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Revision as of 09:18, 7 December 2017 editFresheneesz (talk | contribs)Extended confirmed users9,055 edits →Commentary: -> Criticism + two additional critical sources← Previous edit Latest revision as of 21:51, 26 December 2024 edit undoWotwotwoot (talk | contribs)197 edits Undid revision 1265417274 by The Smartest Prize (talk) Article given does not mention later-no-harm; specifically, the author refers to Borda passing his two-party inducement criterion, but Borda fails later-no-harm.Tag: Undo 
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{{Short description|Property of electoral systems}}
{{more footnotes|date=October 2011}}
{| class="wikitable sortable floatright"
|+ Voting system
!Name||Comply?
|-
|]
|Yes{{NoteTag|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.|name=p}}
|-
|]
|Yes
|-
|]
|Yes
|-
|]
|Yes
|-
|]
|Yes
|-
|DSC
|Yes
|-
|]||No{{cn|date=November 2024}}
|-
|]||N/A
|-
|]||No
|-
|]||No
|-
|]||No
|-
|]||No
|-
|]||No
|-
|]||No
|-
|]||No
|-
|]
|No
|}


'''Later-no-harm''' is a property of some ] systems, first described by ]. In later-no-harm systems, increasing the rating or rank of a candidate ranked below the winner of an election cannot cause a higher-ranked candidate to lose. It is a common property in the ] of voting systems.


For example, say a group of voters ranks Alice 2nd and Bob 6th, 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 6th-place to 3rd. Later-no-harm says that this increased support from Alice's voters should not allow Bob to win.<ref name=":0" />
The '''later-no-harm criterion''' is a ] 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 ] known as ], which can be used to deny victory to a sincere ].


Later-no-harm may be confused as implying ], since later-no-harm is a defining characteristic of ] and ], and descending solid coalitions (DSC), systems that have similar mechanics that are based on first preference counting. These systems pass later-no-harm compliance by making sure the results either do not depend on lower preferences at all (]) or only depend on them if all higher preferences have been eliminated (] and DSC), and thus exhibit a ] effect. <ref name=":02">{{Cite journal |last=Lewyn |first=Michael |date=2012 |title=Two Cheers for Instant Runoff Voting |url=https://papers.ssrn.com/abstract=2276015 |journal=6 Phoenix L. Rev. |language=en |location=Rochester, NY |volume=117 |pages= |ssrn=2276015 |quote=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 |via=}}</ref><ref>{{Cite journal |last=Stensholt |first=Eivind |date=2015-10-07 |title=What Happened in Burlington? |url=https://ideas.repec.org//p/hhs/nhhfms/2015_026.html |journal=Discussion Papers |language=en |pages=13 |quote=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.}}</ref> However, this does not mean that methods that pass later-no-harm must be vulnerable to center squeezes. The properties are distinct, as ] also passes later-no-harm.
{{TOC limit|limit=3}}


Later-no-harm is also often confused with immunity to a kind of ] called ] or ].<ref>{{cite web |last1=The Non-majority Rule Desk |date=July 29, 2011 |title=Why Approval Voting is Unworkable in Contested Elections - FairVote |url=http://www.fairvote.org/why-approval-voting-is-unworkable-in-contested-elections |accessdate=11 October 2016 |website=FairVote Blog}}</ref> Satisfying later-no-harm does not provide immunity to such strategies. Systems like ] that pass later-no-harm but fail ] still incentivize truncation or bullet voting in some situations.<ref>{{Cite journal |last1=Graham-Squire |first1=Adam |last2=McCune |first2=David |date=2023-06-12 |title=An Examination of Ranked-Choice Voting in the United States, 2004–2022 |url=https://www.tandfonline.com/doi/full/10.1080/00344893.2023.2221689 |journal=Representation |language=en |pages=1–19 |arxiv=2301.12075 |doi=10.1080/00344893.2023.2221689 |issn=0034-4893}}</ref><ref>{{Cite journal | last=Brams |first=Steven |title= The AMS nomination procedure is vulnerable to 'truncation of preferences' |year=1982 |journal=Notices of the American Mathematical Society |volume=29 |pages=136–138 |issn=0002-9920 |publisher=American Mathematical Society |url=https://www.ams.org/cgi-bin/notices/nxgnotices.pl?fm=main&current=198202}}</ref><ref>{{Cite journal |last1=Fishburn |first1=Peter C. |last2=Brams |first2=Steven J. |date=1984-01-01 |title=Manipulability of voting by sincere truncation of preferences |url=https://doi.org/10.1007/BF00119689 |journal=Public Choice |language=en |volume=44 |issue=3 |pages=397–410 |doi=10.1007/BF00119689 |issn=1573-7101}}</ref>{{rp|401}}
== Complying methods ==
], ], ], ], ] (a pairwise opposition variant which does not satisfy the Condorcet Criterion), and Descending Solid Coalitions, a variant of Woodall's ] rule, satisfy the later-no-harm criterion.


