Approval voting: Difference between revisions

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==Usage==
 
==Usage==
Approval voting has been and is currenly used in many places over the years. Recently there has been some interest in the [[United States]] for municipal elections.<ref>{{Cite web|date=2021-11-17|title=The New Frontier: Seattle Approves Launches a Ballot Initiative Campaign|url=https://electionscience.org/ces-updates/the-new-frontier-seattle-approves-launches-a-ballot-initiative-campaign/|access-date=2021-12-13|website=The Center for Election Science|language=en-US}}</ref> The following is a non-exhaustive list of prominent use of approval voting.
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Approval voting has been and is currently used in many places over the years. Recently there has been some interest in the [[United States]] for municipal elections.<ref>{{Cite web|date=2021-11-17|title=The New Frontier: Seattle Approves Launches a Ballot Initiative Campaign|url=https://electionscience.org/ces-updates/the-new-frontier-seattle-approves-launches-a-ballot-initiative-campaign/|access-date=2021-12-13|website=The Center for Election Science|language=en-US}}</ref> The following is a non-exhaustive list of prominent use of approval voting.
   
 
* [[Fargo, North Dakota|'''Fargo, North Dakota''']] (as of 2018) — Fargo used approval voting in June 2020 to elect two at-large seats on its city council,<ref name="Ballotpedia Fargo">[https://ballotpedia.org/Fargo,_North_Dakota,_Measure_1,_Approval_Voting_Initiative_(November_2018) Fargo, North Dakota, Measure 1, Approval Voting Initiative (November 2018)], November 7, 2018 ''[[Ballotpedia]]''</ref><ref name="Fargo approves">[https://ivn.us/2018/11/06/one-americas-famous-towns-becomes-first-nation-adopt-approval-voting/ One of America’s Most Famous Towns Becomes First in the Nation to Adopt Approval Voting] {{Webarchive|url=https://web.archive.org/web/20181107185459/https://ivn.us/2018/11/06/one-americas-famous-towns-becomes-first-nation-adopt-approval-voting/|date=2018-11-07}}, accessed November 7, 2018</ref><ref name="Fargo votes">{{cite web |url=https://www.publicnewsservice.org/2020-06-10/civic-engagement/fargo-becomes-first-u-s-city-to-try-approval-voting/a70495-1 |title=Fargo Becomes First U.S. City to Try Approval Voting |last=Moen |first=Mike |date=June 10, 2020 |work=Public News Service |access-date=December 3, 2020 }}</ref><ref name="St. Louis approves">{{cite web|last=|first=|date=November 4, 2020|title=St. Louis Voters Approve Nonpartisan Elections|url=https://www.usnews.com/news/best-states/missouri/articles/2020-11-04/st-louis-voters-approve-nonpartisan-elections|url-status=live|archive-url=|archive-date=|access-date=December 3, 2020|work=US News and World Report}}</ref>
 
* [[Fargo, North Dakota|'''Fargo, North Dakota''']] (as of 2018) — Fargo used approval voting in June 2020 to elect two at-large seats on its city council,<ref name="Ballotpedia Fargo">[https://ballotpedia.org/Fargo,_North_Dakota,_Measure_1,_Approval_Voting_Initiative_(November_2018) Fargo, North Dakota, Measure 1, Approval Voting Initiative (November 2018)], November 7, 2018 ''[[Ballotpedia]]''</ref><ref name="Fargo approves">[https://ivn.us/2018/11/06/one-americas-famous-towns-becomes-first-nation-adopt-approval-voting/ One of America’s Most Famous Towns Becomes First in the Nation to Adopt Approval Voting] {{Webarchive|url=https://web.archive.org/web/20181107185459/https://ivn.us/2018/11/06/one-americas-famous-towns-becomes-first-nation-adopt-approval-voting/|date=2018-11-07}}, accessed November 7, 2018</ref><ref name="Fargo votes">{{cite web |url=https://www.publicnewsservice.org/2020-06-10/civic-engagement/fargo-becomes-first-u-s-city-to-try-approval-voting/a70495-1 |title=Fargo Becomes First U.S. City to Try Approval Voting |last=Moen |first=Mike |date=June 10, 2020 |work=Public News Service |access-date=December 3, 2020 }}</ref><ref name="St. Louis approves">{{cite web|last=|first=|date=November 4, 2020|title=St. Louis Voters Approve Nonpartisan Elections|url=https://www.usnews.com/news/best-states/missouri/articles/2020-11-04/st-louis-voters-approve-nonpartisan-elections|url-status=live|archive-url=|archive-date=|access-date=December 3, 2020|work=US News and World Report}}</ref>
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*Knoxville: 32 total votes
 
