Approval voting

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On an approval ballot, the voter can select any number of candidates. Approval ballot.svg
On an approval ballot, the voter can select any number of candidates.

Approval voting is a single-winner electoral system in which voters mark all the candidates they support, instead of just choosing one. The candidate with the highest approval rating is elected. Approval voting is currently in use for government elections in St. Louis, Missouri and Fargo, North Dakota.

Contents

Effect on elections

Research by social choice theorists Steven Brams and Dudley R. Herschbach found that approval voting would increase voter participation, prevent minor-party candidates from being spoilers, and reduce negative campaigning. [1] Brams' research concluded that approval can be expected to elect majority-preferred candidates in practical election scenarios, avoiding the center squeeze common to ranked-choice voting and primary elections. [2] [3]

One study showed that approval would not have chosen the same two winners as plurality voting (Chirac and Le Pen) in the first round of the 2002 French presidential election; it instead would have chosen Chirac and Jospin as the top two candidates to proceed to the runoff.

In the actual election, Le Pen lost by an overwhelming margin in the runoff, 82.2% to 17.8%, a sign that the true top two candidates had not been found. In the approval voting survey primary, Chirac took first place with 36.7%, compared to Jospin at 32.9%. Le Pen, in that study, received 25.1% and so would not have made the cut to the second round. In the real primary election, the top three were Chirac, 19.9%, Le Pen, 16.9%, and Jospin, 16.2%. [4] A study of various "evaluative voting" methods (approval and score voting) during the 2012 French presidential election showed that "unifying" candidates tended to do better, and polarizing candidates did worse, as compared to under plurality voting. [5]

Operational impacts

Usage

Current electoral use

Latvia

The Latvian parliament uses a modified version of approval voting within open list proportional representation, in which voters can cast either positive (approval) votes, negative votes or neither for any number of candidates. [6]

United States

Missouri

In November 2020, St. Louis, Missouri, passed Proposition D with 70% voting to authorize a variant of approval (unified primary) for municipal offices. [7] In 2021, the first mayoral election with approval voting saw Tishaura Jones and Cara Spencer move on to the general with 57% and 46% support. Lewis Reed and Andrew Jones were eliminated with 39% and 14% support, resulting in an average of 1.6 candidates supported by each voter in the 4 person race. [8]

North Dakota

In 2018, Fargo, North Dakota, passed a local ballot initiative adopting approval for the city's local elections, becoming the first United States city and jurisdiction to adopt approval. [9] [10] Previously in 2015, a Fargo city commissioner election had suffered from six-way vote-splitting, resulting in a candidate winning with an unconvincing 22% plurality of the vote. [11]

The first election was held June 9, 2020, selecting two city commissioners, from seven candidates on the ballot. [12] Both winners received over 50% approval, with an average 2.3 approvals per ballot, and 62% of voters supported the change to approval in a poll. [13] A poll by opponents of approval was conducted to test whether voters had in fact voted strategically according to the Burr dilemma. [14] They found that 30% of voters who bullet voted did so for strategic reasons, while 57% did so because it was their sincere opinion. [15] [16] Fargo's second approval election took place in June 2022, for mayor and city commission. The incumbent mayor was re-elected from a field of 7 candidates, with an estimated 65% approval, with voters expressing 1.6 approvals per ballot, and the two commissioners were elected from a field of 15 candidates, with 3.1 approvals per ballot. [17]

In 2023, the North Dakota legislature passed a bill which intended to ban approval voting. The bill was vetoed by governor Doug Burgum, citing the importance of "home rule" and allowing citizens control over their local government. The legislature attempted to overrule the veto but failed. [18]

Use by organizations

Approval has been used in privately administered nomination contests by the Independent Party of Oregon in 2011, 2012, 2014, and 2016. Oregon is a fusion voting state, and the party has cross-nominated legislators and statewide officeholders using this method; its 2016 presidential preference primary did not identify a potential nominee due to no candidate earning more than 32% support. [19] [20] [21] The party switched to using STAR voting in 2020. [22] [23]

It is also used in internal elections by the American Solidarity Party; [24] the Green Parties of Texas [25] [26] and Ohio; [27] the Libertarian National Committee; [28] the Libertarian parties of Texas, [29] Colorado, [30] [31] Arizona, [32] and New York; [33] Alliance 90/The Greens in Germany; [34] and the Czech [35] and German Pirate Party. [36] [37]

Approval has been adopted by several societies: the Society for Social Choice and Welfare (1992), [38] Mathematical Association of America (1986), [39] the American Mathematical Society, [40] the Institute of Management Sciences (1987) (now the Institute for Operations Research and the Management Sciences), [41] the American Statistical Association (1987), [42] and the Institute of Electrical and Electronics Engineers (1987). The IEEE board in 2002 rescinded its decision to use approval. IEEE Executive Director Daniel J. Senese stated that approval was abandoned because "few of our members were using it and it was felt that it was no longer needed." [3]

