Approval voting

"Approve" redirects here. For other uses, see Approval (disambiguation).
A political cartoon about the Greek referendum, 1920, depicting a voter at the ballot box being watched by the spirit of War. Shows the form of ballot boxes used in 19th-20th century Greece for approval voting.
On an approval ballot, the voter can select any number of candidates

Approval voting is a single-winner voting method used for elections. Each voter may "approve" of (i.e., select) any number of candidates. The winner is the most-approved candidate.

Guy Ottewell first described the system in 1977.[1] and also by Robert J. Weber, who coined the term "Approval Voting."[2] It was more fully published in 1978 by political scientist Steven Brams and mathematician Peter Fishburn.[3]

Description

Approval voting can be considered a form of score voting, with the range restricted to two values, 0 and 1—or a form of majority judgment, with grades restricted to good and poor. Approval Voting can also be compared to plurality voting, without the rule that discards ballots that vote for more than one candidate.

By treating each candidate as a separate question, "Do you approve of this person for the job?" approval voting lets each voter indicate support for one, some, or all candidates. All votes count equally, and everyone gets the same number of votes: one vote per candidate, either for or against. Final tallies show how many voters support each candidate, and the winner is the candidate whom the most voters support.

Approval voting ballots show, for each office being contested, a list of the candidates running for that seat. Next to each name is a checkbox, or another similar way to mark "Yes" or "No" for that candidate. This "check yes or no" approach means approval voting provides one of the simplest ballots for a voter to understand.

Ballots on which the voter marked every candidate the same (whether yes or no) have no effect on the outcome of the election. Each ballot can, therefore, be viewed as a small "delta" that separates two groups of candidates, those supported and those that are not. Each candidate approved is considered preferred to any candidate not approved, while the voter's preferences among approved candidates is unspecified, and likewise the voter's preferences among unapproved candidates is also unspecified.

Uses

Approval voting has been adopted by the Mathematical Association of America (1986),[4] the American Mathematical Society,[5] the Institute of Management Sciences (1987) (now the Institute for Operations Research and the Management Sciences),[6] the American Statistical Association (1987),[7] and the Institute of Electrical and Electronics Engineers (1987). The IEEE board in 2002 rescinded its decision to use approval voting. IEEE Executive Director Daniel J. Senese stated that approval voting was abandoned because "few of our members were using it and it was felt that it was no longer needed."[8] Because none of these associations report results to their members and the public, it is difficult to evaluate Senese's claim and whether it is may also be true of the other associations; Steven Brams analysis of the 1987 Mathematical Association of American shows that 79 percent of voters cast a ballot for one candidate in a five candidate race for president, and the winner earned the approval of by 1,267 (32.3%) of 3,924 voters.[9]

Approval voting was used for Dartmouth Alumni Association elections for seats on the College Board of Trustees, but after some controversy[10] it was replaced with traditional runoff elections by an alumni vote of 82% to 18% in 2009.[11] Dartmouth students started to use approval voting to elect their student body president in 2011. In the first election, the winner secured the support of 41% of voters against several write-in candidates.[12] In 2012, Suril Kantaria won with the support of 32% of the voters.[13] In 2013, 2014 and 2016, the winners also earned the support of under 40% of the voters.[14][15][16] Results reported in The Dartmouth show that in the 2014 and 2016 elections, more than 80 percent of voters approved of only one candidate.[15][16]

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

The idea of approval was adopted by X. Hu and Lloyd Shapley in 2003[22] in studying authority distribution in organizations.

Approval voting also can be used in social scenarios as a fairer, but still quick system compared to an FPTP equivalent.

Consider this situation:

Ten friends have to choose between three places for lunch: Kombucha Kick (K), Meatlover's Mansion (M) and Super Sushi (S), and they are deciding via a vote. If they vote for their favourite place, and each person gets only one vote, the results could end up with: 4 votes for Kombucha, 3 for Meatlover's and 3 for Sushi. This would result in the group going to Kombucha, even though 6 of the 10 people did not vote for Kombucha (especially if some people despise Kombucha). An approval voting system would work by asking the group on which places they are ok with, allowing them multiple votes and simply tallying up which place has the most votes, appropriate for a social situation, as figuring out preferences and proportion can take too long for simple decisions such as lunch.

