Condorcet loser criterion

In single-winner voting system theory, the Condorcet loser criterion 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.

Compliance

Compliant methods include: two-round system, instant-runoff voting, contingent vote, Borda count, Schulze method, ranked pairs, and Kemeny 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

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 only A or A and B or A, B and L or all candidates or 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 the 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 Approval winner. However, since approval voting elects the Condorcet loser in the example, approval voting fails the Condorcet loser criterion.

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:
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

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, all three Minimax methods elect the same winners:

indicates voters who preferred the candidate listed in the column caption to the candidate listed in the row caption
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

Here, Memphis has a plurality of the first preferences, so would be the winner under simple plurality voting. However, the majority 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

This example shows that range voting violates the Condorcet loser criterion. Assume two candidates A and L and 3 voters with the following opinions:
The total scores would be:
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.