Condorcet winner criterion: Difference between revisions
→Commentary
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Range voting does not comply because it allows for the difference between 'rankings' to matter. E.g. 51 people might rate A at 100, and B at 90, while 49 people rate A at 0, and B at 100. Condorcet would consider this 51 people voting A>B, and 49 voting B>A, and A would win. Range voting would see this as A having support of 5100/100 = 51%, and B support of (51*90+49*100)/100 = 94.9%; range voting advocates would probably say that in this case the Condorcet winner is not the socially ideal winner. In general however, it is expected that the Condorcet winner (and Smith Set candidates in general) will almost always be very high-utility when compared to the utilitarian winner.
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)
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3 B1(>A)</blockquote>Now both of A and B1 are weak CWs, because they both pairwise tie each other. In this particular example, since there is nothing that distinguishes either candidate from the other, the neutrality criterion requires that both A and B1 must have an equal probability of winning i.e. both must have a 50% chance. This means that removing clones of B1 increased B1's chances of winning (which were originally at 0%, since A was guaranteed to win earlier i.e. had a 100% chance of winning.) <ref>https://arxiv.org/abs/1804.02973v6 p. 206-207</ref>
Weak CWs have also been called Condorcet non-losers, with the requirement that they always win when they exist being called Exclusive-Condorcet. <ref>{{Cite web|url=http://www.votingmatters.org.uk/ISSUE3/P5.HTM#r2|title=Voting matters, Issue 3: pp 8-15|last=|first=|date=|website=www.votingmatters.org.uk|url-status=live|archive-url=|archive-date=|access-date=2020-05-09|quote=Exclusive-Condorcet (see Fishburn[2]). If there is a Condorcet non-loser, then at least one Condorcet non-loser should be elected.}}</ref>
▲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.
== Multi-winner generalizations ==
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* It implies failure of the [[Favorite Betrayal|Favorite Betrayal criterion]] (possibly leading to [[two-party domination]])
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>
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This criticism can easily be averted if the voters on either side of the Democrats simply refuse to rank the Democrats above the other side..
== Notes ==
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One strategy common to most Condorcet methods is to prevent a candidate from being a Condorcet winner by [[burying]] them (giving them a pairwise defeat against another candidate).
=== Alternative definitions ===
'''Alternative definition of "beat" that is claimed to be more consistent with the preferences, intent and wishes of equal-top-ranking voters:'''
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