Effect on elections

Approval voting advocates Steven Brams and Dudley R. Herschbach predict that approval voting should increase voter participation, prevent minor-party candidates from being spoilers, and reduce negative campaigning.[23] The effect of this system as an electoral reform measure is not without critics, however. FairVote has a position paper arguing that approval voting has three flaws that undercut it as a method of voting and political vehicle.[24] They argue that it can result in the defeat of a candidate who would win an absolute majority in a plurality election, can allow a candidate to win who might not win any support in a plurality election, and has incentives for tactical voting.

One study showed that approval voting would not have chosen the same two winners as plurality voting (Chirac and Le Pen) in France's presidential election of 2002 (first round) – it instead would have chosen Chirac and Jospin as the top two to proceed to a runoff.[25] Le Pen lost by a very high margin in the runoff, 82.2% to 17.8%, a sign that the true top two had not been found. Straight approval voting without a runoff, from the study, still would have selected Chirac, but with an approval percentage of only 36.7%, compared to Jospin at 32.9%. Le Pen, in that study, would have received 25.1%. In the real primary election, the top three were Chirac, 19.9%, Le Pen, 16.9%, and Jospin, 16.2%.[26] A study of various "evaluative voting" methods (approval voting and score voting) during the French presidential election, 2012 showed that "unifying" candidates tended to do better, and polarizing candidates did worse, via the evaluative voting methods than via the plurality system.[27]

A generalized version of the Burr dilemma applies to approval voting when two candidates are appealing to the same subset of voters. Although approval voting differs from the voting system used in the Burr dilemma, approval voting can still leave candidates and voters with the generalized dilemma of whether to compete or cooperate.[28][29]

While in the modern era there have been relatively few competitive approval voting elections where tactical voting is more likely, Brams argues that approval voting usually elects Condorcet winners in practice.[30] Critics of the use of approval voting in the alumni elections for the Dartmouth Board of Trustees in 2009 placed its ultimately successful repeal before alumni voters, arguing that the system has not been electing the most centrist candidates. The Dartmouth editorialized that "When the alumni electorate fails to take advantage of the approval voting process, the three required Alumni Council candidates tend to split the majority vote, giving petition candidates an advantage. By reducing the number of Alumni Council candidates, and instituting a more traditional one-person, one-vote system, trustee elections will become more democratic and will more accurately reflect the desires of our alumni base."[31]

Strategic voting

Overview

Approval voting is vulnerable to Bullet Voting and Compromising, while it is 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.

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.

Strategic Approval voting differs from ranked choice voting methods where voters might reverse the preference order of two options. Strategic Approval voting, with more than two options, involves the voter changing their approval threshold. The voter decides which options to give the same rating, despite having a strict preference order between them.

Sincere voting

Approval voting experts describe sincere votes as those "... that directly reflect the true preferences of a voter, i.e., that do not report preferences 'falsely.'"[32] 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:

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:

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 voting includes sincere voting, rather than being an alternative to it.[33] 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 voting election can change, depending on which sincere votes are used. In some cases, approval voting 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 voting can be considered vulnerable to sincere, strategic voting.[34] In one sense, conditions where this can happen are robust and are not isolated cases.[35] On the other hand, the variety of possible outcomes has also been portrayed as a virtue of approval voting, representing the flexibility and responsiveness of approval voting, not just to voter ordinal preferences, but cardinal utilities as well.[36]

Dichotomous preferences

Approval voting avoids the issue of multiple sincere votes in special cases when voters have dichotomous preferences. For a voter with dichotomous preferences, approval voting is strategy-proof (also known as strategy-free).[37] When all voters have dichotomous preferences and vote the sincere, strategy-proof vote, approval voting is guaranteed to elect the Condorcet winner, if one exists.[38] 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.[33]

