Condorcet winner criterion: Difference between revisions

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.

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>

Schulze has proposed a generalization of the Condorcet criterion for multi-winner methods:<ref name="Schulze 2018">{{cite web | last=Schulze | first=Markus | title=The Schulze Method of Voting | website=arXiv.org | date=2018-03-15 | url=https://arxiv.org/abs/1804.02973v6 | access-date=2020-02-11|page=351}}</ref> Suppose all but M+1 candidates are eliminated from the ballots, and the remaining candidates include candidate ''b''. If ''b'' is always a winner when electing M winners from the M+1 remaining candidates, no matter who the other M remaining candidates are, then ''b'' is an M-seat Condorcet winner.

See also https://arxiv.org/abs/1701.08023.

Analogues to the Condorcet criterion have been proposed in non-voting contexts; it appears in many places when discussing how to aggregate ranked information. It has been used to discuss search engine rankings <ref>{{Cite web|url=http://www10.org/cdrom/papers/577/|title=Rank Aggregation Methods for the Web|last=|first=|date=|website=|url-status=live|archive-url=|archive-date=|access-date=}}</ref> and metasearch <ref>{{Cite web|url=http://www.ccs.neu.edu/home/jaa/IS4200.10X1/resources/condorcet.pdf|title=Condorcet Fusion for Improved Retrieval|last=|first=|date=|website=|url-status=live|archive-url=|archive-date=|access-date=}}</ref>. The concept of a [[Smith set ranking]] (which is sometimes referred to as the "extended/generalised Condorcet criterion") helps structure how the ranking of all options should be, rather than only the winner.