Laternoharm criterion
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The laternoharm criterion is a voting system criterion formulated by Douglas Woodall. The criterion is satisfied if, in any election, a voter giving an additional ranking or positive rating to a lesspreferred candidate cannot cause a morepreferred candidate to lose.
Contents
 1 Complying methods
 2 Noncomplying methods
 3 Examples
 4 Commentary
 5 References
Complying methods
Single transferable vote (including traditional forms of Instant Runoff Voting and Contingent vote), Minimax Condorcet (a pairwise opposition variant which does not satisfy the Condorcet Criterion), and Descending Solid Coalitions, a variant of Woodall's Descending Acquiescing Coalitions rule, satisfy the laternoharm criterion.
Noncomplying methods
Approval voting, Borda count, Range voting, Schulze method and Bucklin voting do not satisfy laternoharm. The Condorcet criterion is incompatible with laternoharm.
When plurality is used to fill two or more seats in a single district (pluralityatlarge) it fails laternoharm.
The laternoharm criterion is by definition inapplicable to any voting system in which a voter is not allowed to express more than one choice, such as plurality voting and most party list forms of proportional representation.
Examples
Approval voting
Since Approval voting does not allow voters to differentiate their views about candidates for whom they choose to vote and the laternoharm criterion explicitly requires the voter's ability to express later preferences on the ballot, the criterion using this definition is not applicable for Approval voting.
However, if the laternoharm criterion is expanded to consider the preferences within the mind of the voter to determine whether a preference is "later" instead of actually expressing it as a later preference as demanded in the definition, Approval would not satisfy the criterion.
This can be seen with the following example with two candidates A and B and 3 voters:
# of voters  Preferences 

2  A > B 
1  B 
Express "later" preference
Assume that the two voters supporting A (marked bold) would also approve their later preference B.
Result: A is approved by two voters, B by all three voters. Thus, B is the Approval winner.
Hide "later" preference
Assume now that the two voters supporting A (marked bold) would not approve their last preference B on the ballots:
# of voters  Preferences 

2  A 
1  B 
Result: A is approved by two voters, B by only one voter. Thus, A is the Approval winner.
Conclusion
By approving an additional less preferred candidate the two A > B voters have caused their favourite candidate to lose. Thus, Approval voting fails the Laternoharm criterion.
Borda count
This example shows that the Borda count violates the Laternoharm criterion. Assume three candidates A, B and C and 5 voters with the following preferences:
# of voters  Preferences 

3  A > B > C 
2  B > C > A 
Express later preferences
Assume that all preferences are expressed on the ballots.
The positions of the candidates and computation of the Borda points can be tabulated as follows:
candidate  #1.  #2.  #last  computation  Borda points 

A  3  0  2  3*2 + 0*1  6 
B  2  3  0  2*2 + 3*1  7 
C  0  2  3  0*2 + 2*1  2 
Result: B wins with 7 Borda points.
Hide later preferences
Assume now that the three voters supporting A (marked bold) would not express their later preferences on the ballots:
# of voters  Preferences 

3  A 
2  B > C > A 
The positions of the candidates and computation of the Borda points can be tabulated as follows:
candidate  #1.  #2.  #last  computation  Borda points 

A  3  0  2  3*2 + 0*1  6 
B  2  0  3  2*2 + 0*1  4 
C  0  2  3  0*2 + 2*1  2 
Result: A wins with 6 Borda points.
Conclusion
By hiding their later preferences about B, the three voters could change their first preference A from loser to winner. Thus, the Borda count fails the Laternoharm criterion.
Copeland
This example shows that Copeland's method violates the Laternoharm criterion. Assume four candidates A, B, C and D with 4 potential voters and the following preferences:
# of voters  Preferences 

2  A > B > C > D 
1  B > C > A > D 
1  D > C > B > A 
Express later preferences
Assume that all preferences are expressed on the ballots.
The results would be tabulated as follows:
X  
A  B  C  D  
Y  A  [X] 2 [Y] 2 
[X] 2 [Y] 2 
[X] 1 [Y] 3 