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.[39] 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 voting, the strategy-proof vote, if it exists, is a sincere vote.[32]

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.[33]

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 voting. Without providing specifics, he argues that the pragmatic judgements of voters about which candidates are acceptable should take precedence over the Condorcet criterion and other social choice criteria.[40]

Strategy with cardinal utilities

Voting strategy under approval is guided by two competing features of approval voting. On the one hand, approval voting fails the later-no-harm criterion, so voting for a candidate can cause that candidate to win instead of a more preferred candidate. On the other hand, approval voting 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 voting strategy that votes for those candidates that have a positive prospective rating.[41] 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. However, an optimal vote can require voting for a candidate and not voting for a more preferred candidate if there 4 candidates or more.[42]

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

Another strategy is to vote for the top half of the candidates, the candidates that have an above-median utility. When the voter thinks that others are balancing their votes randomly and evenly, the strategy maximizes the voter's power or efficacy, meaning that it maximizes the probability that the voter will make a difference in deciding which candidate wins.[44]

Optimal strategic approval voting fails to satisfy the Condorcet criterion and can elect a Condorcet loser. Strategic approval voting can guarantee electing the Condorcet winner in some special circumstances. For example, if all voters are rational and cast a strategically optimal vote based on a common knowledge of how all the other voters vote except for small-probability, statistically independent errors in recording the votes, then the winner will be the Condorcet winner, if one exists.[45]

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
  Candidates  
Fraction of Voters
(living close to)		 Memphis Nashville Chattanooga Knoxville Average
Memphis (42%) 100 15 10 0 31.25
Nashville (26%) 0 100 20 15 33.75
Chattanooga (15%) 0 15 100 35 37.5
Knoxville (17%) 0 15 40 100 38.75

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
  Candidate vote totals
Strategy scenario Winner Runner-up Memphis Nashville Chattanooga Knoxville
Zero-info Memphis Chattanooga 42 26 32 17
Memphis leading Chattanooga Three-way tie 42 58 58 58
Chattanooga leading Knoxville Chattanooga Nashville 42 68 83 17
Chattanooga leading Nashville Nashville Memphis 42 68 32 17
Nashville leading Memphis Nashville Memphis 42 58 32 32

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 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. While this extreme appears unrealistic, it actually reflects reality in the way that many voters become disenfranchised and apathetic if they see no candidates they approve of. In this way, there is evidence to suggest that 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 Electorate Approval of Candidate A Approval of Candidate B Approval of Candidate C Approval of Candidate D Average 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 Electorate Approval of Candidate A Approval of Candidate B Approval of Candidate C Approval of Candidate D Average 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 Electorate Approval of Candidate A Approval of Candidate B Approval of Candidate C Approval of Candidate D Average 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 Electorate Approval of Candidate A Approval of Candidate B Approval of Candidate C Approval of Candidate D Average 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 Electorate Approval of Candidate A Approval of Candidate B Approval of Candidate C Approval of Candidate D Average 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 voting 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:

Unrestricted domain Non-dictatorship Pareto efficiency Majority Monotone Consistency & Participation Condorcet Condorcet loser IIA Clone independence Reversal symmetry
Cardinal preferences Zero information, rational voters Yes Yes No[47] No Yes Yes No No No No Yes
Imperfect information, rational voters Yes Yes No No Yes Yes No No No No Yes
Strong Nash equilibrium (Perfect information, rational voters, and perfect strategy) Yes Yes Yes Yes Yes No Yes No[48] No Yes Yes
Absolute dichotomous cutoff Yes No[49] Yes[50] No Yes Yes No No Yes[51] Yes Yes
Dichotomous preferences Rational voters No[52] Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes

Other issues and comparisons

Multiple winners

Approval voting can be extended to multiple winner elections. A simple way to do so is 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. This does not provide proportional representation and is subject to the Burr dilemma, among other problems.

Other ways of extending Approval voting to multiple winner elections have been devised. Among these are satisfaction approval voting and proportional approval voting[53] for determining a proportional assembly, and minimax approval[54] for determining a consensus assembly where the least satisfied voter is satisfied the most.