*Knoxville: 32 total votes
   
==Criterion compliances==
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==Criterion compliance==
 
Approval voting satisfies the [[unanimous consensus criterion]] and [[greatest possible consensus criterion]]. It is strongly promoted by advocates of consensus democracy for single-winner elections.
 
Approval voting satisfies the [[unanimous consensus criterion]] and [[greatest possible consensus criterion]]. It is strongly promoted by advocates of consensus democracy for single-winner elections.
   
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*Chattanooga: 32
 
*Chattanooga: 32
 
*Knoxville: 32
 
*Knoxville: 32
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==== Indeterminacy of Outcome ====
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In certain elections, honest voters merely varying the cut-off where they give approval can lead to any particular candidate winning.<ref>{{Cite journal|last=Saari|first=Donald G.|last2=Jill|first2=Van Newenhizen|year=1988|title=The problem of indeterminancy in approval, multiple, and truncated voting systems|journal=Public Choice|volume=59|issue=2|pages=101–120|doi=10.1007/BF00054447|jstor=30024954}}</ref> Consider an election with 15 voters deciding among three candidates (A, B, C). The voters have the preferences
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{|
  +
|{A: 2, B: 1, C: 0} × 6
  +
|-
  +
|{B: 2, C: 1, A: 0} × 5
  +
|-
  +
|{C: 2, B: 1, A: 0} × 4.
  +
|}
  +
Even if all voters vote honestly, any candidate can win, dependent on which voters choose to approve a second candidate. If no voters approve of a second candidate, A wins. If CBA voters approve of C and B, and the other voters only approve their favorite, then B wins. If all BCA voters approve of B and C, and the other voters only approve their favorite, then C wins. Thus, as noted above, in such elections, voters have an incentive to strategically vary the number of candidates they approve of.
  +
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Approval voting advocates say this is a positive feature of approval voting, saying that the above example "demonstrates that AV responds positively to distinctions voters make among candidates that ordinal preference rankings do not mirror".<ref>{{Cite journal|last=Brams|first=Steven J.|last2=Fishburn|first2=Peter C.|last3=Merrill|first3=Samuel|author-mask3=Merrill, Samuel, III|year=1988|title=The responsiveness of approval voting: Comments on Saari and Van Newenhizen|journal=Public Choice|volume=59|pages=121–131|doi=10.1007/BF00054448}}</ref> That is, approval voting allows voters to better express their degree of approval. One example of such a situation is where we replace the CBA voter preferences with {C: 2.1, B: 2, A: 0}; in this case, it would be appropriate for B to win, as the CBA voters think C and B nearly equivalent.
   
 
==Effect on elections==
 
==Effect on elections==
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====Strategically electing a pairwise-preferred candidate====
 