B  [X] 2 [Y] 2 
[X] 1 [Y] 3 
[X] 1 [Y] 3 

C  [X] 2 [Y] 2 
[X] 3 [Y] 1 
[X] 1 [Y] 3 

D  [X] 3 [Y] 1 
[X] 3 [Y] 1 
[X] 3 [Y] 1 

Pairwise election results (wontiedlost):  120  210  111  003 
Result: B has two wins and no defeat, A has only one win and no defeat. Thus, B is elected Copeland winner.
Hide later preferences
Assume now, that the two voters supporting A (marked bold) would not express their later preferences on the ballots:
# of voters  Preferences 

2  A 
1  B > C > A > D 
1  D > C > B > A 
The results would be tabulated as follows:
X  
A  B  C  D  
Y  A  [X] 2 [Y] 2 
[X] 2 [Y] 2 
[X] 1 [Y] 3 

B  [X] 2 [Y] 2 
[X] 1 [Y] 1 
[X] 1 [Y] 1 

C  [X] 2 [Y] 2 
[X] 1 [Y] 1 
[X] 1 [Y] 1 

D  [X] 3 [Y] 1 
[X] 1 [Y] 1 
[X] 1 [Y] 1 

Pairwise election results (wontiedlost):  120  030  030  021 
Result: A has one win and no defeat, B has no win and no defeat. Thus, A is elected Copeland winner.
Conclusion
By hiding their later preferences, the two voters could change their first preference A from loser to winner. Thus, Copeland's method fails the Laternoharm criterion.
Instant runoff voting variant with majority requirement
Traditional forms of instant runoff voting satisfy the laternoharm criterion. But if a method permits incomplete ranking of candidates, and if a majority of initial round votes is required to win and avoid another election, that variant does not satisfy Laternoharm. A lower preference vote cast may create a majority for that lower preference, whereas if the vote was not cast, the election could fail, proceed to a runoff, repeated ballot or other process, and the favored candidate could possibly win.
Assume, the votes are as follows:
40: A  39: B>A  21: C 
In the Preferential voting method, described as an example in Robert's Rules of Order, elimination continues iteratively until "one pile contains more than half the ballots." Eliminating one of the two final candidate never changes who wins, but can change how many votes that final candidate receives. Thus, C would be eliminated, then B, and the B ballots would be counted for A, who would thereby obtain a majority and be elected.
Now, suppose the B voters would hide their second preference for A:
40: A  39: B  21: C 
This failure to win a majority in the final round results in a runoff between A and B, which B could win.
By adding a second preference vote for A, the B voters eliminated the election possibility for B. Thus, this variant of instant runoff voting with majority requirement fails the laternoharm criterion. In traditional IRV, A would have been elected by 40% of the voters, and the laternoharm criterion would not have been violated.
Also, compliance of LNH can be reestablished by stopping the elimination process when there are two candidates left. Applying this to the example, the second preferences of B are ignored either case. Thus, the B voters would not violate laternoharm by indicating A as a second choice.
Kemeny–Young method
This example shows that the Kemeny–Young method violates the Laternoharm criterion. Assume three candidates A, B and C and 9 voters with the following preferences:
# of voters  Preferences 

3  A > C > B 
1  A > B > C 
3  B > C > A 
2  C > A > B 
Express later preferences
Assume that all preferences are expressed on the ballots.
The Kemeny–Young method arranges the pairwise comparison counts in the following tally table:
All possible pairs of choice names 
Number of votes with indicated preference  

Prefer X over Y  Equal preference  Prefer Y over X  
X = A  Y = B  6  0  3 
X = A  Y = C  4  0  5 
X = B  Y = C  4  0  5 
The ranking scores of all possible rankings are:
Preferences  1. vs 2.  1. vs 3.  2. vs 3.  Total 

