Condorcet loser criterion

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Condorcet loser criterion

Suppose the voters had been polled on their preferences among the candidates, and the following preferences in head-to-head matchups are produced between French Fries (FF), Hamburger (H), and Cookies (C) (FF>C shows the number of voters who prefer FF over C, for example):

FF>H:51, FF>C:60

H>FF:49, H>C:70

C>FF:40, C>H:20

If for each pair of candidates, we subtract the number of votes preferring the second candidate over the first from the number of votes preferring the first to the second, then we'll know which one won the head-to-head matchup.

(Margins)

FF>H:2, FF>C:20

H>FF:-2 (Loss), H>C:20

C>FF:-20 (Loss), C>H:-50 (Loss)

The Condorcet loser (if one exists) will be the candidate who got a minority of votes (as indicated by the negative margin) in all of their head-to-head matchups.

C (Cookies) is the CL here.

Example of finding the Condorcet loser

In single-winner voting system theory, the Condorcet loser criterion (or anti-Condorcet 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 (anti-Condorcet candidate) 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. 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.)

Compliant methods include: two-round system, instant-runoff voting (AV), contingent vote, borda count, Schulze method, ranked pairs, and Kemeny-Young method.

Noncompliant methods include: plurality voting, supplementary voting, Sri Lankan contingent voting, approval voting, range voting, Bucklin voting and minimax Condorcet.

Implications

In an election with only two candidates, the Condorcet loser criterion implies the majority criterion. Given a three-candidate Condorcet cycle, it's always possible to eliminate a candidate who didn't win so that the winner changes. Thus the Condorcet loser criterion is incompatible with independence of irrelevant alternatives.

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 (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

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

?

Median point

L

A

B

Excellent

Good

Fair

Poor

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:

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

Pairwise election results
		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 (won-tied-lost):		2-0-1	2-0-1	2-0-1	0-0-3
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

Tennessee's four cities are spread throughout the state

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)
Memphis Nashville Chattanooga Knoxville	Nashville Chattanooga Knoxville Memphis	Chattanooga Knoxville Nashville Memphis	Knoxville Chattanooga Nashville Memphis

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

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

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.

Runoffs and honest Condorcet loser compliance

Any voting method or algorithm that operates by eventually reducing the number of candidates to two, where the candidate who beats the other one-on-one becomes the winner of the election, will always pass the Condorcet loser criterion. By definition, the Condorcet loser can't win the finalist match-up even if chosen to be one of the two finalists. Such methods include IRV, STAR, the contingent vote, etc.

Voting methods with a manual two-candidate runoff round pass a strategy-immune Condorcet loser criterion: if the voters are honest, they pass Condorcet loser as usual. But if the voters are strategic, and everybody participates in both rounds, then the method as a whole will never elect the honest Condorcet loser. This happens because two-candidate majority elections are strategy-proof: there's never any incentive to vote for X when you prefer Y to X, if the only two candidates are Y and X. The honest Condorcet loser thus can't win even if he gets to the second round. One example of such a method is top-two runoff.

If the voters change their minds between the rounds, the "honest Condorcet loser" is the candidate who, according to the voters' opinion at the time they vote in the second round, is the Condorcet loser.

Notes

Note that it is very unlikely for a candidate with any significant amount of support to be a CL, even if they have significantly less support than other candidates, because they will still beat any candidates with negligible support i.e. Write-in candidates marked on only one ballot. Because of this, it is common to discuss the CL criterion when only looking at matchups between the major candidates. ^[1]

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.^[2] 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 non-Smith efficient methods that pass the Condorcet loser criterion fail the Smith loser criterion.