Historical

Rows of secret approval vote boxes from early 1900s Greece, where the voter drops a marble to the right or left of the box, through a tube, one for each candidate standing The story of the greatest nations, from the dawn of history to the twentieth century - a comprehensive history, founded upon the leading authorities, including a complete chronology of the world, and (14765071792).jpg
Rows of secret approval vote boxes from early 1900s Greece, where the voter drops a marble to the right or left of the box, through a tube, one for each candidate standing

Robert J. Weber coined the term "Approval Voting" in 1971. [43] It was more fully published in 1978 by political scientist Steven Brams and mathematician Peter Fishburn. [44]

Historically, several voting methods that incorporate aspects of approval have been used:

The idea of approval was adopted by X. Hu and Lloyd Shapley in 2003 in studying authority distribution in organizations. [52]

Strategic voting

Overview

Approval voting allows voters to select all the candidates whom they consider to be reasonable choices.

Strategic approval differs from ranked voting (aka preferential voting) methods where voters are generally forced to reverse the preference order of two options, which if done on a larger scale can cause an unpopular candidate to win. Strategic approval, with more than two options, involves the voter changing their approval threshold. The voter decides which options to give the same rating, even if they were to have a preference order between them. This leaves a tactical concern any voter has for approving their second-favorite candidate, in the case that there are three or more candidates. Approving their second-favorite means the voter harms their favorite candidate's chance to win. Not approving their second-favorite means the voter helps the candidate they least desire to beat their second-favorite and perhaps win.

Approval technically allows for but is strategically immune to push-over and burying.

Bullet voting occurs when a voter approves only candidate "a" instead of both "a" and "b" for the reason that voting for "b" can cause "a" to lose. The voter would be satisfied with either "a" or "b" but has a moderate preference for "a". Were "b" to win, this hypothetical voter would still be satisfied. If supporters of both "a" and "b" do this, it could cause candidate "c" to win. This creates the "chicken dilemma", as supporters of "a" and "b" are playing chicken as to which will stop strategic voting first, before both of these candidates lose.

Compromising occurs when a voter approves an additional candidate who is otherwise considered unacceptable to the voter to prevent an even worse alternative from winning.

Sincere voting

Approval experts describe sincere votes as those "... that directly reflect the true preferences of a voter, i.e., that do not report preferences 'falsely.'" [53] They also give a specific definition of a sincere approval vote in terms of the voter's ordinal preferences as being any vote that, if it votes for one candidate, it also votes for any more preferred candidate. This definition allows a sincere vote to treat strictly preferred candidates the same, ensuring that every voter has at least one sincere vote. The definition also allows a sincere vote to treat equally preferred candidates differently. When there are two or more candidates, every voter has at least three sincere approval votes to choose from. Two of those sincere approval votes do not distinguish between any of the candidates: vote for none of the candidates and vote for all of the candidates. When there are three or more candidates, every voter has more than one sincere approval vote that distinguishes between the candidates.

Examples

Based on the definition above, if there are four candidates, A, B, C, and D, and a voter has a strict preference order, preferring A to B to C to D, then the following are the voter's possible sincere approval votes:

  • vote for A, B, C, and D
  • vote for A, B, and C
  • vote for A and B
  • vote for A
  • vote for no candidates

If the voter instead equally prefers B and C, while A is still the most preferred candidate and D is the least preferred candidate, then all of the above votes are sincere and the following combination is also a sincere vote:

  • vote for A and C

The decision between the above ballots is equivalent to deciding an arbitrary "approval cutoff." All candidates preferred to the cutoff are approved, all candidates less preferred are not approved, and any candidates equal to the cutoff may be approved or not arbitrarily.

Sincere strategy with ordinal preferences

A sincere voter with multiple options for voting sincerely still has to choose which sincere vote to use. Voting strategy is a way to make that choice, in which case strategic approval includes sincere voting, rather than being an alternative to it. [54] This differs from other voting systems that typically have a unique sincere vote for a voter.

When there are three or more candidates, the winner of an approval election can change, depending on which sincere votes are used. In some cases, approval can sincerely elect any one of the candidates, including a Condorcet winner and a Condorcet loser, without the voter preferences changing. To the extent that electing a Condorcet winner and not electing a Condorcet loser is considered desirable outcomes for a voting system, approval can be considered vulnerable to sincere, strategic voting. [55] In one sense, conditions where this can happen are robust and are not isolated cases. [56] On the other hand, the variety of possible outcomes has also been portrayed as a virtue of approval, representing the flexibility and responsiveness of approval, not just to voter ordinal preferences, but cardinal utilities as well. [57]

Dichotomous preferences

Approval avoids the issue of multiple sincere votes in special cases when voters have dichotomous preferences. For a voter with dichotomous preferences, approval is strategyproof. [58] When all voters have dichotomous preferences and vote the sincere, strategy-proof vote, approval is guaranteed to elect the Condorcet winner, if one exists. [59] However, having dichotomous preferences when there are three or more candidates is not typical. It is an unlikely situation for all voters to have dichotomous preferences when there are more than a few voters. [54]

Having dichotomous preferences means that a voter has bi-level preferences for the candidates. All of the candidates are divided into two groups such that the voter is indifferent between any two candidates in the same group and any candidate in the top-level group is preferred to any candidate in the bottom-level group. [60] A voter that has strict preferences between three candidates—prefers A to B and B to C—does not have dichotomous preferences.

