Condorcet method: Difference between revisions

If one candidate is preferred by more voters than all other candidates (when [[pairwise counting|compared one-on-one]]), that candidate is the [[Condorcet Criterion|Condorcet Winner]], abbreviated as CW. This can be determined through use of ranked or rated ballots (i.e. if a voter ranks or rates one candidate higher than another). On rare occasions, there is no Condorcet winner (because of either [[pairwise counting#Terminology|ties]] in the head-to-head matchups or the [[Condorcet paradox]]. In that case it is necessary to use some tiebreaking procedure; the most common minimum standard for a Condorcet method's tiebreaking procedure is that it should be [[Smith-efficient]], that is, always elect someone from the [[Smith set]], the smallest group of candidates that win all their [[head-to-head matchup|head-to-head matchups]] against all candidates not in the group.

SequentialThere comparisonare isvarious the fastest wayways to determinefind the Condorcet winner, when one exists, from the pairwise matrix. 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.

* '''[[Independence of clone alternatives|Cloneproof]]''': a method that is immune to the presence of '''clones''' (candidates which are essentially identical to each other). In some voting methods, a party can increase its odds of selection if it provides a large number of "identical" options. A cloneproof voting method prevents this attack. See [[strategic nomination]].

* *'''[[Smith/IRV]]''' is [[instant-runoff voting]] with the candidates restricted to the Smith set.

* **'''[[MinmaxSmith//Minimax|Smith/Minimax]]''' (also called '''Simpson''') choosesrestricts the candidateMinimax whosealgorithm worstto pairwisethe defeat[[Smith is less bad than that of all other candidatesset]].¹

* *'''[[Approval-Condorcet Hybrids]]''', such as '''[[Definite Majority Choice]]''', use an [[Approval Cutoff]] to augment the Condorcet pair wise array. Many believe that such a method would make a good first-round public proposal.

**'''[[Smith//Score]]''' chooses the candidate with the highest summed or average score in the Smith Set. '''[[Condorcet//Score]]''' chooses the [[Utilitarian winner|Score winner]] when no Condorcet winner exists. (These can only be done with rated ballots, or with ranked/rated ballots modified to include approval thresholds).