Condorcet method: Difference between revisions

Any election method conforming to the [[Condorcet criterion]] - that is, one which always elects the [[beats-all winner]], a candidate who can beat any other candidate in a runoff, if one exists - is known as a '''Condorcet method'''. The name comes from the 18th century mathematician and philosopher [[Marquis de Condorcet]], although the method was previously described by [[Ramon Llull]] in the 13th century. Many Condorcet advocates agree that a further criterion that Condorcet methods should pass is the [[Smith criterion]], which means the Condorcet method will always elect someone from the [[Smith set]] when there is no beats-all winner (due to the [[Condorcet paradox]]).

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" or cycle resolution/completion procedure; thea mostvery 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.

Each voter fills out a [[preferential voting|ranked ballot]] or [[Cardinal voting|rated ballot]]. 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.

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.

If a candidate is preferred over all other candidates i.e. wins all of their matchups, that candidate is the [[Condorcet winner|Condorcet candidate]] (Condorcet winner). However, a Condorcet candidate may not exist, due to a fundamental [[Voting paradox|paradox]]: It is possible for the electorate to prefer A over B, B over C, and C over A in each head-to-head matchup simultaneously. This is called a majority rule cycle or Condorcet cycle, and it must be resolved by some other mechanism (usually either by using another voting method, or [[:Category:Defeat-dropping Condorcet methods|"dropping" weak defeats]] in order to end the cycle).

The sum of all ballot matrices, the '''Condorcet pairwise matrix''', is the primary piece of data used to resolve majority rule cycles in [[:Category:Defeat-dropping Condorcet methods|defeat-dropping Condorcet methods]], and can be used to find the Condorcet winner and [[Smith set]] in any Condorcet method.

There are various ways to find the Condorcet winner from the pairwise matrix. [[: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.

*[[Defeat strength|'''Defeat strength''']]: Different ways of measuring how "strong" a pairwise defeat is. Useful when deciding which defeats to "drop" in [[:Category:Defeat-dropping Condorcet methods|defeat-dropping Condorcet methods]].

The text below describes (variants of) thesethe defeat-dropping methods in more detail.

* '''weak [[Condorcet winnerloser]]''': athe candidate who beatsis orless tiespreferred withthan every other candidate in a [[pairwise matchup. There can be more than one weak Condorcet winner]].

* '''weak Condorcet loserwinner''': a candidate who is defeated bybeats or ties with every other candidate in a pair wisepairwise matchup. Similarly, thereThere can be more than one weak Condorcet loserwinner. Because of Condorcet cycles, it is possible for the Smith set to, for example, have one weak Condorcet winner, and three candidates that pairwise beat each other but pairwise tie with the WCW; for this reason, it is not always best to elect the weak Condorcet winner over the candidates that tie with them.

An alternative way of demonstrating this (using [[ISDA]]-based logic) is that a [[Mutual majority criterion|mutual majority]] of voters prefer any city other than Memphis, so that knocks Memphis out of contention. When looking at Memphis voter's new 1st choice among the candidates, it is Nashville, resulting in Nashville having a 68% majority of 1st choices and thus pairwise beating all others.

Alternative formatting of the pairwise matrix (shows the margins in each matchup by subtracting number of votes for one candidate from the votes for the other):

* [http://condorcet.ericgorr.net Condorcet Voting Calculator] by Eric Gorr (includes [[Ranked Pairs]], and [[Schulze]])