A > B > C  6  4  4  14 
A > C > B  4  6  5  15 
B > A > C  3  4  4  11 
B > C > A  4  3  5  12 
C > A > B  5  5  6  16 
C > B > A  5  5  3  13 
Result: The ranking C > A > B has the highest ranking score. Thus, the Condorcet winner C wins ahead of A and B.
Hide later preferences
Assume now that the three voters supporting A (marked bold) would not express their later preferences on the ballots:
# of voters  Preferences 

3  A 
1  A > B > C 
3  B > C > A 
2  C > A > B 
The Kemeny–Young method arranges the pairwise comparison counts in the following tally table:
All possible pairs of choice names 
Number of votes with indicated preference  

Prefer X over Y  Equal preference  Prefer Y over X  
X = A  Y = B  6  0  3 
X = A  Y = C  4  0  5 
X = B  Y = C  4  0  2 
The ranking scores of all possible rankings are:
Preferences  1. vs 2.  1. vs 3.  2. vs 3.  Total 

A > B > C  6  4  4  14 
A > C > B  4  6  2  12 
B > A > C  3  4  4  11 
B > C > A  4  3  5  12 
C > A > B  5  2  6  13 
C > B > A  2  5  3  10 
Result: The ranking A > B > C has the highest ranking score. Thus, A wins ahead of B and C.
Conclusion
By hiding their later preferences about B and C, the three voters could change their first preference A from loser to winner. Thus, the KemenyYoung method fails the Laternoharm criterion. Note, that IRV  by ignoring the Condorcet winner C in the first case  would choose A in both cases.
Majority Judgment
Considering, that an unrated candidate is assumed to be receiving the worst possible rating, this example shows that Majority Judgment violates the laternoharm criterion. Assume two candidates A and B with 3 potential voters and the following ratings:
Candidates/ # of voters 
A  B 

1  Excellent  Good 
1  Poor  Excellent 
1  Fair  Poor 
Express later preferences
Assume that all ratings are expressed on the ballots.
The sorted ratings would be as follows:
Candidate 


A 


B 



Result: A has the median rating of "Fair" and B has the median rating of "Good". Thus, B is elected Majority Judgment winner.
Hide later ratings
Assume now that the voter supporting A (marked bold) would not express his later ratings on the ballot. Note, that this is handled as if the voter would have rated that candidate with the worst possible rating "Poor":
Candidates/ # of voters 
A  B 

1  Excellent  (Poor) 
1  Poor  Excellent 
1  Fair  Poor 
The sorted ratings would be as follows:
Candidate 


A 


B 



Result: A has still the median rating of "Fair". Since the voter revoked his acceptance of the rating "Good" for B, B now has the median rating of "Poor". Thus, A is elected Majority Judgment winner.
Conclusion
By hiding his later rating for B, the voter could change his highestrated favorite A from loser to winner. Thus, Majority Judgment fails the Laternoharm criterion. Note, that Majority Judgment's failure to laternoharm only depends on the handling of notrated candidates. If all notrated candidates would receive the bestpossible rating, Majority Judgment would satisfy the laternoharm criterion, but fail laternohelp.
If Majority Judgment would just ignore not rated candidates and compute the median just from the values that the voters expressed, a failure to laternoharm could only help candidates for whom the voter has a higher honest opinion than the society has.
Minimax
This example shows that the Minimax method violates the Laternoharm criterion in its two variants winning votes and margins. Note that the third variant of the Minimax method (pairwise opposition) meets the laternoharm criterion. Since all the variants are identical if equal ranks are not allowed, there can be no example for Minimax's violation of the laternoharm criterion without using equal ranks. Assume four candidates A, B, C and D and 23 voters with the following preferences:
# of voters  Preferences 

4  A > B > C > D 
2  A = B = C > D 
2  A = B = D > C 
1  A = C > B = D 
1  A > D > C > B 
1  B > D > C > A 
1  B = D > A = C 
2  C > A > B > D 
2  C > A = B = D 
1  C > B > A > D 
1  D > A > B > C 
2  D > A = B = C 
3  D > C > B > A 
Express later preferences
Assume that all preferences are expressed on the ballots.
The results would be tabulated as follows:
X  
A  B  C  D  
Y  A  [X] 6 [Y] 9 
[X] 9 [Y] 8 
[X] 8 [Y] 11 

