Later-no-harm criterion

Last updated February 24, 2024

Voting system
Name	Comply?
Plurality	Yes^{[note 1]}
Two-round system	Yes
Nonpartisan primary	Yes
Instant-runoff voting	Yes
Minimax Condorcet	Yes^{[note 2]}
Descending solid coalitions	Yes
Anti-plurality	No
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
Usual judgment	No

The later-no-harm criterion is a voting system criterion first formulated by Douglas Woodall. Woodall defined the criterion by saying that "[a]dding a later preference to a ballot should not harm any candidate already listed."^[1] For example, a ranked voting method in which a voter adding a 3^rd preference could reduce the likelihood of their 1^st preference being selected, fails later-no-harm.

Voting systems that fail the later-no-harm criterion can sometimes be vulnerable to the tactical voting strategies called bullet voting and burying, which can deny victory to a sincere Condorcet winner. However, both strategies can also be successful in criteria that pass later-no-harm (including instant runoff voting),^[2] and cardinal voting systems seem to be more resistant to these strategies in practice.^[2] Moreover, the fact that all cardinal voting methods can fail the later-no-harm criterion in theory is essential to their favoring consensus options (broad, moderate support) over pluralitarian options (narrow, strong support);^[3] voting systems that pass later-no-harm are unable to consider weak (secondary) preferences when evaluating candidates. As a result, many social choice theorists question whether the criterion is even desirable in the first place.^[2]

Complying methods

The plurality vote, 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.

Plurality voting is typically considered to satisfy later-no-harm because it can be thought of as a ranked voting system where only the first preference matters.

Noncomplying methods

Approval voting, score voting, highest medians, Borda count, ranked pairs, the Schulze method, the 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 resolvability criterion, i.e. that 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.

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

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.

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

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

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

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
All possible pairs of choice names		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
All possible pairs of choice names		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

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

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

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

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

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

Woodall, author of the Later-no-harm 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,^[4] 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".^[5]

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.
↑ Minimax can occasionally violate later-no-harm if tied ranks are allowed.

Related Research Articles

Approval voting is an electoral system in which voters can select any number of candidates instead of selecting only one.

A Condorcet method is an election method that elects the candidate who wins a majority of the vote in every head-to-head election against each of the other candidates, that is, a candidate preferred by more voters than any others, whenever there is such a candidate. A candidate with this property, the pairwise champion or beats-all winner, is formally called the Condorcet winner. The head-to-head elections need not be done separately; a voter's choice within any given pair can be determined from the ranking.

Copeland's method, also called Llull's method or round-robin voting, is a ranked-choice voting system based on scoring pairwise wins and losses.

An electoral system satisfies the Condorcet winner criterion if it always chooses the Condorcet winner when one exists. The candidate who wins a majority of the vote in every head-to-head election against each of the other candidates – that is, a candidate preferred by more voters than any others – is the Condorcet winner, although Condorcet winners do not exist in all cases. It is sometimes simply referred to as the "Condorcet criterion", though it is very different from the "Condorcet loser criterion". Any voting method conforming to the Condorcet winner criterion is known as a Condorcet method. The Condorcet winner is the person who would win a two-candidate election against each of the other candidates in a plurality vote. For a set of candidates, the Condorcet winner is always the same regardless of the voting system in question, and can be discovered by using pairwise counting on voters' ranked preferences.

The Smith criterion is a voting system criterion that formalizes the concept of a majority rule. A voting system satisfies the Smith criterion if it always elects a candidate from the Smith set, which generalizes the idea of a "Condorcet winner" to cases where there may be cycles or ties, by allowing for several who together can be thought of as being "Condorcet winners." A Smith method will always elect a candidate from the Smith set.

The participation criterion, also called vote or population monotonicity, is a voting system criterion that says that a candidate should never lose an election because they have "too much support." It says that adding voters who support A over B should not cause A to lose the election to B.

The majority criterion is a voting system criterion. The criterion states that "if only one candidate is ranked first by a majority of voters, then that candidate must win."

The mutual majority criterion is a criterion used to compare voting systems. It is also known as the majority criterion for solid coalitions and the generalized majority criterion. The criterion says if there is a subset S of candidates, with more than half of voters strictly preferring any member of S to every candidate outside of S, the winner must come from S. This is similar to but stricter than the majority criterion, where the requirement applies only to the case that S contains a single candidate. This is also stricter than the majority loser criterion, where the requirement applies only to the case that S contains all but one candidate. The mutual majority criterion is the single-winner case of the Droop proportionality criterion.

In single-winner voting system theory, the Condorcet loser criterion (CLC) is a measure for differentiating voting systems. It implies the majority loser criterion but does not imply the Condorcet winner criterion.

