Laternoharm criterion
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 does not cause a morepreferred candidate to lose. Voting systems that fail the laternoharm criterion are vulnerable to a certain type of tactical voting known as bullet voting, which can be used to deny victory to a sincere Condorcet winner.
Complying methods
Tworound system, Single transferable vote, Instant Runoff Voting, 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.
When a voter is allowed to choose only one preferred candidate, as in plurality voting, laternoharm can be either considered satisfied (as the voter's later preferences can not harm their chosen candidate) or not applicable.
Noncomplying methods
Approval voting, Borda count, Range voting, Majority Judgment, Bucklin voting, Ranked Pairs, Schulze method, KemenyYoung method, Copeland's method, and Nanson's method do not satisfy laternoharm. The Condorcet criterion is incompatible with laternoharm (assuming the discrimination axiom, according to which any tie can be removed by some single voter changing her rating).^{[1]}
Pluralityatlarge voting which allows the voter to select up to a certain number of candidates, fails laternoharm when used to fill two or more seats in a single district.
Checking Compliance
Checking for failures of the Laternoharm criterion requires ascertaining the probability of a voter’s preferred candidate being elected before and after adding a later preference to the ballot, to determine any decrease in probability. Laternoharm presumes that later preferences are added to the ballot sequentially, so that candidates already listed are preferred to a candidate added later.
Examples
Antiplurality
Antiplurality elects the candidate the least number of voters rank last when submitting a complete ranking of the candidates.
LaterNoHarm can be considered not applicable to AntiPlurality if the method is assumed to not accept truncated preference listings from the voter. On the other hand, LaterNoHarm can be applied to AntiPlurality if the method is assumed to apportion the last place vote among unlisted candidates equally, as shown in the example below.
Truncated Ballot Profile
Assume four voters (marked bold) submit a truncated preference listing A > B = C by apportioning the possible orderings for B and C equally. Each vote is counted A > B > C, and A > C > B:
# of voters  Preferences 

2  A ( > B > C) 
2  A ( > C > B) 
1  B > A > C 
1  B > C > A 
1  C > A > B 
1  C > B > A 
Result: A is listed last on 2 ballots; B is listed last on 3 ballots; C is listed last on 3 ballots. A is listed last on the least ballots. A wins.
Adding Later Preferences
Now assume that the four voters supporting A (marked bold) add later preference C, as follows:
# of voters  Preferences 

4  A > C > B 
1  B > A > C 
1  B > C > A 
1  C > A > B 
1  C > B > A 
Result: A is listed last on 2 ballots; B is listed last on 5 ballots; C is listed last on 1 ballot. C is listed last on the least ballots. C wins. A loses.
Conclusion
The four voters supporting A decrease the probability of A winning by adding later preference C to their ballot, changing A from the winner to a loser. Thus, Antiplurality fails the Laternoharm criterion when truncated ballots are considered to apportion the last place vote amongst unlisted candidates equally.
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.
Coombs' method
Coombs' method repeatedly eliminates the candidate listed last on most ballots, until a winner is reached. If at any time a candidate wins an absolute majority of first place votes among candidates not eliminated, that candidate is elected.
LaterNoHarm can be considered not applicable to Coombs if the method is assumed to not accept truncated preference listings from the voter. On the other hand, LaterNoHarm can be applied to Coombs if the method is assumed to apportion the last place vote among unlisted candidates equally, as shown in the example below.
Truncated Ballot Profile
Assume ten voters (marked bold) submit a truncated preference listing A > B = C by apportioning the possible orderings for B and C equally. Each vote is counted A > B > C, and A > C > B:
# of voters  Preferences 

5  A ( > B > C) 
5  A ( > C > B) 
14  A > B > C 
13  B > C > A 
4  C > B > A 
9  C > A > B 
Result: A is listed last on 17 ballots; B is listed last on 14 ballots; C is listed last on 19 ballots. C is listed last on the most ballots. C is eliminated, and A defeats B pairwise 33 to 17. A wins.
Adding Later Preferences
Now assume that the ten voters supporting A (marked bold) add later preference C, as follows:
# of voters  Preferences 