Being strategy-proof for a voter means that there is a unique way for the voter to vote that is a strategically best way to vote, regardless of how others vote. In approval, the strategy-proof vote, if it exists, is a sincere vote. [53]

Approval threshold

Another way to deal with multiple sincere votes is to augment the ordinal preference model with an approval or acceptance threshold. An approval threshold divides all of the candidates into two sets, those the voter approves of and those the voter does not approve of. A voter can approve of more than one candidate and still prefer one approved candidate to another approved candidate. Acceptance thresholds are similar. With such a threshold, a voter simply votes for every candidate that meets or exceeds the threshold. [54]

With threshold voting, it is still possible to not elect the Condorcet winner and instead elect the Condorcet loser when they both exist. However, according to Steven Brams, this represents a strength rather than a weakness of approval. Without providing specifics, he argues that the pragmatic judgments of voters about which candidates are acceptable should take precedence over the Condorcet criterion and other social choice criteria. [61]

Strategy with cardinal utilities

Voting strategy under approval is guided by two competing features of approval. On the one hand, approval fails the later-no-harm criterion, so voting for a candidate can cause that candidate to win instead of a candidate more preferred by that voter. On the other hand, approval satisfies the monotonicity criterion, so not voting for a candidate can never help that candidate win, but can cause that candidate to lose to a less preferred candidate. Either way, the voter can risk getting a less preferred election winner. A voter can balance the risk-benefit trade-offs by considering the voter's cardinal utilities, particularly via the von Neumann–Morgenstern utility theorem, and the probabilities of how others vote.

A rational voter model described by Myerson and Weber specifies an approval strategy that votes for those candidates that have a positive prospective rating. [62] This strategy is optimal in the sense that it maximizes the voter's expected utility, subject to the constraints of the model and provided the number of other voters is sufficiently large.

An optimal approval vote always votes for the most preferred candidate and not for the least preferred candidate, which is a dominant strategy. An optimal vote can require supporting one candidate and not voting for a more preferred candidate if there 4 candidates or more; however, such situations are inherently unstable, suggesting such strategy should be rare. [63]

Other strategies are also available and coincide with the optimal strategy in special situations. For example:

Strategy examples

In the example election described here, assume that the voters in each faction share the following von Neumann–Morgenstern utilities, fitted to the interval between 0 and 100. The utilities are consistent with the rankings given earlier and reflect a strong preference each faction has for choosing its city, compared to weaker preferences for other factors such as the distance to the other cities.

Voter utilities for each candidate city
Fraction of voters (living close to)CandidatesAverage
MemphisNashvilleChattanoogaKnoxville
Memphis (42%)1001510031.25
Nashville (26%)0100201533.75
Chattanooga (15%)0151003537.5
Knoxville (17%)0154010038.75

Using these utilities, voters choose their optimal strategic votes based on what they think the various pivot probabilities are for pairwise ties. In each of the scenarios summarized below, all voters share a common set of pivot probabilities.

Approval Voting results for scenarios using optimal strategic voting
Strategy scenarioWinnerRunner-upCandidate vote totals
MemphisNashvilleChattanoogaKnoxville
Zero-infoMemphisChattanooga42263217
Memphis leading ChattanoogaThree-way tie42585858
Chattanooga leading KnoxvilleChattanoogaNashville42688317
Chattanooga leading NashvilleNashvilleMemphis42683217
Nashville leading MemphisNashvilleMemphis42583232

In the first scenario, voters all choose their votes based on the assumption that all pairwise ties are equally likely. As a result, they vote for any candidate with an above-average utility. Most voters vote for only their first choice. Only the Knoxville faction also votes for its second choice, Chattanooga. As a result, the winner is Memphis, the Condorcet loser, with Chattanooga coming in second place. In this scenario, the winner has minority approval (more voters disapproved than approved) and all the others had even less support, reflecting the position that no choice gave an above-average utility to a majority of voters.

In the second scenario, all of the voters expect that Memphis is the likely winner, that Chattanooga is the likely runner-up, and that the pivot probability for a Memphis-Chattanooga tie is much larger than the pivot probabilities of any other pair-wise ties. As a result, each voter votes for any candidate they prefer more than the leading candidate, and also vote for the leading candidate if they prefer that candidate more than the expected runner-up. Each remaining scenario follows a similar pattern of expectations and voting strategies.

