Condorcet loser criterion explained

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.[1] (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.)

Compliance

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.

Examples

Approval voting

See main article: 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:

  1. of voters
Preferences
1 A > B > L > C
1 B > C > L > A
1 C > A > L > B
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. For example, the first voter could approve (i) only A or (ii) A and B or (iii) A, B and L or (iv) all candidates or (v) none of them. Let's assume, that all voters approve three candidates and disapprove only the last one. The approval ballots would be:

  1. of voters
Approvals Disapprovals
1 A, B, L C
1 B, C, L A
1 A, C, L B

Result: L is approved by all three voters, whereas the three other candidates are approved by only two voters. Thus, the Condorcet loser L is elected Approval winner.

Note, that if any voter would set the threshold between approvals and disapprovals at any other place, the Condorcet loser L would not be the (single) Approval winner. However, since Approval voting elects the Condorcet loser in the example, Approval voting fails the Condorcet loser criterion.

Majority Judgment

See main article: Majority Judgment.

This example shows that Majority Judgment violates the Condorcet loser criterion. Assume three candidates A, B and L and 3 voters with the following opinions:

Candidates/
# of voters
A B L
1 Excellent Bad Good
1 Bad Excellent Good
1 Fair Poor Bad

The sorted ratings would be as follows:

align=right Candidate   
width=49%  width=2% textalign=center width=49% Median point
align=right L
bgcolor=YellowGreen width=67%  bgcolor=red width=33%
align=right A
bgcolor=green width=33%  bgcolor=yellow width=34%  bgcolor=red width=33%
align=right B
bgcolor=green width=33%  bgcolor=orange width=34%  bgcolor=red width=33%
  
 
      bgcolor=green   Excellent  bgcolor=YellowGreen   Good  bgcolor=Yellow   Fair  bgcolor=Orange   Poor  bgcolor=Red   Bad  

L has the median rating "Good", A has the median rating "Fair" and B has the median rating "Poor". Thus, L is the Majority Judgment winner.

Now, the Condorcet loser is determined. If all informations are removed that are not considered to determine the Condorcet loser, we have:

  1. of voters
Preferences
1 A > L > B
1 B > L > A
1 A > B > L
A is preferred over L by two voters and B is preferred over L by two voters. Thus, L is the Condorcet loser.

Result: L is the Condorcet loser. However, while the voter least preferring L also rates A and B relatively low, the other two voters rate L close to their favorites. Thus, L is elected Majority Judgment winner. Hence, Majority Judgment fails the Condorcet loser criterion.

Minimax

See main article: Minimax Condorcet.

This example shows that the Minimax method violates the Condorcet loser criterion. Assume four candidates A, B, C and L with 9 voters with the following preferences:

  1. of voters
Preferences
1 A > B > C > L
1 A > B > L > C
3 B > C > A > L
1 C > L > A > B
1 L > A > B > C
2 L > C > A > B

Since all preferences are strict rankings (no equals are present), all three Minimax methods (winning votes, margins and pairwise opposite) elect the same winners:

Pairwise election results
X
ABCL
YA[X] 3
[Y] 6
[X] 6
[Y] 3
[X] 4
[Y] 5
B[X] 6
[Y] 3
[X] 3
[Y] 6
[X] 4
[Y] 5
C[X] 3
[Y] 6
[X] 6
[Y] 3
[X] 4
[Y] 5
L[X] 5
[Y] 4
[X] 5
[Y] 4
[X] 5
[Y] 4
Pairwise election results (won-tied-lost):2-0-12-0-12-0-10-0-3
worst pairwise defeat (winning votes):6665
worst pairwise defeat (margins):3331
worst pairwise opposition:6665

Result: L loses against all other candidates and, thus, is Condorcet loser. However, the candidates A, B and C form a cycle with clear defeats. L benefits from that since it loses relatively closely against all three and therefore L's biggest defeat is the closest of all candidates. Thus, the Condorcet loser L is elected Minimax winner. Hence, the Minimax method fails the Condorcet loser criterion.

Plurality voting

See main article: Plurality voting system.

Here, Memphis has a plurality (42%) of the first preferences, so would be the winner under simple plurality voting. However, the majority (58%) of voters have Memphis as their fourth preference, and if two of the remaining three cities were not in the running to become the capital, Memphis would lose all of the contests 58–42. Hence, Memphis is the Condorcet loser.

Range voting

See main article: Range voting.

This example shows that Range voting violates the Condorcet loser criterion. Assume two candidates A and L and 3 voters with the following opinions:

Scores
  1. of voters
A L
2 6 5
1 0 10
The total scores would be:
Scores
candidate Sum Average
A 12 4
L 20 6.7
Hence, L is the Range voting winner.

Now, the Condorcet loser is determined. If all informations are removed that are not considered to determine the Condorcet loser, we have:

  1. of voters
Preferences
2 A > L
1 L > A
Thus, L would be the Condorcet loser.

Result: L is preferred only by one of the three voters, so L is the Condorcet loser. However, while the two voters preferring A over L rate both candidates nearly equal and L's supporter rates him clearly over A, L is elected Range voting winner. Hence, Range voting fails the Condorcet loser criterion.

See also

Notes and References

  1. https://arxiv.org/pdf/1801.05911 "We say that an alternative is a Condorcet loser if it would be defeated by every other alternative in a kind of one-on-one contest that takes place in a sequential pairwise voting with a fixed agenda4.– Condorcet loser criterion (CLC), [...] we say that a social choice procedure satisfies the Condorcet loser criterion (CLC) provided that a Condorcet loser is never among the social choices."