The votes cannot be compressed by summing, as in other election methods.

Summability of various methods

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 the Borda count, each ballot is worth a number of points for each candidate. These points can be summed across ballots in the obvious way.

In many Condorcet methods, each vote can be represented as 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.