Later-no-harm criterion

Property of electoral systems

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Voting system
Name	Comply?
Plurality	Yes^{[note 1]}
Two-round system	Yes
Partisan primary	Yes
Instant-runoff voting	Yes
Minimax Opposition	Yes
DSC	Yes
Anti-plurality	Yes
Approval	No
Borda	No
Dodgson	No
Copeland	No
Kemeny–Young	No
Ranked Pairs	No
Schulze	No
Score	No
Majority judgment	No

Later-no-harm is a property of some ranked-choice voting systems, first described by Douglas Woodall. In later-no-harm systems, modifying the rating or rank of a candidate ranked below the winner of an election cannot change the result.^[1]

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 if all higher preferences have been exhausted (as in instant-runoff).^[2]^[3] This tends to favor candidates with strong (but narrow) support over candidates closer to the center of public opinion, which can lead to a phenomenon known as center-squeeze.^[4]^[5]^[6] Cardinal and Condorcet methods, by contrast, tend to select candidates whose ideology is a closer match to that of the median voter.^[4]^[5]^[6] 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 property.^[7]^[6]^[8]

Later-no-harm is sometimes confused with resistance to a kind of strategic voting called truncation or bullet voting.^[9] However, later-no-harm does not provide resistance to such voting strategies; some systems (like instant runoff) that pass later-no-harm but fail participation are still vulnerable to truncation and bullet voting.^[7]^[10]

Example

Say a group of voters ranks Alice 2^nd and Bob 6^th, and Alice wins the election. In the next election, Bob focuses on expanding his appeal with this group of voters, but does not manage to defeat Alice—Bob's rating increases from 6^th-place to 3^rd. Later-no-harm says that this increased support from Alice's voters should not allow Alice to win.^[1]

Later-no-harm methods

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

First-preference plurality satisfies later-no-harm trivially, because later preferences (every preference after the first) are not taken into account at all.

Non-LNH methods

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

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 (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

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

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

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".^[11]

Notes

^ 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.

References

^ ^a ^b ^c Douglas Woodall (1997): Monotonicity of Single-Seat Election Rules, Theorem 2 (b)
^ Lewyn, Michael (2012). "Two Cheers for Instant Runoff Voting". 6 Phoenix L. Rev. 117. Rochester, NY. third place Candidate C is a centrist who is in fact the second choice of Candidate A's left-wing supporters and Candidate B's right-wing supporters. ... In such a situation, Candidate C would prevail over both Candidates A ... and B ... in a one-on-one runoff election. Yet, Candidate C would not prevail under IRV because he or she finished third and thus would be the first candidate eliminated
^ Stensholt, Eivind (2015-10-07). "What Happened in Burlington?". Discussion Papers: 13. There is a Condorcet ranking according to distance from the center, but Condorcet winner M, the most central candidate, was squeezed between the two others, got the smallest primary support, and was eliminated.
^ ^a ^b Hillinger, Claude (2005). "The Case for Utilitarian Voting". SSRN Electronic Journal. doi:10.2139/ssrn.732285. ISSN 1556-5068. S2CID 12873115. Retrieved 2022-05-27.
^ ^a ^b Merrill, Samuel (1984). "A Comparison of Efficiency of Multicandidate Electoral Systems". American Journal of Political Science. 28 (1): 23. doi:10.2307/2110786. ISSN 0092-5853. However, squeezed by surrounding opponents, a centrist candidate may receive few first-place votes and be eliminated under Hare.
^ ^a ^b ^c Merrill, Samuel (1985). "A statistical model for Condorcet efficiency based on simulation under spatial model assumptions". Public Choice. 47 (2): 389–403. doi:10.1007/bf00127534. ISSN 0048-5829. the 'squeeze effect' that tends to reduce Condorcet efficiency if the relative dispersion (RD) of candidates is low. This effect is particularly strong for the plurality, runoff, and Hare systems, for which the garnering of first-place votes in a large field is essential to winning
^ ^a ^b "Later-No-Harm Criterion". The Center for Election Science. Retrieved 2024-02-02.
^ Woodall, Douglas, Properties of Preferential Election Rules, Voting matters - Issue 3, December 1994
^ The Non-majority Rule Desk (July 29, 2011). "Why Approval Voting is Unworkable in Contested Elections - FairVote". FairVote Blog. Retrieved 11 October 2016.
^ Graham-Squire, Adam; McCune, David (2023-06-12). "An Examination of Ranked-Choice Voting in the United States, 2004–2022". Representation: 1–19. arXiv:2301.12075. doi:10.1080/00344893.2023.2221689. ISSN 0034-4893.
^ 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

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