Summability criterion: Difference between revisions

=== Positional 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.

Any [[weighted positional method]] can be summed this way, but with different one-dimensional arrays depending on the method. Alternatively, precincts may sum up the number of times each candidate was ranked at each of the <math>c</math> possible ranks. This ''positional matrix'' can then be used to compute the result for any weighted positional method after the fact, or for [[Bucklin voting]].

=== Condorcet methods ===

In [[Schulze method|Schulze]] and many other summable [[Condorcet method|Condorcet methods]], 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. The precincts' matrices may be added together to get the matrix for the whole electorate, just like a precinct's voters' matrices may be added together to get the matrix for that precinct.

=== Instant-runoff voting ===

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.

== References ==

<references />

== See also ==

*[[Voting system]]

*[[Monotonicity criterion]]

*[[Condorcet Criterion]]

*[[Generalized Condorcet criterion]]

*[[Favorite betrayal criterion]]

*[[Participation criterion]]

[[Category:Voting system criteria]]