Summability criterion: Difference between revisions

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 ''cn^k''. If there is no value of ''k'' for which the method is ''k''th-order summable, the method is ''non-summable''.

The summability criterion addresses implementation logistics. Election methods with lower summability numberslevels are substantially easier to implement with integrity than thosemethods with higher summability levels or methods that doare notnon-summable.

Suppose, for example, that the number of candidates is ten. Under first-order summable methods like [[plurality voting|plurality]] or [[Approval voting]], the votes at any level (precinct, ward, county, etc.) can be compressed into a list of ten numbers. For [[Cloneproof Schwartz Sequential Dropping]], a 10x1010×10 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.