{{TOC limit}}
When a voter is allowed to choose only one preferred candidate, as in ] 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 == == Later-no-harm methods ==
The ], ], ], and descending solid coalitions satisfy the later-no-harm criterion. ] satisfies later-no-harm trivially, by ignoring every preference after the first.<ref name=":0" />


=== Non-LNH methods ===
], ], ], ], ], ], ], ], ], and ] do not satisfy later-no-harm. The ] 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).<ref>Douglas Woodall (1997): , Theorem 2 (b)</ref>
Nearly all voting methods other than first-past-the-post do not pass LNH, including ], ], ], and all ]s. The ] is incompatible with later-no-harm (assuming the ], i.e. any tie can be removed by a single voter changing their rating).<ref name=":0">Douglas Woodall (1997): , Theorem 2 (b)</ref>


] 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. ], which allows a voter to select multiple candidates, does not satisfy later-no-harm when used to fill two or more seats in a single district, although the ] does.

== 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 == == Examples ==
Line 25: Line 67:
{{Main|Anti-plurality voting}} {{Main|Anti-plurality voting}}


Anti-plurality elects the candidate the least number of voters rank last when submitting a complete ranking of the candidates. 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. 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.


{| class="wikitable collapsible collapsed"
==== Truncated Ballot Profile ====
!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 <math>\tfrac{1}{2}</math> A > B > C, and <math>\tfrac{1}{2}</math> A > C > B: 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 <math>\tfrac{1}{2}</math> A > B > C, and <math>\tfrac{1}{2}</math> A > C > B:
{| class="wikitable" {| class="wikitable"
Line 49: Line 95:
'''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. '''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 ==== ; Adding Later Preferences
Now assume that the four voters supporting A (marked bold) add later preference C, as follows: Now assume that the four voters supporting A (marked bold) add later preference C, as follows:


Line 68: Line 114:
'''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. '''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 ==== ;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. 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 ===
{{Main|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:
{| class="wikitable"
! # 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:

{| class="wikitable"
! # 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 === === Borda count ===
{{Main|Borda count}} {{Main|Borda count}}


{| class="wikitable collapsible collapsed"
!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: 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:
{| class="wikitable" {| class="wikitable"
Line 120: Line 134:
|} |}


==== Express later preferences ==== ; Express later preferences
Assume that all preferences are expressed on the ballots. Assume that all preferences are expressed on the ballots.


Line 136: Line 150:
'''Result''': '''B''' wins with 7 Borda points. '''Result''': '''B''' wins with 7 Borda points.


==== Hide later preferences ==== ;Hide later preferences
Assume now that the three voters supporting A (marked bold) would not express their later preferences on the ballots: Assume now that the three voters supporting A (marked bold) would not express their later preferences on the ballots:
{| class="wikitable" {| class="wikitable"
Line 159: Line 173:
'''Result''': '''A''' wins with 6 Borda points. '''Result''': '''A''' wins with 6 Borda points.


==== Conclusion ==== ;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. 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 ===
{{Main|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 <math>\tfrac{1}{2}</math> A > B > C, and <math>\tfrac{1}{2}</math> A > C > B:
{| class="wikitable"
! # 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:

{| class="wikitable"
! # 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 === === Copeland ===
{{Main|Copeland's method}} {{Main|Copeland's method}}
{| class="wikitable collapsible collapsed"

!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: 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:
{| class="wikitable" {| class="wikitable"
Line 225: Line 194:
|} |}


==== Express later preferences ==== ;Express later preferences
Assume that all preferences are expressed on the ballots. Assume that all preferences are expressed on the ballots.