B  [X] 9 [Y] 6 
[X] 10 [Y] 9 
[X] 7 [Y] 10 

C  [X] 8 [Y] 9 
[X] 9 [Y] 10 
[X] 11 [Y] 12 

D  [X] 11 [Y] 8 
[X] 10 [Y] 7 
[X] 12 [Y] 11 

Pairwise election results (wontiedlost):  201  102  300  003  
worst pairwise defeat (winning votes):  9  10  0  12  
worst pairwise defeat (margins):  1  3  0  3  
worst pairwise opposition:  9  10  11  12 
 [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: C has the closest biggest defeat. Thus, C is elected Minimax winner for variants winning votes and margins. Note, that with the pairwise opposition variant, A is Minimax winner, since A has in no duel an opposition that equals the opposition C had to overcome in his victory against D.
Hide later preferences
Assume now that the four voters supporting A (marked bold) would not express their later preferences over C and D on the ballots:
# of voters  Preferences 

4  A > B 
2  A = B = C > D 
2  A = B = D > C 
1  A = C > B = D 
1  A > D > C > B 
1  B > D > C > A 
1  B = D > A = C 
2  C > A > B > D 
2  C > A = B = D 
1  C > B > A > D 
1  D > A > B > C 
2  D > A = B = C 
3  D > C > B > A 
The results would be tabulated as follows:
X  
A  B  C  D  
Y  A  [X] 6 [Y] 9 
[X] 9 [Y] 8 
[X] 8 [Y] 11 

B  [X] 9 [Y] 6 
[X] 10 [Y] 9 
[X] 7 [Y] 10 

C  [X] 8 [Y] 9 
[X] 9 [Y] 10 
[X] 11 [Y] 8 

D  [X] 11 [Y] 8 
[X] 10 [Y] 7 
[X] 8 [Y] 11 

Pairwise election results (wontiedlost):  201  102  201  102  
worst pairwise defeat (winning votes):  9  10  11  11  
worst pairwise defeat (margins):  1  3  3  3  
worst pairwise opposition:  9  10  11  11 
Result: Now, A has the closest biggest defeat. Thus, A is elected Minimax winner in all variants.
Conclusion
By hiding their later preferences about C and D, the four voters could change their first preference A from loser to winner. Thus, the variants winning votes and margins of the Minimax method fails the Laternoharm criterion.
Ranked pairs
For example in an election conducted using the Condorcet compliant method Ranked pairs the following votes are cast:
49: A  25: B  26: C>B 
B is preferred to A by 51 votes to 49 votes. A is preferred to C by 49 votes to 26 votes. C is preferred to B by 26 votes to 25 votes.
There is no Condorcet winner and B is the Ranked pairs winner.
Suppose the 25 B voters give an additional preference to their second choice C.
The votes are now:
49: A  25: B>C  26: C>B 
C is preferred to A by 51 votes to 49 votes. C is preferred to B by 26 votes to 25 votes. B is preferred to A by 51 votes to 49 votes.
C is now the Condorcet winner and therefore the Ranked pairs winner. By giving a second preference to candidate C the 25 B voters have caused their first choice to be defeated.
Similar examples can be constructed for any Condorcetcompliant method, as the Condorcet and laternoharm criteria are incompatible. Minimax is generally classed as a Condorcet method, but the pairwise opposition variant which meets laternoharm actually fails the Condorcet criterion.
Range voting
This example shows that Range voting violates the Laternoharm criterion. Assume two candidates A and B and 2 voters with the following preferences:
Scores  Reading  

# of voters  A  B  
1  10  8  Slightly prefers A (by 2) 
1  0  4  Slightly prefers B (by 4) 
Express later preferences
Assume that all preferences are expressed on the ballots.
The total scores would be:
candidate  Average Score 

