Condorcet loser criterion
In singlewinner voting system theory, the Condorcet loser criterion is a measure for differentiating voting systems. It implies the majority loser 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 headtohead 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 headtohead competitions. However, there is always a Smith loser set, which is the smallest group of candidates such that any of them can be defeated by any candidate not in the group.)
A slightly weaker (easier to pass) version is the majority Condorcet loser criterion, which requires that a candidate who can be defeated by a majority in a headtohead competition against each other candidate, lose. It is possible for a system, such as Majority Judgment, which allows voters not to state a preference between two candidates, to pass the MCLC but not the CLC.^{[citation needed]}
Compliant methods include: tworound system, instantrunoff voting (AV), contingent vote, borda count, Schulze method, ranked pairs, and KemenyYoung method.
Noncompliant methods include: plurality voting, supplementary voting, Sri Lankan contingent voting, approval voting, range voting, Bucklin voting and minimax Condorcet.
Contents
Examples[edit  edit source]
Approval voting[edit  edit source]
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:
# 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:
# 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[edit  edit source]
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:
Candidate 
 
L 
 
A 
 
B 
 

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:
# 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[edit  edit source]
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:
# 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:
X  
A  B  C  L  
Y  A  [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 (wontiedlost):  201  201  201  003  
worst pairwise defeat (winning votes):  6  6  6  5  
worst pairwise defeat (margins):  3  3  3  1  
worst pairwise opposition:  6  6  6  5 
 [X] indicates voters who preferred the candidate listed in the column caption to the candidate listed in the row caption
 [Y] indicates voters who preferred the candidate listed in the row caption to the candidate listed in the column caption
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[edit  edit source]
Imagine that Tennessee is having an election on the location of its capital. The population of Tennessee is concentrated around its four major cities, which are spread throughout the state. For this example, suppose that the entire electorate lives in these four cities, and that everyone wants to live as near the capital as possible.
The candidates for the capital are:
 Memphis, the state's largest city, with 42% of the voters, but located far from the other cities
 Nashville, with 26% of the voters, near the center of Tennessee
 Knoxville, with 17% of the voters
 Chattanooga, with 15% of the voters
The preferences of the voters would be divided like this:
42% of voters (close to Memphis) 
26% of voters (close to Nashville) 
15% of voters (close to Chattanooga) 
17% of voters (close to Knoxville) 





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[edit  edit source]
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  

# 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:
# 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.
Ranked pairs[edit  edit source]
Ranked pairs work by "locking in" strong victories, starting with the strongest, unless that would contradict an earlier lock. Assume that the Condorcet loser is X. For X to win, ranked pairs must lock a preference of X over some other candidate Y (for at least one Y) before it locks Y over X. But since X is the Condorcet loser, the victory of Y over X will be greater than that of X over Y, and therefore Y over X will be locked first, no matter what other candidate Y is. Therefore, X cannot win.
Notes[edit  edit source]
Any voting method that operates by having or being able to be reduced to a final runoff will always pass the Condorcet loser criterion, since either the Condorcet loser is not in the runoff and thus can't win, or is in the runoff and is by definition defeated by their opponent; this includes IRV, STAR, Toptwo runoff, etc. A generalization of the Condorcet loser criterion is the Smith loser criterion: a candidate in the Smith loser set (the smallest group of candidates such that more voters prefer anyone not in the group over anyone in the group) should never win unless all candidates are in the Smith loser set. The Smith criterion implies the Smith loser criterion, since the Smith set only overlaps with the Smith loser set when both sets include all candidates.^{[1]} The Smith loser criterion implies the Condorcet loser criterion, since a Condorcet loser, when they exist, will always be the only candidate in the Smith loser set. Many nonSmith efficient methods that pass the Condorcet loser criterion fail the Smith loser criterion.
See also[edit  edit source]
 ↑ Talk:Condorcet ranking Look for "I think I can prove" on the message with a timestamp date of 21 February 2020.