The results would be tabulated as follows: The results would be tabulated as follows:
{| class=wikitable border=1 {| class=wikitable
|+ Pairwise election results |+ Pairwise election results
|- |-
Line 274: Line 243:
'''Result''': B has two wins and no defeat, A has only one win and no defeat. Thus, '''B''' is elected Copeland winner. '''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 ==== ;Hide later preferences
Assume now, that the two voters supporting A (marked bold) would not express their later preferences on the ballots: Assume now, that the two voters supporting A (marked bold) would not express their later preferences on the ballots:
{| class="wikitable" {| class="wikitable"
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The results would be tabulated as follows: The results would be tabulated as follows:
{| class=wikitable border=1 {| class=wikitable
|+ Pairwise election results |+ Pairwise election results
|- |-
Line 332: Line 301:
'''Result''': A has one win and no defeat, B has no win and no defeat. Thus, '''A''' is elected Copeland winner. '''Result''': A has one win and no defeat, B has no win and no defeat. Thus, '''A''' is elected Copeland winner.


==== Conclusion ==== ;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. 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 ===
{{Main|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 <math>\tfrac{1}{2}</math> A > B > C, and <math>\tfrac{1}{2}</math> A > C > B:
{| class="wikitable"
! # of voters !! Preferences
|-
| '''5''' || '''A''' ( > B > C)
|-
| '''5''' || '''A''' ( > C > B)
|-
| 7 || B > A > C
|-
| 7 || C > B > A
|-
| 3 || C > A > B
|} |}


=== Schulze method ===
{| class="wikitable" style="text-align:center"
{{Main|Schulze method}}
|+ Pairwise Contests
|-
! !! Against A !! Against B !! Against C
|-
! For A
| || bgcolor=#ffdddd|13 || bgcolor=#ddffdd|17
|-
! For B
| bgcolor=#ddffdd|14 || || bgcolor=#ffdddd|12
|-
! For C
| bgcolor=#ffdddd|10 || bgcolor=#ddffdd|15 ||
|}


{| class="wikitable collapsible collapsed"
'''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.
!Examples

==== Adding Later Preferences ====
Now assume that the ten voters supporting A (marked bold) add later preference B, as follows:

{| class="wikitable"
! # of voters !! Preferences
|- |-
| '''10''' || '''A > B > C'''
|-
| 7 || B > A > C
|-
| 7 || C > B > A
|-
| 3 || C > A > B
|}

{| class="wikitable" style="text-align:center"
|+ Pairwise Contests
|-
! !! Against A !! Against B !! Against C
|-
! For A
| || bgcolor=#ffdddd|13 || bgcolor=#ddffdd|17
|-
! For B
| bgcolor=#ddffdd|14 || || bgcolor=#ddffdd|17
|-
! For C
| bgcolor=#ffdddd|10 || bgcolor=#ffdddd|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.

<!-- commented out as per talk page "Proposal to Remove "Certain IRV-variant" Example"
=== Instant runoff voting variant with majority requirement ===
{{Main|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 ] 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:
{| border="1" cellspacing="0" cellpadding="10"
| 40: A
| 39: B>A
| 21: C
|-
|}

In the ] method, described as an example in ], 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 ] and be elected.

Now, suppose the B voters would hide their second preference for A:
{| border="1" cellspacing="0" cellpadding="10"
| 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 ===
{{Main|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:
{| class="wikitable"
! # 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:
{| class="wikitable"
! colspan=2 rowspan=2|All possible pairs<br/>of choice names !! colspan=3|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:
{| class="wikitable"
! Preferences !! 1. vs 2. !! 1. vs 3. !! 2. vs 3. !! Total
|-
| A > B > C || 6 || 4 || 4 || bgcolor=#ffbbbb|''14''
|-
| A > C > B || 4 || 6 || 5 || bgcolor=#ffbbbb|''15''
|-
| B > A > C || 3 || 4 || 4 || bgcolor=#ffbbbb|''11''
|-
| B > C > A || 4 || 3 || 5 || bgcolor=#ffbbbb|''12''
|-
| C > A > B || 5 || 5 || 6 || bgcolor=#bbffbb|'''16'''
|-
| C > B > A || 5 || 5 || 3 || bgcolor=#ffbbbb|''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:
{| class="wikitable"
! # 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:
{| class="wikitable"
! colspan=2 rowspan=2|All possible pairs<br/>of choice names !! colspan=3|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:
{| class="wikitable"
! Preferences !! 1. vs 2. !! 1. vs 3. !! 2. vs 3. !! Total
|-
| A > B > C || 3 || 1 || 4 || bgcolor=#ffbbbb|''8''
|-
| A > C > B || 1 || 3 || 2 || bgcolor=#ffbbbb|''6''
|-
| B > A > C || 3 || 4 || 1 || bgcolor=#ffbbbb|''8''
|-
| B > C > A || 4 || 3 || 5 || bgcolor=#bbffbb|'''12'''
|-
| C > A > B || 5 || 2 || 3 || bgcolor=#ffbbbb|''10''
|-
| C > B > A || 2 || 5 || 3 || bgcolor=#ffbbbb|''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 ===
{{Main|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:
{| class="wikitable"
! Candidates/<br /># of voters !! A !! B
|-
| '''1''' || bgcolor="green"|'''Excellent''' || bgcolor="YellowGreen"|'''Good'''
|-
| 1 || bgcolor="orangered"|Poor || bgcolor="green"|Excellent
|-
| 1 || bgcolor="yellow"|Fair || bgcolor="orangered"|Poor
|}