Ballot types

Approval ballots can be of at least four semi-distinct forms. The simplest form is a blank ballot on which voters hand-write the names of the candidates they support. A more structured ballot lists all candidates, and voters mark each candidate they support. A more explicit structured ballot can list the candidates and provide two choices by each. (Candidate list ballots can include spaces for write-in candidates as well.)

All four ballots are theoretically equivalent. 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. The "single bubble" format is incapable of producing invalid ballots (which might otherwise be rejected in counting).

Unless the second or fourth format is used, fraudulently adding votes to an approval voting ballot does not invalidate the ballot (that is, it does not make it appear inconsistent). Thus, approval voting raises the importance of ensuring that the "chain of custody" of ballots is secure.

See also

Notes

  1. ↑ Ottewell, Guy (2004) [1987]. "Arithmetic of Voting". Universal Workshop. Retrieved 2010-05-08.
  2. ↑ Brams, Steven J.; Fishburn, Peter C. (2007), Approval Voting, Springer-Verlag, p. xv, ISBN 978-0-387-49895-9
  3. ↑ Brams, Steven; Fishburn, Peter (1978). "Approval Voting". American Political Science Review. 72 (3): 831–847. doi:10.2307/1955105. JSTOR 1955105.
  4. ↑ "MAA Bylaws". Maa.org. 2010-08-07. Retrieved 2014-11-06.
  5. ↑ "2015 American Mathematical Society Elections" (PDF). ams.org. Retrieved 2015-08-19.
  6. ↑ INFORMS bylaws Archived February 11, 2009, at the Wayback Machine., p. 7
  7. ↑ "ASA Bylaws". ASA. 2009-12-04. Retrieved 2014-11-06.
  8. ↑ "Going from Theory to Practice: The Mixed Success of Approval Voting" (PDF). 31 May 2010. Retrieved 2010-05-08.
  9. ↑ "Going from Theory to Practice: The Mixed Success of Approval Voting" (PDF). 17 July 2016. Retrieved 2010-05-08.
  10. ↑ VERBUM ULTIMUM Making Amends, April 2, 2009, Editorial, The Dartmouth
  11. ↑ Dartmouth Alumni Association Election Results: New Executive Committee Elected; Constitutional Amendment Passes, May 9, 2009, Dartmouth Office of Alumni Relations
  12. ↑ Hix '12, Dartmouth Student Body, both shafted in student election, April 16, 2011, The Little Green Blog
  13. ↑ Kantaria, Danford win Student Assembly elections, April 17, 2012, The Dartmouth
  14. ↑ Ferrari, Zhu elected to lead Student Assembly, April 16, 2013, The Dartmouth
  15. 1 2 Dennis, Cunningham to lead Assembly , April 15, 2014, The Dartmouth
  16. 1 2 Harrington will be Student Assembly president, April 17, 2016, The Dartmouth
  17. ↑ 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.
  18. ↑ Lines, Marji (1986). "Approval Voting and Strategy Analysis: A Venetian Example". Theory and Decision. 20 (2): 155–172. doi:10.1007/BF00135090.
  19. ↑ Mowbray, Miranda; Gollmann, Dieter (2007). "Electing the Doge of Venice: analysis of a 13th Century protocol" (PDF).
  20. ↑ "The Normative Turn in Public Choice" (PDF). 2006. p. 4. Archived (PDF) from the original on 31 May 2010. Retrieved 2010-05-08.
  21. ↑ "The "Wisnumurti Guidelines" for Selecting a Candidate for Secretary-General" (PDF).
  22. ↑ 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.
  23. ↑ 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.
  24. ↑ Why IRV
  25. ↑ 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.
  26. ↑ The French Approval Voting study, Rangevoting.org
  27. ↑ Baujard, Antoinette; Igersheim, Herrade; Lebon, Isabelle; Gavrel, Frédéric; Laslier, Jean-François (2014-06-01). "Who's favored by evaluative voting? An experiment conducted during the 2012 French presidential election". Electoral Studies. 34: 131–145. doi:10.1016/j.electstud.2013.11.003.
  28. ↑ Nagel, J. H. (2007). "The Burr Dilemma in Approval Voting" (PDF). The Journal of Politics. 69 (1): 43–58. doi:10.1111/j.1468-2508.2007.00493.x.
