The later-no-harm 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 less-preferred candidate can not cause a more-preferred candidate to lose. Voting systems that fail the later-no-harm criterion are vulnerable to the tactical voting strategy called bullet voting, which can deny victory to a sincere Condorcet winner.
Checking for satisfaction of the Later-no-harm 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. Later-no-harm presumes that later preferences are added to the ballot sequentially, so that candidates already listed are preferred to a candidate added later.
Examples
Anti-plurality
Anti-plurality elects the candidate the fewest voters rank last when submitting a complete ranking of the candidates. Later-No-Harm can be considered not applicable to Anti-Plurality if the method is assumed to not accept truncated preference listings from the voter. On the other hand, Later-No-Harm can be applied to Anti-Plurality if the method is assumed to apportion the last place vote among unlisted candidates equally, as shown in the example below.
Examples
-
-
; Truncated Ballot Profile Assume four voters 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:
Since Approval voting does not allow voters to differentiate their views about candidates for whom they choose to vote and the later-no-harm 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 later-no-harm 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. Under Approval voting, this may in some cases encourage the tactical voting strategy called bullet voting.
Examples
-
-
This can be seen with the following example with two candidates A and B and 3 voters:
Borda count
Examples
-
-
This example shows that the Borda count violates the Later-no-harm criterion. Assume three candidates A, B and C and 5 voters with the following preferences:
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. Later-No-Harm 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, Later-No-Harm 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.
Examples
-
-
;Truncated Ballot Profile Assume ten voters 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:
Copeland
Examples
-
-
This example shows that Copeland's method violates the Later-no-harm criterion. Assume four candidates A, B, C and D with 4 potential voters and the following preferences:
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 fewest ordinal preference swaps on voters' ballots. Later-No-Harm 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, Later-No-Harm can be applied to Dodgson if the method is assumed to apportion possible rankings among unlisted candidates equally, as shown in the example below.
Examples
-
-
; Truncated Ballot Profile Assume ten voters 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:
This example shows that the Kemeny–Young method violates the Later-no-harm criterion. Assume three candidates A, B and C and 9 voters with the following preferences:
Majority judgment
Examples
-
-
-
Considering, that an unrated candidate is assumed to be receiving the worst possible rating, this example shows that majority judgment violates the later-no-harm criterion. Assume two candidates A and B with 3 potential voters and the following ratings:
Minimax
Examples
-
-
This example shows that the Minimax method violates the Later-no-harm criterion in its two variants winning votes and margins. Note that the third variant of the Minimax method meets the later-no-harm criterion. Since all the variants are identical if equal ranks are not allowed, there can be no example for Minimax's violation of the later-no-harm criterion without using equal ranks. Assume four candidates A, B, C and D and 23 voters with the following preferences:
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 doesn't satisfy the Later-no-harm criterion.
For example, in an election conducted using the Condorcet compliant method Ranked pairs the following votes are cast:
Range voting
Schulze method
Examples
-
-
This example shows that the Schulze method doesn't satisfy the Later-no-harm criterion. Assume three candidates A, B and C and 16 voters with the following preferences:
Hide later preferences
Assume now that the three voters supporting A 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
d
d
d
5
7
d
6
6
d
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.
p
p
p
p
7
7
p
6
6
p
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 doesn't satisfy the Later-no-harm criterion.
Criticism
Woodall, author of the Later-no-harm writes: Warren Smith writes that the Later-no-harm criterion is "a silly criterion" saying that "objectively, LNH is not even a desirable property with honest voters". He argues that rating a candidate higher should allow the possibility of that candidate winning if the candidates collective ratings are high enough. The Center for Election Science, founded by Smith, also voices their opinion that the name itself is "misleading" and raises the concern that while "a voter can't harm a candidate by ranking additional less preferred candidates,.. voters can still hurt themselves by doing so. This begs the question of whether the later-no-harm criterion actually has value."