==== Express later preferences ====
Assume that all ratings are expressed on the ballots.

The sorted ratings would be as follows:
{|
| align=right | Candidate&nbsp;&nbsp;&nbsp;
| |
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:
{| cellpadding=0 width=500 border=0 cellspacing=0
| width=49% |&nbsp;
| width=2% textalign=center | ↓
| width=49% | Median point
|}
|-
| align=right | A
|
{| cellpadding=0 width=500 border=0 cellspacing=0
| bgcolor=green width=33% |&nbsp;
| bgcolor=yellow width=34% |&nbsp;
| bgcolor=orangered width=33% |
|}
|-
| align=right | B
|
{| cellpadding=0 width=500 border=0 cellspacing=0
| bgcolor=green width=33% |&nbsp;
| bgcolor=yellowgreen width=34% |&nbsp;
| bgcolor=orangered width=33% |
|}
|-
| &nbsp;
| &nbsp;
|-
| &nbsp;
|
{| cellpadding=1 border=0 cellspacing=1
| &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
| bgcolor=green |&nbsp;
| &nbsp;Excellent &nbsp;
| bgcolor=YellowGreen | &nbsp;
| &nbsp;Good &nbsp;
| bgcolor=Yellow | &nbsp;
| &nbsp;Fair &nbsp;
| bgcolor=Orangered | &nbsp;
| &nbsp;Poor &nbsp;
|}
|}

'''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":
{| class="wikitable"
! Candidates/<br /># of voters !! A !! B
|-
| '''1''' || bgcolor="green"|'''Excellent''' || bgcolor="orangered"|''(Poor)''
|-
| 1 || bgcolor="orangered"|Poor || bgcolor="green"|Excellent
|-
| 1 || bgcolor="yellow"|Fair || bgcolor="orangered"|Poor
|}

The sorted ratings would be as follows:
{|
| align=right | Candidate&nbsp;&nbsp;&nbsp;
|
{| cellpadding=0 width=500 border=0 cellspacing=0
| width=49% |&nbsp;
| width=2% textalign=center | ↓
| width=49% | Median point
|}
|-
| align=right | A
|
{| cellpadding=0 width=500 border=0 cellspacing=0
| bgcolor=green width=33% |&nbsp;
| bgcolor=yellow width=34% |&nbsp;
| bgcolor=orangered width=33% |
|}
|-
| align=right | B
|
{| cellpadding=0 width=500 border=0 cellspacing=0
| bgcolor=green width=33% |&nbsp;
| bgcolor=orangered width=67% |
|}
|-
| &nbsp;
| &nbsp;
|-
| &nbsp;
|
{| cellpadding=1 border=0 cellspacing=1
| &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
| bgcolor=green |&nbsp;
| &nbsp;Excellent &nbsp;
| bgcolor=YellowGreen | &nbsp;
| &nbsp;Good &nbsp;
| bgcolor=Yellow | &nbsp;
| &nbsp;Fair &nbsp;
| bgcolor=Orangered | &nbsp;
| &nbsp;Poor &nbsp;
|}
|}

'''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 ===
{{Main|Minimax Condorcet}}

