Condorcet winner criterion: Difference between revisions

The '''Condorcet criterion''' for a [[voting system]] is that it chooses the beats-all winner when one exists. Any method conforming to the Condorcet criterion is known as a [[Condorcet method]].

 

 

 

== A more general wording of Condorcet criterion definition ==

 

# The voting system must allow the voter to vote as many transitive pairwise preferences as desired. (Typically that's in the form of an unlimited ranking)

 

'''Traditional definition of "beat":'''

 

==Commentary==

On a one-dimensional [[political spectrum]], the beats-all winner will be at the position of the median voter. Example (also see https://i.stack.imgur.com/z4bjy.png):<graph>{

"version": 2,

}</graph>Supposing this is the voter distribution, with 3 candidates Leftist, Centrist, Rightist who are each points on the left, center, and right of the distribution respectively, with all voters distributed from left to right, and voters on, for example, the left preferring Leftist>Centrist>Rightist, and the same holding vice versa. The Condorcet winner is the Centrist, because a majority prefer them over the "other side", for whichever side you look at.

 

The Condorcet winner/[[Smith set]] is a common [[equilibrium]] point in many voting methods. This is because a majority/plurality of voters have no incentive to deviate towards another candidate. Example for [[Approval voting]]:

 

B is the CW. If voters approve everyone they ranked before the "|", then B is approved by all voters, and wins. If any of the three groups of voters here raises their approval threshold (only approves their 1st choice), then another group has an incentive to maintain their approval threshold where it is i.e. if C-top voters stop approving B, then the 69 voters who prefer B>C have an incentive to move their approval thresholds between B and C to ensure B is approved by a majority and C is not. Note that this requires both accurate polling and coordinated [[Strategic voting|strategic voting]].

 

Non-ranking methods such as [[plurality voting|plurality]] and [[approval voting|approval]] cannot comply with the Condorcet criterion because they do not allow each voter to fully specify their preferences. But instant-runoff voting allows each voter to rank the candidates, yet it still does not comply. A simple example will prove that IRV fails to comply with the Condorcet criterion.

 

according to the Condorcet criteria, B should win. By contrast, according to the rules of IRV, B is ranked first by the fewest voters and is eliminated, and C wins with the transferred voted from B; in plurality voting A wins with the most first choices. Note that B and C are a [[Mutual majority criterion|mutual majority]], so most majority rule-based methods would rule A out of winning. If A drops out, then B becomes the majority's 1st choice; so this is an example of IRV failing [[Independence of irrelevant alternatives|independence of irrelevant alternatives]].

 

 

Note that the Condorcet criterion also implies the following criterion which is somewhat related to Independence of Irrelevant Alternatives: removing losing candidates can't change the result of an election if there is a Condorcet winner. <ref>https://arxiv.org/abs/1804.02973 The Schulze Method of Voting p.351 "The Condorcet criterion for single-winner elections (section 4.7) is important because, when there is a Condorcet winner b ∈ A, then it is still a Condorcet winner when alternatives a1,...,an ∈ A \ {b} are removed. So an alternative b ∈ A doesn’t owe his property of being a Condorcet winner to the presence of some other alternatives. Therefore, when we declare a Condorcet winner b ∈ A elected whenever a Condorcet winner exists, we know that no other alternatives a1,...,an ∈ A \ {b} have changed the result of the election without being elected."</ref> In addition, adding candidates who are pairwise beaten by the Condorcet winner (when one exists) can't change the result of the election.

 

=== Weak Condorcet winners ===

 

Sometimes there is no Condorcet winner, but there may be candidate(s) who are preferred by at least as many voters as all other candidates (i.e. they beat '''or''' tie all other candidates; as many voters rank or score them higher or equally as each of the other candidates as the other way around), who are known as weak Condorcet winners. While it may thus seem reasonable that a Condorcet method should pass a condition of always electing solely from the set of weak Condorcet winners when no regular Condorcet winner exists and at least one weak Condorcet winner exists, this guaranteeably leads to failures of reversal symmetry and clone immunity, and so it may be better to say that the set of weak Condorcet winners should have some, but not total priority to win. Example (parentheses are used to indicate implied rankings):<blockquote>3 A(>B1=B2=B3)

 

It has been argued that [[Condorcet methods]] may elect the CW less often than other voting methods, generally [[rated method]]<nowiki/>s.<ref>{{Cite web|url=https://rangevoting.org/CondStratProb.html|title=RangeVoting.org - How Condorcet voting can fail to elect Condorcet Winner|website=rangevoting.org|access-date=2020-05-05}}</ref> [[:Category:Condorcet-cardinal hybrid methods|Category:Condorcet-cardinal hybrid methods]] generally avoid this issue.

 

Here is one [https://www.fairvote.org/why-the-condorcet-criterion-is-less-important-than-it-seems critique] by FairVote, with some analysis/rebuttals:<blockquote>If there is a Condorcet winner, it means that he or she is preferred to every other candidate – not necessarily liked more than other candidates and not necessarily ready to represent the constituents.</blockquote>