====Strategically electing a pairwise-preferred candidate====
Supposing rational voters (see [[Approval cutoff#Rationality restrictions]] for examples; chiefly, supposing voters who equally prefer two candidates approve both or neither of them), voters can "simulate" a [[head-to-head matchup]] in Approval voting in the sense that if, between two candidates, the voters who prefer the candidate who pairwise wins the matchup move their [[approval threshold]] between the two candidates, then they can guarantee that the candidate who pairwise loses the matchup is not elected (or if there was a pairwise tie between the two candidates, then they can guarantee a tie between the two candidates). This is because all voters who equally prefer the two candidates will not create an approval-based margin between the two candidates, and because there are more voters who prefer the pairwise winner of the matchup over the other candidate, the pairwise winner will guaranteeably have more approvals (specifically, they will have at least as high an approval-based margin as they do in their pairwise margin over the other candidate). Note however that they can '''not''' always make the pairwise winner of the matchup, or a candidate preferred more than or equally to the pairwise winner by any of the voters who prefer the pairwise winner over the pairwise loser, win. This is most easily seen in [[chicken dilemma]]-type situations; see [[Equilibrium#Notes]] for an example. However, this is true when the winner of the pairwise matchup [[majority-beat]]<nowiki/>s all other candidates.
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Supposing rational voters (see [[Approval cutoff#Rationality restrictions]] for examples; chiefly, supposing voters who equally prefer two candidates approve both or neither of them), voters can "simulate" a [[head-to-head matchup]] in Approval voting in the sense that if, between two candidates, the voters who prefer the candidate who pairwise wins the matchup move their [[approval threshold]] between the two candidates, then they can guarantee that the candidate who pairwise loses the matchup is not elected (or if there was a pairwise tie between the two candidates, then they can guarantee a tie between the two candidates). This is because all voters who equally prefer the two candidates will not create an approval-based margin between the two candidates, and because there are more voters who prefer the pairwise winner of the matchup over the other candidate, the pairwise winner will be guaranteed to have more approvals (specifically, they will have at least as high an approval-based margin as they do in their pairwise margin over the other candidate). Note however that they can '''not''' always make the pairwise winner of the matchup, or a candidate preferred more than or equally to the pairwise winner by any of the voters who prefer the pairwise winner over the pairwise loser, win. This is most easily seen in [[chicken dilemma]]-type situations; see [[Equilibrium#Notes]] for an example. However, this is true when the winner of the pairwise matchup [[majority-beat]]<nowiki/>s all other candidates.
   
 
==See also==
 
==See also==

Revision as of 13:22, 1 July 2022

Wikipedia has an article on:
On an approval ballot, the voter can select any number of candidates.

Approval voting is a single-winner electoral system where each voter may select ("approve") any number of candidates. The winner is the candidate approved by the largest number of voters. It is distinct from plurality voting, in which a voter may choose only one option among several (where the option with the most selections is declared the winner). It is related to score voting in which voters give each option a score on a scale, and the option with the highest total of scores is selected.

Approval voting can also be used in multiwinner elections. See "multiwinner approval voting" on English Wikipedia to learn more about the multi-winner variant of approval voting.

Approval voting has been implemented in municipal elections in the United States. Voters approved of the method in referendums in Fargo, North Dakota and St. Louis, Missouri.

Robert J. Weber coined the term "Approval Voting" in 1971.[1] It was more fully published in 1978 by political scientist Steven Brams and mathematician Peter Fishburn.[2]

Usage

Approval voting has been and is currently used in many places over the years. Recently there has been some interest in the United States for municipal elections.[3] The following is a non-exhaustive list of prominent use of approval voting.

  • The United Nations currently uses Approval Voting to elect its secretary general.
  • China's National People's Congress (NPC), the largest Parliament body in the world, has been elected via, essentially, Approval Voting since 1979.
  • The Greek parliament was elected by means of Approval Voting during 1864-1926 with which they replaced their previous SMP system.
  • Approval voting was widely used with introducing democracy in the Soviet Union started by M.S. Gorbachev.
  • The Catholic Popes were elected via approval voting in 1294-1621, but with revotes and extra nominations until somebody attained 2/3 supermajority approval level.

Procedures

Under approval voting, each voter in the electorate may vote for as many or as few candidates as the voter chooses. Each voter may vote for as many options as they wish (but at most once per option). This is equivalent to saying that each voter may "approve" or "disapprove" each option by voting or not voting for it The votes for each option are tallied, and the option with the most approval marks wins the election.

Approval voting is typically used for single-winner elections but can be extended to multiple winners. The system is sometimes described as a limited form of score voting, where the score that voters are allowed to express is extremely constrained: "acceptable" (with one point awarded to the candidate) or "not acceptable" (with no points awarded).