In the second scenario, there is a three-way tie for first place. This happens because the expected winner, Memphis, was the Condorcet loser and was also ranked last by any voter that did not rank it first.

Only in the last scenario does the actual winner and runner-up match the expected winner and runner-up. As a result, this can be considered a stable strategic voting scenario. In the language of game theory, this is an "equilibrium." In this scenario, the winner is also the Condorcet winner.

Dichotomous cutoff

As this voting method is cardinal rather than ordinal, it is possible to model voters in a way that does not simplify to an ordinal method. Modelling voters with a 'dichotomous cutoff' assumes a voter has an immovable approval cutoff, while having meaningful cardinal preferences. This means that rather than voting for their top 3 candidates, or all candidates above the average approval (which may result in their vote changing if one candidate drops out, resulting in a system that does not satisfy IIA), they instead vote for all candidates above a certain approval 'cutoff' that they have decided. This cutoff does not change, regardless of which and how many candidates are running, so when all available alternatives are either above or below the cutoff, the voter votes for all or none of the candidates, despite preferring some over others. This could be imagined to reflect a case where many voters become disenfranchised and apathetic if they see no candidates they approve of. In a case such as this, many voters may have an internal cutoff, and would not simply vote for their top 3, or the above average candidates, although that is not to say that it is necessarily entirely immovable.

For example, in this scenario, voters are voting for candidates with approval above 50% (bold signifies that the voters voted for the candidate):

Proportion of electorateApproval of Candidate AApproval of Candidate BApproval of Candidate CApproval of Candidate DAverage approval
25%90%60%40%10%50%
35%10%90%60%40%50%
30%40%10%90%60%50%
10%60%40%10%90%50%

C wins with 65% of the voters' approval, beating B with 60%, D with 40% and A with 35%

If voters' threshold for receiving a vote is that the candidate has an above average approval, or they vote for their two most approved of candidates, this is not a dichotomous cutoff, as this can change if candidates drop out. On the other hand, if voters' threshold for receiving a vote is fixed (say 50%), this is a dichotomous cutoff, and satisfies IIA as shown below:

A drops out, candidates voting for above average approval
Proportion of electorateApproval of Candidate AApproval of Candidate BApproval of Candidate CApproval of Candidate DAverage approval
25%–60%40%10%37%
35%–90%60%40%63%
30%–10%90%60%53%
10%–40%10%90%47%

B now wins with 60%, beating C with 55% and D with 40%

A drops out, candidates voting for approval > 50%
Proportion of electorateApproval of Candidate AApproval of Candidate BApproval of Candidate CApproval of Candidate DAverage approval
25%–60%40%10%37%
35%–90%60%40%63%
30%–10%90%60%53%
10%–40%10%90%47%

With dichotomous cutoff, C still wins.

D drops out, candidates voting for top 2 candidates
Proportion of electorateApproval of Candidate AApproval of Candidate BApproval of Candidate CApproval of Candidate DAverage approval
25%90%60%40%–63%
35%10%90%60%–53%
30%40%10%90%–47%
10%60%40%10%–37%

B now wins with 70%, beating C and A with 65%

D drops out, candidates voting for approval > 50%
Proportion of electorateApproval of Candidate AApproval of Candidate BApproval of Candidate CApproval of Candidate DAverage approval
25%90%60%40%–63%
35%10%90%60%–53%
30%40%10%90%–47%
10%60%40%10%–37%

With dichotomous cutoff, C still wins.

Compliance with voting system criteria

Most of the mathematical criteria by which voting systems are compared were formulated for voters with ordinal preferences. In this case, approval voting requires voters to make an additional decision of where to put their approval cutoff (see examples above). Depending on how this decision is made, approval satisfies different sets of criteria.

There is no ultimate authority on which criteria should be considered, but the following are criteria that many voting theorists accept and consider desirable:

Voting model: Pareto efficiency Majority Favorite Monotone and Participation Condorcet and Smith IIA Clone independence Reversal symmetry Strategyproof
Zero information No [lower-alpha 1] NoYesNoNoNoYesNo
Strong Nash equilibrium YesYesYesYesNoYesYesNo
Trembling hand equilibrium YesYesYesYesNoYesYesNo
Absolute cutoffNo [lower-alpha 2] NoYesNoYes [lower-alpha 3] YesYesNo
Dichotomous preferences [lower-alpha 4] YesYesYesYesYesYesYesYes

See also

Some variants and generalizations of approval voting are:

Notes

  1. ↑ When the criterion is failed, the result is always a tie between the alternative preferred by all voters and one or more other alternatives. The criterion can only be failed when the tied candidates are approved on every ballot cast in the election.
  2. ↑ When the criterion is failed, the result is always a tie between the alternative preferred by all voters and one or more other alternatives. The criterion can only be failed when the tied candidates are approved on every ballot cast in the election.
  3. ↑ The model assumes a voter has an immovable approval cutoff while also having meaningful cardinal preferences. When all available alternatives are either above or below the cutoff, the voter votes for all or none of the candidates, despite preferring some over others.
  4. ↑ Binary (or "dichotomous") preferences refer to situations where voters' opinions take the form of only "approve" or "disapprove" for each candidate, with no in-between. This situation is common for countries with a two-party system.