In voting systems, the Minimax Condorcet method is a single-winner ranked-choice voting method that always elects the majority (Condorcet) winner. Minimax compares all candidates against each other in a round-robin tournament, then ranks candidates by their worst election result. The candidate with the largest (maximum) margin of victory in their worst (minimum) matchup is declared the winner.

Reversal symmetry is a voting system criterion which requires that if candidate A is the unique winner, and each voter's individual preferences are inverted, then A must not be elected. Methods that satisfy reversal symmetry include Borda count, ranked pairs, Kemeny–Young method, and Schulze method. Methods that fail include Bucklin voting, instant-runoff voting and Condorcet methods that fail the Condorcet loser criterion such as Minimax.

The plurality criterion is a voting system criterion devised by Douglas R. Woodall for ranked voting methods with incomplete ballots. It is stated as follows:

In voting systems theory, the independence of clones criterion measures an election method's robustness to strategic nomination. Nicolaus Tideman was the first to formulate this criterion, which states that the winner must not change due to the addition of a non-winning candidate who is similar to a candidate already present. To be more precise, a subset of the candidates, called a set of clones, exists if no voter ranks any candidate outside the set between any candidates that are in the set. If a set of clones contains at least two candidates, the criterion requires that deleting one of the clones must not increase or decrease the winning chance of any candidate not in the set of clones.

The Borda count is a family of positional voting rules which gives each candidate, for each ballot, a number of points corresponding to the number of candidates ranked lower. In the original variant, the lowest-ranked candidate gets 0 points, the next-lowest gets 1 point, etc., and the highest-ranked candidate gets n − 1 points, where n is the number of candidates. Once all votes have been counted, the option or candidate with the most points is the winner. The Borda count is intended to elect broadly acceptable options or candidates, rather than those preferred by a majority, and so is often described as a consensus-based voting system rather than a majoritarian one.

The majority loser criterion is a criterion to evaluate single-winner voting systems. The criterion states that if a majority of voters prefers every other candidate over a given candidate, then that candidate must not win.

<span class="mw-page-title-main">Ranked voting</span> Family of electoral systems

The term ranked voting, also known as preferential voting or ranked-choice voting, pertains to any voting system where voters indicate a rank to order candidates or options—in a sequence from first, second, third, and onwards—on their ballots. Ranked voting systems vary based on the ballot marking process, how preferences are tabulated and counted, the number of seats available for election, and whether voters are allowed to rank candidates equally.

The later-no-help 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 win. Voting systems that fail the later-no-help criterion are vulnerable to the tactical voting strategy called mischief voting, which can deny victory to a sincere Condorcet winner.

Homogeneity is a common property for voting systems. The property is satisfied if, in any election, the result depends only on the proportion of ballots of each possible type. Specifically, if every ballot is replicated the same number of times, then the result should not change.

Comparison of electoral systems is the result of comparative politics for electoral systems. Electoral systems are the rules for conducting elections, a main component of which is the algorithm for determining the winner from the ballots cast. This article discusses methods and results of comparing different electoral systems, both those that elect a unique candidate in a 'single-winner' election and those that elect a group of representatives in a multiwinner election.

STAR voting is an electoral system for single-seat elections. Variations also exist for multi-winner and proportional representation elections. The name stands for "Score then Automatic Runoff", referring to the fact that this system is a combination of score voting, to pick two finalists with the highest total scores, followed by an "automatic runoff" in which the finalist who is preferred on more ballots wins. It is a type of cardinal voting electoral system.

References

1 2 Douglas Woodall (1997): Monotonicity of Single-Seat Election Rules, Theorem 2 (b)
1 2 3 "Later-No-Harm Criterion". The Center for Election Science. Retrieved 2024-02-02.
↑ 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.
↑ 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
Tony Anderson Solgard and Paul Landskroener, Bench and Bar of Minnesota, Vol 59, No 9, October 2002.
Brown v. Smallwood, 1915

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[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] 1 2 Douglas Woodall (1997): Monotonicity of Single-Seat Election Rules, Theorem 2 (b)

[:1-4] 1 2 3 "Later-No-Harm Criterion". The Center for Election Science. Retrieved 2024-02-02.

[5] 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.

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

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

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Later-no-harm criterion

Contents

Complying methods

Noncomplying methods

Examples

Anti-plurality

Approval voting

Borda count

Coombs' method

Copeland

Dodgson's method

Kemeny–Young method

Majority judgment

Minimax

Conclusion

Ranked pairs

Score voting

Schulze method

Hide later preferences

Criticism

See also

Notes

Related Research Articles

References