Example

Tennessee's four cities are spread throughout the state

Imagine that Tennessee is having an election on the location of its capital. The population of Tennessee is concentrated around its four major cities, which are spread throughout the state. For this example, suppose that the entire electorate lives in these four cities, and that everyone wants to live as near the capital as possible.

The candidates for the capital are:

  • Memphis, the state's largest city, with 42% of the voters, but located far from the other cities
  • Nashville, with 26% of the voters, near the center of Tennessee
  • Knoxville, with 17% of the voters
  • Chattanooga, with 15% of the voters

The preferences of the voters would be divided like this:

42% of voters
(close to Memphis)
26% of voters
(close to Nashville)
15% of voters
(close to Chattanooga)
17% of voters
(close to Knoxville)
  1. Memphis
  2. Nashville
  3. Chattanooga
  4. Knoxville
  1. Nashville
  2. Chattanooga
  3. Knoxville
  4. Memphis
  1. Chattanooga
  2. Knoxville
  3. Nashville
  4. Memphis
  1. Knoxville
  2. Chattanooga
  3. Nashville
  4. Memphis

Supposing that voters voted for their two favorite candidates, the results would be as follows (a more sophisticated approach to voting is discussed below):

  • Memphis: 42 total votes
  • Nashville: 68 total votes
  • Chattanooga: 58 total votes
  • Knoxville: 32 total votes

Criterion compliance

Approval voting satisfies the unanimous consensus criterion and greatest possible consensus criterion. It is strongly promoted by advocates of consensus democracy for single-winner elections.

Approval voting satisfies a form of the monotonicity criterion. It also satisfies the participation criterion, the Consistency Criterion, the summability criterion, the Weak Defensive Strategy criterion, Independence of irrelevant alternatives, and the non-compulsory support criterion.

Potential for tactical voting

Approval voting passes a form of the monotonicity criterion, in that voting for a candidate never lowers that candidate's chance of winning. Indeed, there is never a reason for a voter to tactically vote for a candidate X without voting for all candidates he or she prefers to candidate X.

While independence of irrelevant alternatives usually implies a complete lack of a spoiler effect, in Approval voting the implication does not necessarily hold. If honest voters are expected to translate their preferences into an Approval ballot by making use of a particular rule, the rule may lead the election outcome to depend on what non-winning candidates were present. See implications of IIA.

Means of expressing sincere preferences

As approval voting does not offer a single method of expressing sincere preferences, but rather a plethora of them, voters are encouraged to analyze their fellow voters' preferences and use that information to decide which candidates to vote for. Some strategies include:

  1. Vote for every candidate you prefer to the leading candidate, and to also vote for the leading candidate if that candidate is preferred to the current second place candidate.
  2. For each candidate C, if the winner is more likely to come from the set of candidates that are worse than C than from the set of candidates that are better than C, then approve C, else don't.

In the above election, if Chattanooga is perceived as the strongest challenger to Nashville, voters from Nashville will only vote for Nashville, because it is the leading candidate and they prefer no alternative to it. Voters from Chattanooga and Knoxville will withdraw their support from Nashville, the leading candidate, because they do not support it over Chattanooga. The new results would be:

  • Memphis: 42
  • Nashville: 68
  • Chattanooga: 32
  • Knoxville: 32

If, however, Memphis were perceived as the strongest challenger, voters from Memphis would withdraw their votes from Nashville, whereas voters from Chattanooga and Knoxville would support Nashville over Memphis. The results would then be:

  • Memphis: 42
  • Nashville: 58
  • Chattanooga: 32
  • Knoxville: 32

Indeterminacy of Outcome

In certain elections, honest voters merely varying the cut-off where they give approval can lead to any particular candidate winning.[10] Consider an election with 15 voters deciding among three candidates (A, B, C). The voters have the preferences

{A: 2, B: 1, C: 0} × 6
{B: 2, C: 1, A: 0} × 5
{C: 2, B: 1, A: 0} × 4.