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:
{| class="wikitable"
! # 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:
{| class=wikitable border=1
|+ Pairwise election results
|-
| colspan=2 rowspan=2 |
| colspan=4 bgcolor="#c0c0ff" align=center | X
|-
| bgcolor="#c0c0ff" | A
| bgcolor="#c0c0ff" | B
| bgcolor="#c0c0ff" | C
| bgcolor="#c0c0ff" | D
|-
| bgcolor="#ffc0c0" rowspan=4 | Y
| bgcolor="#ffc0c0" | A
|
| bgcolor="#ffe0e0" | 6 <br> 9
| bgcolor="#e0e0ff" | 9 <br> 8
| bgcolor="#ffe0e0" | 8 <br> 11
|-
| bgcolor="#ffc0c0" | B
| bgcolor="#e0e0ff" | 9 <br> 6
|
| bgcolor="#e0e0ff" | 10 <br> 9
| bgcolor="#ffe0e0" | 7 <br> 10
|-
| bgcolor="#ffc0c0" | C
| bgcolor="#ffe0e0" | 8 <br> 9
| bgcolor="#ffe0e0" | 9 <br> 10
|
| bgcolor="#ffe0e0" | 11 <br> 12
|-
| bgcolor="#ffc0c0" | D
| bgcolor="#e0e0ff" | 11 <br> 8
| bgcolor="#e0e0ff" | 10 <br> 7
| bgcolor="#e0e0ff" | 12 <br> 11
|
|-
| colspan=2 bgcolor="#c0c0ff" | Pairwise election results (won-tied-lost):
| 2-0-1
| 1-0-2
| 3-0-0
| 0-0-3
|-
| colspan=2 bgcolor="#c0c0ff" | worst pairwise defeat (winning votes):
| bgcolor=#ffbbbb|''9''
| bgcolor=#ffbbbb|''10''
| bgcolor=#bbffbb|'''0'''
| bgcolor=#ffbbbb|''12''
|-
| colspan=2 bgcolor="#c0c0ff" | worst pairwise defeat (margins):
| bgcolor=#ffbbbb|''1''
| bgcolor=#ffbbbb|''3''
| bgcolor=#bbffbb|'''0'''
| bgcolor=#ffbbbb|''3''
|-
| colspan=2 bgcolor="#c0c0ff" | worst pairwise opposition:
| bgcolor=#bbffbb|'''9'''
| bgcolor=#ffbbbb|''10''
| bgcolor=#ffbbbb|''11''
| bgcolor=#ffbbbb|''12''
|}

* indicates voters who preferred the candidate listed in the column caption to the candidate listed in the row caption
* 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:
{| class="wikitable"
! # 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:
{| class=wikitable border=1
|+ Pairwise election results
|-
| colspan=2 rowspan=2 |
| colspan=4 bgcolor="#c0c0ff" align=center | X
|-
| bgcolor="#c0c0ff" | A
| bgcolor="#c0c0ff" | B
| bgcolor="#c0c0ff" | C
| bgcolor="#c0c0ff" | D
|-
| bgcolor="#ffc0c0" rowspan=4 | Y
| bgcolor="#ffc0c0" | A
|
| bgcolor="#ffe0e0" | 6 <br> 9
| bgcolor="#e0e0ff" | 9 <br> 8
| bgcolor="#ffe0e0" | 8 <br> 11
|-
| bgcolor="#ffc0c0" | B
| bgcolor="#e0e0ff" | 9 <br> 6
|
| bgcolor="#e0e0ff" | 10 <br> 9
| bgcolor="#ffe0e0" | 7 <br> 10
|-
| bgcolor="#ffc0c0" | C
| bgcolor="#ffe0e0" | 8 <br> 9
| bgcolor="#ffe0e0" | 9 <br> 10
|
| bgcolor="#e0e0ff" | 11 <br> 8
|-
| bgcolor="#ffc0c0" | D
| bgcolor="#e0e0ff" | 11 <br> 8
| bgcolor="#e0e0ff" | 10 <br> 7
| bgcolor="#ffe0e0" | 8 <br> 11
|
|-
| colspan=2 bgcolor="#c0c0ff" | Pairwise election results (won-tied-lost):
| 2-0-1
| 1-0-2
| 2-0-1
| 1-0-2
|-
| colspan=2 bgcolor="#c0c0ff" | worst pairwise defeat (winning votes):
| bgcolor=#bbffbb|'''9'''
| bgcolor=#ffbbbb|''10''
| bgcolor=#ffbbbb|''11''
| bgcolor=#ffbbbb|''11''
|-
| colspan=2 bgcolor="#c0c0ff" | worst pairwise defeat (margins):
| bgcolor=#bbffbb|'''1'''
| bgcolor=#ffbbbb|''3''
| bgcolor=#ffbbbb|''3''
| bgcolor=#ffbbbb|''3''
|-
| colspan=2 bgcolor="#c0c0ff" | worst pairwise opposition:
| bgcolor=#bbffbb|'''9'''
| bgcolor=#ffbbbb|''10''
| bgcolor=#ffbbbb|''11''
| bgcolor=#ffbbbb|''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 ===
{{Main|Ranked pairs}}