10  A > C > B 
14  A > B > C 
13  B > C > A 
4  C > B > A 
9  C > A > B 
Result: A is listed last on 17 ballots; B is listed last on 19 ballots; C is listed last on 14 ballots. B is listed last on the most ballots. B is eliminated, and C defeats A pairwise 26 to 24. A loses.
Conclusion
The ten voters supporting A decrease the probability of A winning by adding later preference C to their ballot, changing A from the winner to a loser. Thus, Coombs' method fails the Laternoharm criterion when truncated ballots are considered to apportion the last place vote amongst unlisted candidates equally.
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.
Dodgson's method
Dodgson's' method elects a Condorcet winner if there is one, and otherwise elects the candidate who can become the Condorcet winner after the least number of ordinal preference swaps on voters' ballots.
LaterNoHarm can be considered not applicable to Dodgson if the method is assumed to not accept truncated preference listings from the voter. On the other hand, LaterNoHarm can be applied to Dodgson if the method is assumed to apportion possible rankings among unlisted candidates equally, as shown in the example below.
Truncated Ballot Profile
Assume ten voters (marked bold) submit a truncated preference listing A > B = C by apportioning the possible orderings for B and C equally. Each vote is counted A > B > C, and A > C > B:
# of voters  Preferences 

5  A ( > B > C) 
5  A ( > C > B) 
7  B > A > C 
7  C > B > A 
3  C > A > B 
Against A  Against B  Against C  

For A  13  17  
For B  14  12  
For C  10  15 
Result: There is no Condorcet winner. A is the Dodgson winner, because A becomes the Condorcet Winner with only two ordinal preference swaps (changing B > A to A > B). A wins.
Adding Later Preferences
Now assume that the ten voters supporting A (marked bold) add later preference B, as follows:
# of voters  Preferences 

10  A > B > C 
7  B > A > C 
7  C > B > A 
3  C > A > B 
Against A  Against B  Against C  

For A  13  17  
For B  14  17  
For C  10  10 
Result: B is the Condorcet Winner and the Dodgson winner. B wins. A loses.
Conclusion
The ten voters supporting A decrease the probability of A winning by adding later preference B to their ballot, changing A from the winner to a loser. Thus, Dodgson's method fails the Laternoharm criterion when truncated ballots are considered to apportion the possible rankings amongst unlisted candidates equally.
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  3  0  3 
X = A  Y = C  1  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  3  1  4  8 
A > C > B  1  3  2  6 
B > A > C  3  4  1  8 
B > C > A  4  3  5  12 
C > A > B  5  2  3  10 
C > B > A  2  5  3  10 
Result: The ranking B > C > A has the highest ranking score. Thus, B wins ahead of A and B.
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>B=C  25: B>A=C  26: C>B>A 
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; A, B, and C are all weak Condorcet winners and B is the Ranked pairs winner.
Suppose the 25 B voters give an additional preference to their second choice C, and the 25 C voters give an additional preference to their second choice A.
The votes are now:
49: A>B=C  25: B>C>A  26: C>B>A 
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, and by giving a second preference to candidate B, the 26 C voters have caused their first choice to succeed.
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 usually regard this as a very important property,^{[2]} 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.'"^{[3]}
See also
References
 ↑ Douglas Woodall (1997): Monotonicity of SingleSeat Election Rules, Theorem 2 (b)
 ↑ The Nonmajority Rule Desk (July 29, 2011). "Why Approval Voting is Unworkable in Contested Elections  FairVote". FairVote Blog. Retrieved 11 October 2016.
 ↑ 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
 Tony Anderson Solgard and Paul Landskroener, Bench and Bar of Minnesota, Vol 59, No 9, October 2002.
 Brown v. Smallwood, 1915