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Concept
Condorcet loser criterion
Summary
In single-winner voting system theory, the Condorcet loser criterion (CLC) is a measure for differentiating voting systems. It implies the majority loser criterion but does not imply the Condorcet winner criterion.
A voting system complying with the Condorcet loser criterion will never allow a Condorcet loser to win. A Condorcet loser is a candidate who can be defeated in a head-to-head competition against each other candidate. (Not all elections will have a Condorcet loser since it is possible for three or more candidates to be mutually defeatable in different head-to-head competitions.)
Compliant methods include: two-round system, instant-runoff voting (AV), contingent vote, borda count, Schulze method, ranked pairs, and Kemeny-Young method. Any voting method that ends in a runoff passes the criterion, so long as all voters are able to express their preferences in that runoff i.e. STAR voting passes only when voters can always indicate their ranked preference in their scores; if there are more than 6 candidates, then this is impossible.
Noncompliant methods include: plurality voting, supplementary voting, Sri Lankan contingent voting, approval voting, range voting, Bucklin voting and minimax Condorcet.
The Smith criterion implies the Condorcet loser criterion, because no candidate in the Smith set can lose a head-to-head matchup against a candidate not in the Smith set.
Approval voting
The ballots for Approval voting do not contain the information to identify the Condorcet loser. Thus, Approval Voting cannot prevent the Condorcet loser from winning in some cases. The following example shows that Approval voting violates the Condorcet loser criterion.
Assume four candidates A, B, C and L with 3 voters with the following preferences:
The Condorcet loser is L, since every other candidate is preferred to him by 2 out of 3 voters.
There are several possibilities how the voters could translate their preference order into an approval ballot, i.e. where they set the threshold between approvals and disapprovals.
Official source
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