Even if all voters vote honestly, any candidate can win, dependent on which voters choose to approve a second candidate. If no voters approve of a second candidate, A wins. If CBA voters approve of C and B, and the other voters only approve their favorite, then B wins. If all BCA voters approve of B and C, and the other voters only approve their favorite, then C wins. Thus, as noted above, in such elections, voters have an incentive to strategically vary the number of candidates they approve of.

Approval voting advocates say this is a positive feature of approval voting, saying that the above example "demonstrates that AV responds positively to distinctions voters make among candidates that ordinal preference rankings do not mirror".[11] That is, approval voting allows voters to better express their degree of approval. One example of such a situation is where we replace the CBA voter preferences with {C: 2.1, B: 2, A: 0}; in this case, it would be appropriate for B to win, as the CBA voters think C and B nearly equivalent.

Effect on elections

The effect of this system is disputed. Instant-runoff voting advocates like the Center for Voting and Democracy argue that Approval Voting would lead to the election of "compromise candidates" disliked by few, and liked by few. A study by Approval advocates Steven Brams and Dudley R. Herschbach published in Science in 2000 argued that approval voting was "fairer" than preference voting on a number of criteria. They claimed that a close analysis shows that the hesitation to support a 'compromise candidate' to the same degree as one supports one's first choice (as approval voting requires) actually outweighs the extra votes that such second choices get. Accordingly, preference voting is more biased towards compromise candidates than approval voting - a non-obvious and surprising result. The United States organisation Citizens for Approval Voting was organized in December 2002 to promote the use of approval voting in all public single-winner elections.

Other issues and comparisons

Advocates of approval voting often note that a single simple ballot can serve for single, multiple, or negative choices. It requires the voter to think carefully about who or what they really accept, rather than trusting a system of tallying or compromising by formal ranking or counting. Compromises happen but they are explicit, and chosen by the voter, not by the ballot counting. Some features of approval voting include:

  • Unlike Condorcet method, instant-runoff voting, and other methods that require ranking candidates, approval voting does not require significant changes in ballot design, voting procedures or equipment, and it is easier for voters to use and understand. This reduces problems with mismarked ballots, disputed results and recounts.
  • Increasing options for voters, when compared with the common first-past-the-post system, could increase voter turnout
  • It provides less incentive for negative campaigning than many other systems.
  • It allows voters to express tolerances but not preferences. Some political scientists consider this a major advantage, especially where acceptable choices are more important than popular choices.

Multiple winners

Approval voting can be extended to multiple winner elections, either as block approval voting, a simple variant on block voting where each voter can select an unlimited number of candidates and the candidates with the most approval votes win, or as proportional approval voting which seeks to maximise the overall satisfaction with the final result using approval voting.

Note that the 'Voting Method,' what you do on the ballot, is only a single, and the least important, aspect of a 'Voting System.' Due to tabulation and how the winner is selected, multi-seat Approval Voting systems are fundamentally unrelated to single-seat Approval Voting systems.

Relation to effectiveness of choices

Operations research has shown that the effectiveness of a policy and thereby a leader who sets several policies will be sigmoidally related to the level of approval associated with that policy or leader.[citation needed] There is an acceptance level below which effectiveness is very low and above which it is very high. More than one candidate may be in the effective region, or all candidates may be in the ineffective region. Approval voting attempts to ensure that the most-approved candidate is selected, maximizing the chance that the resulting policies will be effective.

Ballot types

Approval ballots can be of at least four semi-distinct forms. The simplest form is a blank ballot where the names of supported candidates is written in by hand. A more structured ballot will list all the candidates and allow a mark or word to be made by each supported candidate. A more explicit structured ballot can list the candidates and give two choices by each. (Candidate list ballots can include spaces for write-in candidates as well.)

Approvalballotname.png Approvalballotword.png Approvalballotmark.png Approvalballotchoice.png

All four ballots are interchangeable. The more structured ballots may aid voters in offering clear votes so they explicitly know all their choices. The Yes/No format can help to detect an "undervote" when a candidate is left unmarked, and allow the voter a second chance to confirm the ballot markings are correct.