Related Research Articles

Score voting, sometimes called range voting, is an electoral system for single-seat elections. Voters give each candidate a numerical score, and the candidate with the highest average score is elected. Score voting includes the well-known approval voting, but also lets voters give partial (in-between) approval ratings to candidates.

Strategic or tactical voting is voting in consideration of possible ballots cast by other voters in order to maximize one's satisfaction with the election's results. For example, in plurality or instant-runoff, a voter may recognize their favorite candidate is unlikely to win and so instead support a candidate they think is more likely to win.

<span class="mw-page-title-main">Condorcet method</span> Pairwise-comparison electoral system

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

In an election, a candidate is called a majority winner or majority-preferred candidate if more than half of all voters would support them in a one-on-one race against any one of their opponents. Voting systems where a majority winner will always win are said to satisfy the majority-rule principle, because they extend the principle of majority rule to elections with multiple candidates.

The participation criterion, sometimes called votermonotonicity, is a voting system criterion that says candidates should never lose an election as a result of receiving too many votes in support. More formally, it says that adding more voters who prefer Alice to Bob should not cause Alice to lose the election to Bob.

The majority favorite criterion is a voting system criterion that says that, if a candidate would win more than half the vote in a first-preference plurality election, that candidate should win. Equivalently, if only one candidate is ranked first by a over 50% of voters, that candidate must win. It is occasionally referred to simply as the "majority criterion", but this term is more often used to refer to Condorcet's majority-rule principle.

The Borda count electoral system can be combined with an instant-runoff procedure to create hybrid election methods that are called Nanson method and Baldwin method. Both methods are designed to satisfy the Condorcet criterion, and allow for incomplete ballots and equal rankings.

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

<span class="mw-page-title-main">Bullet voting</span> Vote supporting only a single candidate

Bullet, single-shot, or plump voting is when a voter supports only a single candidate, typically to show strong support for a single favorite.

The Kemeny–Young method is an electoral system that uses ranked ballots and pairwise comparison counts to identify the most popular choices in an election. It is a Condorcet method because if there is a Condorcet winner, it will always be ranked as the most popular choice.

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.

The Borda method or order of merit is a positional voting rule which gives each candidate a number of points equal to the number of candidates ranked below them: the lowest-ranked candidate gets 0 points, the second-lowest gets 1 point, and so on. Once all votes have been counted, the option or candidate with the most points is the winner.

Instant-runoff voting (IRV), also known as ranked-choice voting (RCV), preferential voting (PV), or the alternative vote (AV), is a multi-round elimination method where the loser of each round is determined by the first-past-the-post method. In academic contexts, the term instant-runoff voting is generally preferred as it does not run the risk of conflating the method with methods of ranked voting in general.

<span class="mw-page-title-main">Rated voting</span> Electoral systems with independent candidate ratings

Rated, evaluative, graded, or cardinalvotingsystems are a class of voting methods which allow voters to state how strongly they support a candidate, which involves giving each one a grade on a separate scale. Cardinal methods and ordinal methods are the two categories of modern voting systems.

<span class="mw-page-title-main">Ranked voting</span> Voting systems that use ranked ballots

Ranked voting is any voting system that uses voters' orderings (rankings) of candidates to choose a single winner or multiple winners. More formally, a ranked rule is one that depends only on which of two candidates is preferred by a voter, and as such does not incorporate any information about intensity of preferences. Ranked voting systems vary dramatically in how preferences are tabulated and counted, which gives them very different properties.

The later-no-help criterion is a voting system criterion formulated by Douglas Woodall. The criterion is satisfied if, in any election, a voter giving an additional ranking or positive rating to a less-preferred candidate can not cause a more-preferred candidate to win. Voting systems that fail the later-no-help criterion are vulnerable to the tactical voting strategy called mischief voting, which can deny victory to a sincere Condorcet winner.

A major branch of social choice theory is devoted to the comparison of electoral systems, otherwise known as social choice functions. Viewed from the perspective of political science, electoral systems are rules for conducting elections and determining winners from the ballots cast. From the perspective of economics, mathematics, and philosophy, a social choice function is a mathematical function that determines how a society should make choices, given a collection of individual preferences.

