Later-no-harm criterion

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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 cannot cause a more-preferred candidate to lose.

Complying methods[edit]

Single transferable vote (including traditional forms of Instant Runoff Voting and Contingent vote), Minimax Condorcet (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.

Noncomplying methods[edit]

Approval voting, Borda count, Range voting, Schulze method and Bucklin voting do not satisfy later-no-harm. The Condorcet criterion is incompatible with later-no-harm.

When plurality is used to fill two or more seats in a single district (plurality-at-large) it fails later-no-harm.

The later-no-harm criterion is by definition inapplicable to any voting system in which a voter is not allowed to express more than one choice, such as plurality voting and most party list forms of proportional representation.

Examples[edit]

Approval voting[edit]

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 an 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[edit]

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[edit]

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[edit]

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[edit]

Main article: 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[edit]

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[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
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[edit]

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.

Copeland[edit]

Main article: Copeland's method

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[edit]

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[edit]

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[edit]

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.

Instant runoff voting variant with majority requirement[edit]

Main article: Instant-runoff voting

Traditional forms of instant runoff voting satisfy the later-no-harm criterion. But if a method permits incomplete ranking of candidates, and if a majority of initial round votes is required to win and avoid another election, that variant does not satisfy Later-no-harm. A lower preference vote cast may create a majority for that lower preference, whereas if the vote was not cast, the election could fail, proceed to a runoff, repeated ballot or other process, and the favored candidate could possibly win.

Assume, the votes are as follows:

40: A 39: B>A 21: C

In the Preferential voting method, described as an example in Robert's Rules of Order, elimination continues iteratively until "one pile contains more than half the ballots." Eliminating one of the two final candidate never changes who wins, but can change how many votes that final candidate receives. Thus, C would be eliminated, then B, and the B ballots would be counted for A, who would thereby obtain a majority and be elected.

Now, suppose the B voters would hide their second preference for A:

40: A 39: B 21: C

This failure to win a majority in the final round results in a runoff between A and B, which B could win.

By adding a second preference vote for A, the B voters eliminated the election possibility for B. Thus, this variant of instant runoff voting with majority requirement fails the later-no-harm criterion. In traditional IRV, A would have been elected by 40% of the voters, and the later-no-harm criterion would not have been violated.

Also, compliance of LNH can be reestablished by stopping the elimination process when there are two candidates left. Applying this to the example, the second preferences of B are ignored either case. Thus, the B voters would not violate later-no-harm by indicating A as a second choice.

Kemeny–Young method[edit]

Main article: 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[edit]

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[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
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 6 0 3
X = A Y = C 4 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 6 4 4 14
A > C > B 4 6 2 12
B > A > C 3 4 4 11
B > C > A 4 3 5 12
C > A > B 5 2 6 13
C > B > A 2 5 3 10

Result: The ranking A > B > C has the highest ranking score. Thus, A wins ahead of B and C.

Conclusion[edit]

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[edit]

Main article: 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[edit]

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[edit]

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[edit]

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[edit]

Main article: Minimax Condorcet

This example shows that the Minimax method violates the Later-no-harm criterion in his 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[edit]

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[edit]

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[edit]

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[edit]

Main article: Ranked pairs

For example in an election conducted using the Condorcet compliant method Ranked pairs the following votes are cast:

49: A 25: B 26: C>B

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 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 25: B>C 26: C>B

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.

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[edit]

Main article: 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[edit]

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[edit]

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[edit]

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[edit]

Main article: 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[edit]

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[edit]

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[edit]

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[who?] usually regard this as a very important property, 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.'"[1]

References[edit]

  1. ^ 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