Before its recent modification, the ElectionMethods.org website had a page called "Technical Evaluation of Election Methods." Two of the criteria listed on that page were monotonicity and summability. Most of you are familiar with the former, but the latter was my own idea. I have cut some of the text from the definition of summability and included it below for reference.
It occurred to me a while back that the two criteria may be equivalent. That is, if a method passes monotonicity, perhaps it must also pass summability, and vice versa. That's just a hunch. Can anyone prove (or disprove) it?
--Russ
Summability Criterion
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
<p>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).</p>
<p>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.</p>
<p>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.</p>
<p>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.</p>
<p>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.</p>
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