Later-no-harm criterion

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 does not cause a more-preferred candidate to lose. Voting systems that fail the later-no-harm 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

Two-round 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 later-no-harm criterion.

When a voter is allowed to choose only one preferred candidate, as in plurality voting, later-no-harm 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, Kemeny-Young method, Copeland's method, and Nanson's method do not satisfy later-no-harm. The Condorcet criterion is incompatible with later-no-harm (assuming the discrimination axiom, according to which any tie can be removed by some single voter changing her rating).^[1]

Plurality-at-large voting which allows the voter to select up to a certain number of candidates, fails later-no-harm when used to fill two or more seats in a single district.

Checking Compliance

Checking for failures 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 least number of 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.

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, Anti-plurality fails the Later-no-harm 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 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.

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 Later-no-harm criterion.

Borda count

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 fails the Later-no-harm 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.

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.

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 Later-no-harm 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 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 fails the Later-no-harm 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.

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.

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

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 fails the Later-no-harm 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 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 fails 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

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 fails the Later-no-harm criterion. Note, that majority judgment's failure to later-no-harm 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 fail later-no-help.

If majority judgment would just ignore not rated candidates and compute the median just from the values that the voters expressed, a failure to later-no-harm 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 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 fails the Later-no-harm 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 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 actually fails the Condorcet criterion.

Range voting

This example shows that Range voting violates the Later-no-harm 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 less-preferred 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 Later-no-harm 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 less-preferred candidate) is higher than the opinion of the society about that later preference is. Thus, a failure to later-no-harm 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 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 fails the Later-no-harm criterion.

Commentary

Woodall writes about Later-no-harm, "... 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

• Later-no-help criterion

References

• 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