Later-no-harm criterion

Voting system

Name	Comply?
Plurality	Yes
Two-round system	Yes
Nonpartisan primary	Yes
Partisan primary	Yes
Instant-runoff voting	Yes
MMPO	Yes
Descending solid coalitions	Yes
Anti-plurality	Yes
Approval voting	No
Borda count	No
Dodgson's method	No
Copeland's method	No
Kemeny–Young method	No
Ranked Pairs	No
Schulze method	No
Score voting	No
Majority judgment	No

The later-no-harm property is a property of some voting systems first described by Douglas Woodall, who defined it as saying that increasing the rating of a later candidate should not hurt a candidate placed earlier on the ballot. For example, say a group of voters have ranked Alice 1^st and Bob 3^rd. Then, later-no-harm says that if this group increases Bob's rating from 3^rd-place to 2^nd, this will not allow Bob to defeat Alice.

Later-no-harm is often confused with resistance to a kind of strategic voting called truncation or bullet voting; however, such strategies are also effective in instant-runoff voting, along with other systems that pass later-no-harm but fail positive responsiveness.

Later-no-harm is a defining characteristic of plurality and similar systems that compare remaining candidates by how many ballots consider each candidate their "favorite". In later-no-harm systems, the results either do not depend on lower preferences at all (as in plurality) or only depend on them in situations where all higher preferences have been exhausted (as in instant-runoff voting). This tends to favor candidates with strong-but-narrow support over consensus candidates, which can lead to a phenomenon known as center-squeeze.

By contrast, systems that do not satisfy later-no-harm, such as cardinal and Condorcet methods, tend to favor consensus options (broad, moderate support) over pluralitarian ones (narrow, strong support). Later-no-harm systems are unable to consider weak (secondary) preferences when comparing candidates early on, leading to a phenomenon called center squeeze. This has led many social choice theorists to question whether the property is desirable in the first place or should instead be seen as a negative (anti-criterion).

Later-no-harm methods

The plurality vote, two-round system, instant-runoff voting, and descending solid coalitions satisfy the later-no-harm criterion.

Plurality voting is considered to satisfy later-no-harm because later preferences are not taken into account at all; thus, plurality can be thought of as a ranked voting system where only the first preference matters.

Non-LNH methods

Nearly all , including score voting, highest medians, Borda count, and all Condorcet method do not satisfy later-no-harm. The Condorcet criterion is incompatible with later-no-harm (assuming the resolvability criterion, i.e. any tie can be removed by some single voter changing their rating).

Plurality-at-large voting, which allows the voter to select multiple candidates, does not satisfy later-no-harm when used to fill two or more seats in a single district.

Examples

Anti-plurality

Main article: Anti-plurality voting

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 (marked bold) submit a truncated preference listing A > B = C by apportioning the possible orderings for B and C equally. Each vote is counted ${\tfrac {1}{2}}$ A > B > C, and ${\tfrac {1}{2}}$ 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, Anti-plurality doesn't satisfy the Later-no-harm criterion when truncated ballots are considered to apportion the last place vote amongst unlisted candidates equally.

Approval voting

Main article: Approval voting

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: # 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 doesn't satisfy the Later-no-harm criterion.

Borda count

Main article: 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: # 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 doesn't satisfy the Later-no-harm criterion.

Coombs' method

Main article: 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.

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 (marked bold) submit a truncated preference listing A > B = C by apportioning the possible orderings for B and C equally. Each vote is counted ${\tfrac {1}{2}}$ A > B > C, and ${\tfrac {1}{2}}$ 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 doesn't satisfy the Later-no-harm criterion when truncated ballots are considered to apportion the last place vote amongst unlisted candidates equally.

Copeland

Main article: Copeland's method

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: # 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: Pairwise election results 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 (won-tied-lost): 1-2-0 2-1-0 1-1-1 0-0-3 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: Pairwise election results 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 (won-tied-lost): 1-2-0 0-3-0 0-3-0 0-2-1 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 doesn't satisfy the Later-no-harm criterion.

Dodgson's method

Main article: 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 (marked bold) submit a truncated preference listing A > B = C by apportioning the possible orderings for B and C equally. Each vote is counted ${\tfrac {1}{2}}$ A > B > C, and ${\tfrac {1}{2}}$ 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 Pairwise Contests 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 Pairwise Contests 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 doesn't satisfy the Later-no-harm criterion when truncated ballots are considered to apportion the possible rankings amongst unlisted candidates equally.

Kemeny–Young method

Main article: Kemeny–Young method

Examples

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: # 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 Kemeny-Young method doesn't satisfy the Later-no-harm criterion. Note, that IRV - by ignoring the Condorcet winner C in the first case - would choose A in both cases.

Majority judgment

Main article: 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: 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      ↓ Median point A B                 Excellent      Good      Fair      Poor   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      ↓ Median point A B                 Excellent      Good      Fair      Poor   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 highest-rated favorite A from loser to winner. Thus, majority judgment doesn't satisfy the Later-no-harm criterion. Note, that this only depends on the handling of not-rated candidates. If all not-rated candidates would receive the best-possible rating, majority judgment would satisfy the later-no-harm criterion, but not later-no-help. If instead majority judgment ignored unrated candidates and computed the median solely from the values that the voters expressed, a voter in a later-no-harm scenario could only help candidates for whom the voter has a higher honest opinion than the society has.

Minimax

Main article: Minimax Condorcet

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 (pairwise opposition) 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: # 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: Pairwise election results 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 (won-tied-lost): 2-0-1 1-0-2 3-0-0 0-0-3 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: Pairwise election results 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 (won-tied-lost): 2-0-1 1-0-2 2-0-1 1-0-2 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 doesn't satisfy the Later-no-harm criterion.

Ranked pairs

Main article: Ranked pairs

Examples

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. 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 Condorcet-compliant method, as the Condorcet and later-no-harm criteria are incompatible. Minimax is generally classed as a Condorcet method, but the pairwise opposition variant which meets later-no-harm doesn't actually satisfy the Condorcet criterion.

Score voting

Main article: Score voting

Examples

This example shows that Score voting violates the Later-no-harm criterion, and how in theory the tactical voting strategy called bullet voting could be a response. 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 Score 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 Score voting winner. Conclusion By withholding his opinion on the less-preferred B candidate, the voter caused his first preference (A) to win the election. This both proves that Score voting is not immune to strategic voting (as no system is), and shows that Score voting doesn't satisfy the Later-no-harm criterion. It should also be noted that this effect can only occur if the voter's expressed opinion on B (the less-preferred candidate) is higher than the opinion of the electorate about that later preference is. Thus, a later-no-harm scenario can only turn a candidate into a winner if the voter likes that candidate more than the rest of the electorate does.

Schulze method

Main article: 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: # 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: Matrix of pairwise preferences 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: Matrix of pairwise preferences 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). Strengths of the strongest paths 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 doesn't satisfy the Later-no-harm criterion.

Criticism

Douglas Woodall writes:

[U]nder STV 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, although it has to be said that 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".

See also

• Later-no-help criterion
• Monotonicity criterion

References

• 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