For example in an election conducted using the ] compliant method ] the following votes are cast:

{| border="1" cellspacing="0" cellpadding="10"
| 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 ]; A, B, and C are all ] and B is the ] 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:

{| border="1" cellspacing="0" cellpadding="10"
| 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 ] and therefore the ] 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 ===
{{Main|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:
{| class="wikitable"
! !! colspan=2|Scores !! Reading
|-
! # of voters !! A || B ||
|-
| '''1''' || bgcolor=#55ff55|'''10''' || bgcolor=#99ff55| '''8''' || Slightly prefers A (by 2)
|-
| 1 || bgcolor=#ff5555| 0 || bgcolor=#ffdd55| 4 || Slightly prefers B (by 4)
|}

==== Express later preferences ====
Assume that all preferences are expressed on the ballots.

The total scores would be:
{| class="wikitable"
! candidate !! Average Score
|-
| A || bgcolor=#ffdddd|''5''
|-
| B || bgcolor=#ddffdd|'''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:
{| class="wikitable"
! !! colspan=2|Scores !! Reading
|-
! # of voters !! A || B ||
|-
| '''1''' || bgcolor=#55ff55|'''10''' || bgcolor=#cccccc|--- || Greatly prefers A (by 10)
|-
| 1 || bgcolor=#ff5555| 0 || bgcolor=#ffdd55| 4 || Slightly prefers B (by 4)
|}

The total scores would be:
{| class="wikitable"
! candidate !! Average Score
|-
| A || bgcolor=#ddffdd|'''5'''
|-
| B || bgcolor=#ffdddd|''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 (as ]), 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 ===
{{Main|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:
{| class="wikitable" {| class="wikitable"
! # of voters !! Preferences ! # of voters !! Preferences
Line 989: Line 333:
|} |}


==== Express later preferences ==== ; Express later preferences
Assume that all preferences are expressed on the ballots. Assume that all preferences are expressed on the ballots.


Line 1,011: Line 355:


==== Hide later preferences ==== ==== Hide later preferences ====

Assume now that the three voters supporting A (marked bold) would not express their later preferences on the ballots: Assume now that the three voters supporting A (marked bold) would not express their later preferences on the ballots:
{| class="wikitable" {| class="wikitable"
Line 1,066: Line 411:
'''Result''': The full ranking is A > C > B. Thus, '''A''' is elected Schulze winner. '''Result''': The full ranking is A > C > B. Thus, '''A''' is elected Schulze winner.


==== Conclusion ==== ; 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. 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 == == Criticism ==


] writes:
Woodall writes about Later-no-harm, {{quote|nder ] 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 ] nor harm any candidate already listed. Supporters of STV usually regard this as a very important property,<ref>{{cite web|last1=The Non-majority Rule Desk|title=Why Approval Voting is Unworkable in Contested Elections - FairVote|url=http://www.fairvote.org/why-approval-voting-is-unworkable-in-contested-elections|website=FairVote Blog|accessdate=11 October 2016|date=July 29, 2011}}</ref> although it has to be said that not everyone agrees; the property has been described (by ], in a letter to Robert Newland) as "quite unreasonable", and (by an anonymous referee) as "unpalatable".<ref>Woodall, Douglas, Properties of Preferential Election Rules, </ref>}}


{{blockquote|nder ] 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 ] 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 ], in a letter to Robert Newland) as "quite unreasonable", and (by an anonymous referee) as "unpalatable".<ref name="votingmatters.org.uk">Woodall, Douglas, Properties of Preferential Election Rules, </ref>
Mathematics PhD Warren Smith of rangevoting.org writes that the Later-no-harm criterion is "a silly criterion" saying that "objectively, LNH is not even a desirable property with honest voters". He argues that rating a candidate higher should allow the possibility of that candidate winning if the candidates collective ratings are high enough.
}}

The Center for Election Science also voices their opinion that the name itself is "misleading"and that while "a voter can’t harm a candidate by ranking additional less preferred candidates, .. voters can still hurt themselves by doing so. This begs the question of whether the later-no-harm criterion actually has value."