For casual usage

Approval voting is a simple way to make decisions in small groups without the flaws of FPTP:

As mentioned in the video, Approval voting can be calculated by mentioning each option or candidate's name, and asking the voters to raise their hands if they support that option/candidate. The option that got the most hands raised in its favor wins. (Score voting could potentially be done if voters were allowed to raise, say, anywhere from 0 to 5 fingers, or had specially colored cards that they could raise that indicated how many points they wanted to give each option).

Notes

Approval voting passes Favorite Betrayal, so unlike Choose-one FPTP voting, it never hurts a voter to support their favorite. Example where that makes a difference:

10 A>B

41 B>A

49 C>A

In FPTP, most A-top voters would likely support B instead, to avoid electing C. But with Approval, they can support both A and B, and if a few C-top voters support A as well, then A will win. This is also an example of averted center squeeze effect.

However, there is potential for what is known as the chicken dilemma, where one majority subfaction withholds support for the other subtraction to help its candidate win rather than the other subfaction's candidate.

Alternative names

Approval voting is also called set voting or unordered voting, because a voter expresses on their ballot the set of candidates that they prefer above all others, but is not allowed to rank (order) the candidates from 1st to 2nd to 3rd to... to last. In other words, it only allows voters to rank all candidates either 1st or last.

Unlimited number of candidates can be supported

See Number of supportable candidates in various voting methods. One of the major implications of Approval voting in relation to choose-one FPTP voting is that with Approval, there is no way to tell (solely with the vote totals for each candidate) whether a voter who supported one candidate did or didn't support another. For example, in FPTP (assuming no equal ranking was allowed, a la cumulative voting), if Obama gets 30 votes and Romney gets 20, then it is guaranteed that none of the 30 voters who supported Obama also supported Romney, and vice versa. But with Approval, it is possible (albeit unlikely in this example) that everyone who voted for Romney also voted for Obama, and that Obama really only has 10 voters who support him but not Romney. This means that proportional forms of Approval voting are not as precinct-summable as the proportional form of FPTP, SNTV, because not only must one know how many voters approved each candidate to calculate the winner(s) in Approval PR methods, but also which candidates each ballot approved. With choose-one ballots, knowledge of the former yields knowledge of the latter.

Connection to Condorcet methods

The Pairwise counting#Negative vote-counting approach approach is based on Approval voting.

Dichotomous preferences

The Approval voting winner is also always someone from the Smith set if voters' preferences truly are dichotomous (i.e. they don't have ranked preferences, but rather, honestly only support or oppose each candidate).

Equilibrium

Fully strategic Approval voting with perfectly informed voters generally elects the Condorcet winner, and more generally, someone from the Smith set; this is because a plurality of voters have an incentive to set their approval thresholds between the Smith candidate and the most-viable non-Smith candidate, resulting in at least the same approval-based margin as the Smith candidate has in their head-to-head matchup against the non-Smith candidate. A common argument for Approval>Condorcet methods is that when voters are honest, they get a utilitarian outcome, while if they are strategic, they at least get the CW. This is not as much the case with Score voting or STAR voting, but it is not possible to figure out who the CW is from Approval ballots, since only limited pairwise counting information can be inferred.

Using pairwise counting to find the result

Here is an example of finding the Approval voting result, and its ranking of all candidates, using pairwise counting and the Smith set ranking:
30 AB

20 BC

10 ADE

20 BCE
If the pairwise counting is done by looking at the margins on each voter's ballot, rather than the winning votes/approvals directly (i.e. in the A vs B matchup, the AB voters are recorded as having no preference, since they approve both candidates, but do have a preference for A>C in the A vs C matchup because they approve one candidate and not the other), then the table is:
Number of approvals Ranking of candidates B A C E D
70 1st B --- 20 (+10 Win) 30 (+30 Win) 50 (+40 Win) 50 (+40 Win)
40 2nd A

