Later-no-harm criterion

Voting system
Name	Comply?
Plurality	Yes^{[note 1]}
Two-round system	Yes
Nonpartisan primary	Yes
Partisan primary	Yes
Instant-runoff voting	Yes
MMPO	Yes^{[note 2]}
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. Intuitively, a voting system satisfies this property, increasing the rating of a later candidate should not hurt a candidate placed earlier on the ballot.^[1] For example, suppose that a voter has ranked the candidate Alice 1^st and another candidate Bob 3^rd. Then, later-no-harm says that if this voter increases Bob's rating from 3^rd-place to 2^nd, this will not allow Bob to defeat Alice.

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).

Later-no-harm methods[edit]

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[edit]

Nearly all voting methods other than first-past-the-post do not pass LNH, including score voting, highest medians, Borda count, and all Condorcet methods. 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).^[1]

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[edit]

Anti-plurality[edit]

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.

Borda count[edit]

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	32 + 01	6
B	2	3	0	22 + 31	7
C	0	2	3	02 + 21	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	32 + 01	6
B	2	0	3	22 + 01	4
C	0	2	3	02 + 21	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.

Copeland[edit]

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.

Schulze method[edit]

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[edit]

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[edit]

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,^[2] 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".^[3]

Notes[edit]

^ Plurality voting can be thought of as a ranked voting system that disregards preferences after the first; because all preferences other than the first are unimportant, plurality passes later-no-harm as traditionally defined.
^ Minimax can occasionally violate later-no-harm if tied ranks are allowed.

References[edit]

^ ^a ^b Douglas Woodall (1997): Monotonicity of Single-Seat Election Rules, Theorem 2 (b)
^ The Non-majority 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 [1]
Tony Anderson Solgard and Paul Landskroener, Bench and Bar of Minnesota, Vol 59, No 9, October 2002. [2]
Brown v. Smallwood, 1915

[p-1] Plurality voting can be thought of as a ranked voting system that disregards preferences after the first; because all preferences other than the first are unimportant, plurality passes later-no-harm as traditionally defined.

[m-2] Minimax can occasionally violate later-no-harm if tied ranks are allowed.

[:0-3] Douglas Woodall (1997): Monotonicity of Single-Seat Election Rules, Theorem 2 (b)

[4] The Non-majority Rule Desk (July 29, 2011). "Why Approval Voting is Unworkable in Contested Elections - FairVote". FairVote Blog. Retrieved 11 October 2016.

[5] Woodall, Douglas, Properties of Preferential Election Rules, Voting matters - Issue 3, December 1994

[note 1]

[note 2]

[1]

[2]

[3]

Later-no-harm methods[edit]

Non-LNH methods[edit]

Examples[edit]

Anti-plurality[edit]

Borda count[edit]

Copeland[edit]

Schulze method[edit]

Hide later preferences[edit]

Criticism[edit]

See also[edit]

Notes[edit]

References[edit]