==See also== ==See also==


*] *]
*]


== References == == Notes ==
{{reflist}} {{reflist|group=note}}

== Bibliography ==
* D R Woodall, "Properties of Preferential Election Rules", '']'', Issue 3, December 1994 * D R Woodall, "Properties of Preferential Election Rules", '']'', Issue 3, December 1994
* Tony Anderson Solgard and Paul Landskroener, Bench and Bar of Minnesota, Vol 59, No 9, October 2002. * Tony Anderson Solgard and Paul Landskroener, Bench and Bar of Minnesota, Vol 59, No 9, October 2002.
* *


== References ==
{{reflist}}


{{voting systems}} {{voting systems}}

Latest revision as of 21:51, 26 December 2024

Property of electoral systems
Voting system
Name Comply?
Plurality Yes
Two-round system Yes
Partisan primary Yes
Instant-runoff voting Yes
Minimax Opposition Yes
DSC Yes
Anti-plurality No
Approval N/A
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, increasing the rating or rank of a candidate ranked below the winner of an election cannot cause a higher-ranked candidate to lose. It is a common property in the plurality-rule family of voting systems.

For example, say a group of voters ranks Alice 2nd and Bob 6th, 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 6th-place to 3rd. Later-no-harm says that this increased support from Alice's voters should not allow Bob to win.

Later-no-harm may be confused as implying center squeeze, since later-no-harm is a defining characteristic of first-preference plurality (FPP) and instant-runoff voting (IRV), and descending solid coalitions (DSC), systems that have similar mechanics that are based on first preference counting. These systems pass later-no-harm compliance by making sure the results either do not depend on lower preferences at all (plurality) or only depend on them if all higher preferences have been eliminated (IRV and DSC), and thus exhibit a center squeeze effect. However, this does not mean that methods that pass later-no-harm must be vulnerable to center squeezes. The properties are distinct, as Minimax opposition also passes later-no-harm.

Later-no-harm is also often confused with immunity to a kind of strategic voting called strategic truncation or bullet voting. Satisfying later-no-harm does not provide immunity to such strategies. Systems like instant runoff that pass later-no-harm but fail monotonicity still incentivize truncation or bullet voting in some situations.

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, by ignoring every preference after the first.

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 a single voter changing their rating).

Bloc voting, which allows a voter to select multiple candidates, does not satisfy later-no-harm when used to fill two or more seats in a single district, although the single non-transferable vote does.

Examples

Anti-plurality

Main article: Anti-plurality voting

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 1 2 {\displaystyle {\tfrac {1}{2}}} A > B > C, and 1 2 {\displaystyle {\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

Main article: 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 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 doesn't satisfy the Later-no-harm criterion.

Copeland

Main article: Copeland's method
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 2
2 		2
2 		1
3
B 2
2 		1
3 		1
3
C 2
2 		3
1 		1
3
D 3
1 		3
1 		3
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 2
2 		2
2 		1
3
B 2
2 		1
1 		1
1
C 2
2 		1
1 		1
1
D 3
1 		1
1 		1
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

Main article: 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 d d
d 5 7
d 6 9
d 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 d d
d 5 7
d 6 6
d 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 p p
p 7 7
p 6 6
p 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:

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".

See also

Notes

  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.

Bibliography

  • 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


References

  1. ^ Douglas Woodall (1997): Monotonicity of Single-Seat Election Rules, Theorem 2 (b)
  2. Lewyn, Michael (2012). "Two Cheers for Instant Runoff Voting". 6 Phoenix L. Rev. 117. Rochester, NY. SSRN 2276015. 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
  3. 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.
  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. 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.
  6. Brams, Steven (1982). "The AMS nomination procedure is vulnerable to 'truncation of preferences'". Notices of the American Mathematical Society. 29. American Mathematical Society: 136–138. ISSN 0002-9920.
  7. Fishburn, Peter C.; Brams, Steven J. (1984-01-01). "Manipulability of voting by sincere truncation of preferences". Public Choice. 44 (3): 397–410. doi:10.1007/BF00119689. ISSN 1573-7101.
  8. Woodall, Douglas, Properties of Preferential Election Rules, Voting matters - Issue 3, December 1994
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