  29. ↑ Nagel, J. H. (2006). "A Strategic Problem in Approval Voting". Mathematics and Democracy. Studies in Choice and Welfare. Springer: 133–150. doi:10.1007/3-540-35605-3_10.
  30. ↑ Steven J. Brams, Mathematics and Democracy, Princeton University Press, 2008, p. 16, See also S. Brams and P. Fishburn, Going from Theory to Practice: The Mixed Success of Approval Voting (PDF)
  31. ↑ "VERBUM ULTIMUM: Making Amends". TheDartmouth.com. 2009-04-03. Retrieved 2010-05-08.
  32. 1 2 Brams, Steven and Fishburn, Peter (1983). Approval Voting, Boston: Birkhäuser, p. 29
  33. 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.
  34. ↑ 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.
  35. ↑ 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.
  36. ↑ 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.
  37. ↑ Brams, Steven and Fishburn, Peter (1983). Approval Voting, Boston: Birkhäuser, p. 31
  38. ↑ Brams, Steven and Fishburn, Peter (1983). Approval Voting, Boston: Birkhäuser, p. 38
  39. ↑ Brams, Steven and Fishburn, Peter (1983). Approval Voting, Boston: Birkhäuser, pp. 16–17
  40. ↑ 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.
  41. ↑ R., Myerson; R. J., Weber (1993). "A theory of Voting Equilibria". American Political Science Review. 87 (1): 102–114. doi:10.2307/2938959. JSTOR 2938959.
  42. ↑ Smith, Warren Completion of Gibbard-Satterthwaite impossibility theorem; range voting and voter honesty. Actually, there are no known examples of such situations for under 6 candidates, and definitely none for 3 candidates; the situation for 4 or 5 candidates is unknown.
  43. ↑ Brams, Steven and Fishburn, Peter (1983). Approval Voting, Boston: Birkhäuser, p. 85
  44. ↑ Brams, Steven and Fishburn, Peter (1983). Approval Voting, Boston: Birkhäuser, p. 74, 81
  45. ↑ Laslier, J.-F. (2006) "Strategic approval voting in a large electorate," IDEP Working Papers No. 405 (Marseille, France: Institut D'Economie Publique)
  46. ↑ Consistency implies participation, but not vice versa. For example, Score Voting complies with participation and consistency, but median ratings satisfies participation and fails consistency.
  47. ↑ 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.
  48. ↑ The probability of failing this criterion vanishes asymptotically as the number of voters grows.
  49. ↑ When each voter's absolute cutoff is determined using non-objective criteria (i.e., from a unique vantage point), independently of knowing the available alternatives, there can exist two alternatives for which one voter's cardinal preference decides arbitrarily, regardless of another voter's cardinal preference, or strength of preference.
  50. ↑ Pareto efficiency is implied by, and is weaker than the combination of Monotonicity, IIA and Non-Imposition (that every possible societal preference order should be achievable by some set of individual preference orders, which is the case in all of these scenarios). These three conditions were in fact specified in the original statement of Arrow's impossibility theorem
  51. ↑ The model assumes a voter has an immovable dichotomous 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.
  52. ↑ In a dichotomous preference society, voters do not have a preferred ordering for the alternatives, such as A>B>C. Each voter has a binary "yes" or "no" rating for any alternative, while having no degree of preference among alternatives with either rating.
  53. ↑ Plaza, Enric. "Technologies for political representation and accountability" (PDF). CiteSeerX 10.1.1.74.3284Freely accessible. Retrieved 2011-06-17.
  54. ↑ LeGrand, Rob; Markakis, Evangelos; Mehta, Aranyak. "Some Results on Approximating the Minimax Solution in Approval Voting" (PDF). doi:10.1145/1329125.1329365. Retrieved 2011-06-17.

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