In the fields of mechanism design and social choice theory, Gibbard's theorem is a result proven by philosopher Allan Gibbard in 1973. It states that for any deterministic process of collective decision, at least one of the following three properties must hold:

  1. The process is dictatorial, i.e. there is a single voter whose vote chooses the outcome.
  2. The process limits the possible outcomes to two options only.
  3. The process is not straightforward; the optimal ballot for a voter "requires strategic voting", i.e. it depends on their beliefs about other voters' ballots.

The sincere favorite or no favorite-betrayal criterion is a property of some voting systems, that says voters should have no incentive to vote for someone else over their favorite. It protects voters from having to engage in lesser-evil voting or a strategy called "decapitation".

Multiwinner, at-large, or committeevoting refers to electoral systems that elect several candidates at once. Such methods can be used to elect parliaments or committees.

References

  1. ↑ Brams, Steven J.; Herschbach, Dudley R. (2001). "The Science of Elections". Science. 292 (5521): 1449. doi:10.1126/science.292.5521.1449. JSTOR   3083781. PMID   11379606. S2CID   28262658.
  2. ↑ Brams, Steven J. (2008). Mathematics and Democracy: Designing Better Voting and Fair-Division Procedures . Princeton University Press. p.  16. ISBN   9780691133218.
  3. 1 2 Brams, Steven J.; Fishburn, Peter C. (August 2003). Going from Theory to Practice: The Mixed Success of Approval Voting (PDF). Annual Meeting of the American Political Science Association. American Political Science Association. Archived from the original (PDF) on December 18, 2018. Retrieved May 8, 2010.
  4. ↑ Laslier, Jean-François; Vander Straeten, Karine (April 2003). "Approval Voting: An Experiment during the French 2002 Presidential Election" (PDF). p. 6. Archived from the original (PDF) on May 7, 2005. Retrieved July 8, 2014.
  5. ↑ Baujard, Antoinette; Igersheim, Herrade; Lebon, Isabelle; Gavrel, Frédéric; Laslier, Jean-François (June 1, 2014). "Who's favored by evaluative voting? An experiment conducted during the 2012 French presidential election" (PDF). Electoral Studies. 34: 131–145. doi:10.1016/j.electstud.2013.11.003. Archived (PDF) from the original on October 9, 2022.
  6. ↑ "How do elections work in Latvia?". Electoral Reform Society. Retrieved July 28, 2024.
  7. ↑ "St. Louis Voters Approve Nonpartisan Elections". U.S. News & World Report. November 4, 2020. Archived from the original on June 14, 2021. Retrieved December 3, 2020.
  8. ↑ Clancy, Sam (March 2, 2021). "Tishaura Jones, Cara Spencer advance to general election in race for St. Louis Mayor". KSDK. St. Louis, MO. Retrieved May 22, 2024.
  9. ↑ One of America's Most Famous Towns Becomes First in the Nation to Adopt Approval Voting Archived November 7, 2018, at the Wayback Machine , accessed November 7, 2018
  10. ↑ Moen, Mike (June 10, 2020). "Fargo Becomes First U.S. City to Try Approval Voting". Public News Service. Retrieved December 3, 2020.
  11. ↑ Piper, Kelsey (November 15, 2018). "This city just approved a new election system never tried before in America". Vox. Retrieved July 8, 2020.
  12. ↑ Amundson, Barry (June 9, 2020). "Strand and Preston emerge as leaders in Fargo City Commission race". INFORUM. Archived from the original on June 10, 2020. Retrieved July 8, 2020.
  13. ↑ "Fargo's First Approval Voting Election: Results and Voter Experience". The Center for Election Science. June 17, 2020. Retrieved July 8, 2020.
  14. ↑ Nagel, Jack H. (2007). "The Burr Dilemma in Approval Voting" . The Journal of Politics. 69 (1): 43–58. doi:10.1111/j.1468-2508.2007.00493.x. ISSN   0022-3816. S2CID   154179804.
  15. ↑ RCV for Colorado. "Approval Voting Case Study". Google Docs. Archived from the original on November 24, 2021. Retrieved March 4, 2021.
  16. ↑ "Guest opinion: Emma Donahue and Linda S. Templin: Ballot issue 2E just makes sense - Ranked Choice Voting for Colorado". RCV for Colorado. Archived from the original on November 30, 2020. Retrieved March 4, 2021. Triton Polling called Fargo voters and found that a significant number had gamed that voting method by bullet voting, just like the experts said they would.
