Condorcet method: Difference between revisions
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Each voter fills out a [[preferential voting|ranked ballot]] or [[Cardinal voting|rated ballot]] (i.e. they rank the candidates 1st, 2nd, 3rd, or they rate the candidates, for example, a 0 out of 5, a 3 out of 5, etc.) The voter can include less than all candidates under consideration. Usually when a candidate ''is not listed'' on the voter's ballot they are considered less preferred than listed candidates, and ranked accordingly, with the voter considered to have no preference between any of them. However, some variations allow a "no opinion" default option where no for- or against- preference is counted for that candidate. Write-ins are possible, but are somewhat more difficult to implement for automatic counting than in other election methods. This is a counting issue, but results in the frequent omission of the write-in option in ballot software.
==Counting ballots==
{{Image frame|width=300|content=
<youtube width="300" height="230">5RtOCvFqIKk</youtube>
|caption=Procedure for finding the Condorcet winner, or showing that there is none.
}}
Ballots are counted by considering all possible sets of two-candidate elections from all available candidates. That is, each candidate is considered against each and every other candidate. A candidate is considered to "win" against another on a single ballot and receive that ballot's vote in the matchup against their opponent if they are ranked or rated higher than their opponent. All the votes for candidate Alice over candidate Bob are counted, as are all of the votes for Bob over Alice. Whoever has the most votes in each one-on-one election/matchup wins the matchup.
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There are various ways to find the Condorcet winner from the pairwise matrix. The simplest is to look for a single candidate who has a positive margin of votes against all other candidates in each matchup (i.e. if they got 5 votes and another candidate 4 votes in the pair's matchup, then the margin is 1 vote in favor of the first candidate, indicating they won the matchup), if one exists. [[Copeland]] more generally can help find the Smith set by looking for a smallest group of candidates that have victories against all others, by starting from the candidates with the most victories in their head-to-head matchups.
[[:Category:Sequential comparison Condorcet methods|Sequential comparison]] is one such way: order all of the candidates in any manner desired, pairwise compare the first two, eliminate the loser of the matchup, and repeat until only one candidate remains. This requires ((number of candidates) - 1) pairwise comparisons, since for each comparison one candidate is eliminated, and all but one candidate must be eliminated. To check whether a Condorcet winner exists in a given election, do the previous procedure and then check whether the remaining candidate wins all of their pairwise matchups; this requires ((number of candidates) - 2) pairwise comparisons in the worst case, though if the ordering of the candidates in the procedure is done in such a way as to put candidates more likely to be Condorcet winners higher in the ordering, then in the best case 0 pairwise comparisons are required, since if the first candidate in the ordering turns out to be the Condorcet winner, all of their pairwise comparisons have already been done. Condorcet winners may often have a lot of 1st choice votes, especially in less contested elections, so it may be best to order the candidates descending by order of 1st choice votes, then 2nd choice votes, etc. These procedures can be used even for Condorcet PR methods by considering each winner set to be a candidate.
==Key terms in ambiguity resolution==
[[File:Finding_smith_set_ranking.png|thumb|487x487px|All of the candidates in 1st place (Andy, Brianna, Charles) are in the Smith set. See the [[Smith set ranking]] article for more information on this image.]]Handling cases where there is not a single Condorcet winner is called ambiguity resolution in this article, though other phrases such as "cyclic ambiguity resolution" and "Condorcet completion" are used as well.
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