A  5 
B  6 
Result: B is the Range voting winner.
Hide later preferences
Assume now that the voter supporting A (marked bold) would not express his later preference on the ballot:
Scores  Reading  

# of voters  A  B  
1  10    Greatly prefers A (by 10) 
1  0  4  Slightly prefers B (by 4) 
The total scores would be:
candidate  Average Score 

A  5 
B  4 
Result: A is the Range voting winner.
Conclusion
By withholding his opinion on the lesspreferred B candidate, the voter caused his first preference (A) to win the election. This both proves that Range voting is not immune to strategic voting, and shows that Range voting fails the Laternoharm criterion (albeit "harm" in this case means having a winner that less preferred only by a small margin).
This is an important effect to keep in mind when using Range voting in practice. Situations such as this are actually quite likely when voters are instructed to consider each candidate in isolation (then often called Score voting) which can produce ballots consisting of mostly high marks (such as picking a leader among friends) or mostly low marks (as the oppressed protesting a set of election choices).
It should also be noted that this effect can only occur if the voter's expressed opinion on B (the lesspreferred candidate) is higher than the opinion of the society about that later preference is. Thus, a failure to laternoharm can only turn a candidate into a winner, if the voter likes him more than (rest of) the society does.
Schulze method
This example shows that the Schulze method violates the Laternoharm criterion. Assume three candidates A, B and C and 16 voters with the following preferences:
# of voters  Preferences 

3  A > B > C 
1  A = B > C 
2  A = C > B 
3  B > A > C 
1  B > A = C 
1  B > C > A 
4  C > A = B 
1  C > B > A 
Express later preferences
Assume that all preferences are expressed on the ballots.
The pairwise preferences would be tabulated as follows:
d[*,A]  d[*,B]  d[*,C]  

d[A,*]  5  7  
d[B,*]  6  9  
d[C,*]  6  7 
Result: B is Condorcet winner and thus, the Schulze method will elect B.
Hide later preferences
Assume now that the three voters supporting A (marked bold) would not express their later preferences on the ballots:
# of voters  Preferences 

3  A 
1  A = B > C 
2  A = C > B 
3  B > A > C 
1  B > A = C 
1  B > C > A 
4  C > A = B 
1  C > B > A 
The pairwise preferences would be tabulated as follows:
d[*,A]  d[*,B]  d[*,C]  

d[A,*]  5  7  
d[B,*]  6  6  
d[C,*]  6  7 
Now, the strongest paths have to be identified, e.g. the path A > C > B is stronger than the direct path A > B (which is nullified, since it is a loss for A).
p[*,A]  p[*,B]  p[*,C]  

p[A,*]  7  7  
p[B,*]  6  6  
p[C,*]  6  7 
Result: The full ranking is A > C > B. Thus, A is elected Schulze winner.
Conclusion
By hiding their later preferences about B and C, the three voters could change their first preference A from loser to winner. Thus, the Schulze method fails the Laternoharm criterion.
Commentary
Woodall writes about Laternoharm, "... under STV [single transferable vote] the later preferences on a ballot are not even considered until the fates of all candidates of earlier preference have been decided. Thus a voter can be certain that adding extra preferences to his or her preference listing can neither help nor harm any candidate already listed. Supporters of STV^{[who?]} usually regard this as a very important property, although not everyone agrees; the property has been described (by Michael Dummett, in a letter to Robert Newland) as 'quite unreasonable', and (by an anonymous referee) as 'unpalatable.'"^{[1]}
References
 ↑ Woodall, Douglas, Properties of Preferential Election Rules, Voting matters  Issue 3, December 1994
 D R Woodall, "Properties of Preferential Election Rules", Voting matters, Issue 3, December 1994 [1]
 Tony Anderson Solgard and Paul Landskroener, Bench and Bar of Minnesota, Vol 59, No 9, October 2002. [2]
 Brown v. Smallwood, 1915
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