  17. ↑ Hamlin, Aaron (June 16, 2022). "Fargo's Second Approval Voting Election Runs Smoothly". The Center for Election Science. Retrieved July 6, 2022.
  18. ↑ Carvel, Tasha (April 19, 2023). "Fargo approval voting survives after Senate fails to override Burgum veto of ban". The Mighty 790 KFGO. Fargo, ND. Retrieved May 22, 2024.
  19. ↑ Peralta, Sal (June 17, 2016). "Following tense primary, Independent Party plans presidential preference vote (OPINION)". The Oregonian . Retrieved June 26, 2018.
  20. ↑ Radnovich, Connor (April 21, 2018). "Independent Party's internal conflict highlighted in Oregon governor race". Statesman Journal. Retrieved June 26, 2018.
  21. ↑ Richie, Rob (December 14, 2016). "New Lessons from Problems with Approval Voting in Practice". FairVote.
  22. ↑ "STAR VOTING ANNOUNCEMENT". Independent Party of Oregon. Salem. April 7, 2020. Retrieved April 11, 2020.
  23. ↑ "Independent Party of Oregon to utilize STAR system for primary". Herald and News. April 8, 2020. Retrieved April 10, 2020.
  24. ↑ Covich, Skylar. "Vice-Chair's Remarks". American Solidarity Party. Archived from the original on June 22, 2018. Retrieved June 21, 2018.
  25. ↑ "Platform". Green Party of Texas. Retrieved January 31, 2017. We support switching to a preferential voting method, such as approval voting.
  26. ↑ "By-Laws". Green Party of Texas. Archived from the original on January 29, 2017. Retrieved January 31, 2017. The Co-Chair, Treasurer, and Secretary positions shall be filled by using approval voting.
  27. ↑ "Ohio Green Party Presidential Nominating Convention (Columbus)". Ohio Green Party. Archived from the original on February 26, 2021. Retrieved January 31, 2017. We have agreed to use "approval voting" as opposed to "instant runoff voting" to make the Ohio Green Party choice for president
  28. ↑ "MEETING MINUTES LIBERTARIAN NATIONAL COMMITTEE" (PDF). October 4, 2020. Archived (PDF) from the original on October 9, 2022. VOTING METHOD: Approval Voting per OpaVote. Vote for as many candidates as desired.
  29. ↑ "State Platform". Libertarian Party of Texas. Archived from the original on January 30, 2017. Retrieved January 31, 2017. We also believe that all races should be decided by Approval Voting
  30. ↑ "Document:Colorado State Party Bylaws 2021". LPedia. Retrieved September 24, 2022. For any partisan offices, Convention Delegates shall vote by approval voting
  31. ↑ "Colorado Convention 2018". LPedia. Retrieved September 24, 2022. All candidates were selected by approval voting via standing vote.
  32. ↑ "Bylaws". Arizona Libertarian Party. Retrieved September 24, 2022. The remaining Delegates … shall be elected by approval voting … should there be more Delegate Candidates than allotted seats.
  33. ↑ "Rules of the Libertarian Party" (PDF). Libertarian Party of New York. November 7, 2021. Archived (PDF) from the original on October 9, 2022. All elections shall be held with approval voting
  34. ↑ "Grüne München: Satzung" [Statutes of the city association - Alliance 90/The Greens Munich-City District Association] (in German). January 18, 2017. Archived from the original on January 18, 2017. Retrieved March 5, 2023. Delegierte für Bundes, Landes- und Bezirksversammlungen werden per Zustimmungsblockwahl gewählt. Jede*r Stimmberechtigte hat so viele Stimmen, wie Bewerber*innen zur Wahl stehen, und kann jeder*m Bewerber*in eine oder keine Stimme geben.
  35. ↑ "Jednací řád celostátního fóra". Pirati.CZ (in Czech). Retrieved November 16, 2021.
  36. ↑ "Bundesparteitag 2010.1/Geschäftsordnung" (in German). German Pirate Party . Retrieved June 24, 2018.
  37. ↑ "Election Data from the German Pirate Party (Piratenpartei)". The Center for Election Science. Archived from the original on February 19, 2019. Retrieved September 24, 2022.
  38. ↑ "Statutes of the Society for Social Choice and Welfare" (PDF). Society for Social Choice and Welfare. Archived (PDF) from the original on October 9, 2022. Retrieved January 28, 2020.
  39. ↑ "MAA Bylaws". Mathematical Association of America. August 7, 2010. Retrieved November 6, 2014.
  40. ↑ "2015 American Mathematical Society Elections" (PDF). American Mathematical Society. Archived (PDF) from the original on October 9, 2022. Retrieved August 19, 2015.
  41. ↑ "Constitution of the Institute for Operations Research and the Management Sciences" (PDF). Institute for Operations Research and the Management Sciences. August 2017. p. 7. Archived (PDF) from the original on October 9, 2022.