10 (-10 Loss)

--- 40 (Tie) 30 (+10 Win) 30 (+30 Win)
40 2nd C 0 (-30 Loss) 40 (Tie) --- 20 (+10 Win) 40 (+30 Win)
30 3rd E 10 (-40 Loss) 20 (-10 Loss)

10 (-10 Loss)

--- 20 (+20 Win)
10 4th D 10 (-40 Loss) 0 (-30 Loss) 10 (-30 Loss) 0 (-10 Loss) ---

This could also be done by treating each voter's Approval ballot as a ranked ballot where all approved candidates are equally ranked 1st and all other candidates are ranked last. This shows how Approval can be thought of as a Condorcet method where every candidate must be ranked either 1st or last.

Strategically electing a pairwise-preferred candidate

Supposing rational voters (see Approval cutoff#Rationality restrictions for examples; chiefly, supposing voters who equally prefer two candidates approve both or neither of them), voters can "simulate" a head-to-head matchup in Approval voting in the sense that if, between two candidates, the voters who prefer the candidate who pairwise wins the matchup move their approval threshold between the two candidates, then they can guarantee that the candidate who pairwise loses the matchup is not elected (or if there was a pairwise tie between the two candidates, then they can guarantee a tie between the two candidates). This is because all voters who equally prefer the two candidates will not create an approval-based margin between the two candidates, and because there are more voters who prefer the pairwise winner of the matchup over the other candidate, the pairwise winner will be guaranteed to have more approvals (specifically, they will have at least as high an approval-based margin as they do in their pairwise margin over the other candidate). Note however that they can not always make the pairwise winner of the matchup, or a candidate preferred more than or equally to the pairwise winner by any of the voters who prefer the pairwise winner over the pairwise loser, win. This is most easily seen in chicken dilemma-type situations; see Equilibrium#Notes for an example. However, this is true when the winner of the pairwise matchup majority-beats all other candidates.

See also

Footnotes

References

  1. Brams, Steven J.; Fishburn, Peter C. (2007), Approval Voting, Springer-Verlag, p. xv, ISBN 978-0-387-49895-9
  2. Brams, Steven; Fishburn, Peter (1978). "Approval Voting". American Political Science Review. 72 (3): 831–847. doi:10.2307/1955105. JSTOR 1955105.
  3. "The New Frontier: Seattle Approves Launches a Ballot Initiative Campaign". The Center for Election Science. 2021-11-17. Retrieved 2021-12-13.
  4. Fargo, North Dakota, Measure 1, Approval Voting Initiative (November 2018), November 7, 2018 Ballotpedia
  5. One of America’s Most Famous Towns Becomes First in the Nation to Adopt Approval Voting (Archived here: archiveforthis on 2018-11-07), accessed November 7, 2018
  6. Moen, Mike (June 10, 2020). "Fargo Becomes First U.S. City to Try Approval Voting". Public News Service. Retrieved December 3, 2020.
  7. "St. Louis Voters Approve Nonpartisan Elections". US News and World Report. November 4, 2020. Retrieved December 3, 2020.
  8. Rakich, Nathaniel (2021-03-01). "In St. Louis, Voters Will Get To Vote For As Many Candidates As They Want". FiveThirtyEight. Retrieved 2021-03-04.
  9. "March 2, 2021 Non-Partisan Primary Municipal Election". City of St. Louis Board of Election Commissioners. Retrieved 2021-03-04.
  10. Saari, Donald G.; Jill, Van Newenhizen (1988). "The problem of indeterminancy in approval, multiple, and truncated voting systems". Public Choice. 59 (2): 101–120. doi:10.1007/BF00054447. JSTOR 30024954.
  11. Brams, Steven J.; Fishburn, Peter C.; Merrill, Samuel, III (1988). "The responsiveness of approval voting: Comments on Saari and Van Newenhizen". Public Choice. 59: 121–131. doi:10.1007/BF00054448.
  12. Introduction copied from Wikipedia's Approval voting article (oldid=967925338 and oldid=1036612916)
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