  42. ↑ "Bylaws of the American Statistical Association" (PDF). American Statistical Association. September 19, 2018. Retrieved January 28, 2020.
  43. ↑ Brams, Steven J.; Fishburn, Peter C. (2007), Approval Voting, Springer-Verlag, p. xv, ISBN   978-0-387-49895-9
  44. ↑ Brams, Steven; Fishburn, Peter (1978). "Approval Voting". American Political Science Review. 72 (3): 831–847. doi:10.2307/1955105. JSTOR   1955105. S2CID   251092061.
  45. ↑ Colomer, Josep M.; McLean, Iain (1998). "Electing Popes: Approval Balloting and Qualified-Majority Rule". The Journal of Interdisciplinary History. 29 (1): 1–22. doi:10.1162/002219598551616. JSTOR   205972. S2CID   145296691.
  46. ↑ Lines, Marji (1986). "Approval Voting and Strategy Analysis: A Venetian Example". Theory and Decision. 20 (2): 155–172. doi:10.1007/BF00135090. S2CID   121512308.
  47. ↑ Mowbray, Miranda; Gollmann, Dieter (July 2007). Electing the Doge of Venice: analysis of a 13th Century protocol (PDF). IEEE Computer Security Foundations Symposium. Venice, Italy. Archived (PDF) from the original on October 9, 2022.
  48. ↑ Brams, Steven J. (April 1, 2006). The Normative Turn in Public Choice (PDF) (Speech). Presidential Address to Public Choice Society. New Orleans, Louisiana. Archived from the original (PDF) on May 31, 2010. Retrieved May 8, 2010.
  49. ↑ "The "Wisnumurti Guidelines" for Selecting a Candidate for Secretary-General" (PDF). Archived from the original (PDF) on February 27, 2008. Retrieved November 30, 2007.
  50. ↑ Tharoor, Shashi (October 21, 2016). "The inside Story of How I Lost the Race for the UN Secretary-General's Job in 2006". OPEN Magazine. Archived from the original on July 21, 2019. Retrieved March 6, 2019.
  51. ↑ Mavrogordatos, George Th. (1983). Stillborn Republic: Social Coalitions and Party Strategies in Greece 1922–1936. University of California Press. pp. 351–352.
  52. ↑ Hu, Xingwei; Shapley, Lloyd S. (2003). "On Authority Distributions in Organizations". Games and Economic Behavior. 45 (1): 132–170. doi:10.1016/S0899-8256(03)00130-1.
  53. 1 2 Brams & Fishburn 1983, p. 29.
  54. 1 2 3 Niemi, R. G. (1984). "The Problem of Strategic Behavior under Approval Voting". American Political Science Review. 78 (4): 952–958. doi:10.2307/1955800. JSTOR   1955800. S2CID   146976380.
  55. ↑ Yilmaz, M. R. (1999). "Can we improve upon approval voting?". European Journal of Political Economy. 15 (1): 89–100. doi:10.1016/S0176-2680(98)00043-3.
  56. ↑ Saari, Donald G.; Van Newenhizen, Jill (2004). "The problem of indeterminancy in approval, multiple, and truncated voting systems". Public Choice. 59 (2): 101–120. doi:10.1007/BF00054447. JSTOR   30024954. S2CID   154705078.
  57. ↑ Saari, Donald G.; Van Newenhizen, Jill (2004). "Is approval voting an 'unmitigated evil?' A response to Brams, Fishburn, and Merrill". Public Choice. 59 (2): 133–147. doi:10.1007/BF00054449. JSTOR   30024956. S2CID   154007278.
  58. ↑ Brams & Fishburn 1983, p. 31.
  59. ↑ Brams & Fishburn 1983, p. 38.
  60. ↑ Brams & Fishburn 1983, p. 16–17.
  61. ↑ Brams, S. J.; Remzi Sanver, M. (2005). "Critical strategies under approval voting: Who gets ruled in and ruled out". Electoral Studies. 25 (2): 287–305. doi:10.1016/j.electstud.2005.05.007.
  62. ↑ Myerson, R.; Weber, R. J. (1993). "A theory of Voting Equilibria" (PDF). American Political Science Review. 87 (1): 102–114. doi:10.2307/2938959. hdl: 10419/221141 . JSTOR   2938959. S2CID   143828854. Archived (PDF) from the original on October 9, 2022.
  63. ↑ Dutta, B; De Sinopoli, F.; Laslier, J.-F. (2006). "Approval voting: three examples". International Journal of Game Theory. 35: 27–38. CiteSeerX   10.1.1.365.8090 . doi:10.1007/s00182-006-0053-2. S2CID   801286.
  64. ↑ Brams & Fishburn 1983, p. 85.
  65. ↑ Laslier, J.-F. (2009). "The Leader rule: a model of strategic approval voting in a large electorate". Journal of Theoretical Politics. 21 (1): 113–136. doi:10.1177/0951629808097286. S2CID   153790214.
  66. ↑ Brams & Fishburn 1983, p. 74, 81.
  67. ↑ Laslier, J.-F. (2006) "Strategic approval voting in a large electorate," IDEP Working Papers No. 405 (Marseille, France: Institut D'Economie Publique)

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