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Approval voting is a single-winner voting system used for elections. Each voter may vote for (approve of) as many of the candidates as they wish. The winner is the candidate receiving the most votes. Each voter may vote for any combination of candidates and may give each candidate at most one vote.^[1]

Approval voting is a form of range voting with the range restricted to two values, 0 and 1. Approval voting can be compared to plurality voting without the rule that discards ballots which vote for more or less than one candidate.

The system was described in 1976 by Guy Ottewell^[2] and also by Robert J. Weber, who coined the term "approval voting." It was more fully published in 1978 by political scientist Steven Brams and mathematician Peter Fishburn.^[3] Approval voting is used by some professional societies. Voting systems which incorporated aspects of approval voting have been used historically.

Contents 1 Uses 2 Effect on elections 3 Strategic voting 3.1 Sincere voting 3.1.1 Examples 3.2 Strategy with ordinal preferences 3.2.1 Dichotomous preferences 3.2.2 Approval threshold 3.3 Strategy with cardinal utilities 3.4 Strategy examples 4 Compliance with voting system criteria 5 Other issues and comparisons 6 Multiple winners 7 Ballot types 8 See also 9 Notes 10 External links

[edit] Uses

Approval voting has been adopted by the Mathematical Association of America (1986)^[4] The Institute of Management Sciences (1987) (now the Institute for Operations Research and the Management Sciences),^[5] the American Statistical Association (1987),^[6] and Institute of Electrical and Electronics Engineers (1987). According to Steven J. Brams and Peter C. Fishburn, only the IEEE has rescinded the decision. They report the IEEE Executive Director, Daniel J. Senese, as stating that approval voting was abandoned in 2002 because "few of our members were using it and it was felt that it was no longer needed." Approval voting was implemented by the IEEE board and rescinded by the board.^[7] Approval voting also was used for several contested Dartmouth Alumni Association elections for seats on the College Board of Trustees, but replaced with traditional runoff elections by a membership vote of 82% to 18% in May 2009.^[8]

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

• Approving voting was used for papal conclaves between 1294 and 1621, with an average of about forty cardinals engaging in repeated rounds of voting until one candidate was listed on at least two-thirds of ballots.^[9]
• In the 13th through 18th centuries, the Republic of Venice elected the Doge of Venice using a multi-stage process that featured random selection and voting which allowed approval of multiple candidates and required a supermajority^[10][11].
• According to Steven J. Brams, approval voting was used in 19th century England.^[12]
• The selection of the Secretary-General of the United Nations has involved rounds of approval polling to help discover and build a consensus before a formal vote is held in the Security Council.^[13]

[edit] 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.^[14] The effect of this system as an electoral reform measure is not without critics, however. FairVote has a position paper ^[15] arguing that approval voting has three flaws that undercut it as a method of voting and political vehicle. It can result in the defeat of a candidate who would win an absolute majority in a plurality system, can allow a candidate to win who might not win any support in a plurality elections, and has incentives for tactical voting.

One study ^[16] 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. To some, this seemed a more reasonable result^{[citation needed]} since Le Pen was a radical who lost to Chirac by an enormous margin in the second round.

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.^[17][18]

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.^[19] 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."^[20].

[edit] Strategic voting

[edit] 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.' " ^[21] 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.

[edit] 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

[edit] 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.^[22] 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.^[23] In one sense, conditions where this can happen are robust and are not isolated cases.^[24] 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.^[25]

[edit] 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).^[26] 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.^[27] However, having dichotomous preferences when there are three or more candidates would not be typical. It would be an unlikely situation for all voters to have dichotomous preferences when there are more than a few voters.^[22]

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.^[28] 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.^[21]

[edit] 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.^[22]

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 advocates that the pragmatic judgements of voters about which candidates are acceptable should take precedence over the Condorcet criterion and other social choice criteria.^[29]

[edit] 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 von Neumann-Morgenstern utilities, and the probabilities of how others will 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.^[30] 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 will always vote for the most preferred candidate and not vote 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.^[31]

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

• Vote for the candidates that have above average utility. This strategy coincides with the optimal strategy if the voter thinks that all pairwise ties are equally likely ^[32]
• Vote for any candidate that is more preferred than the expected winner and also vote for the expected winner if the expected winner is more preferred than the expected runner-up. This strategy coincides with the optimal strategy if there are three or fewer candidates or if the pivot probability for a tie between the expected winner and expected runner-up is sufficiently large compared to the other pivot probabilities.
• Vote for the most preferred candidate only. This strategy coincides with the optimal strategy when there is only one candidate with a positive prospective rating.

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 the others will balance their votes randomly and evenly, the strategy will maximize the voter's power or efficacy, meaning that it will maximize the probability that the voter will make a difference in deciding which candidate wins.^[33]

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.^[34]

[edit] Strategy examples

In the example election described earlier, 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
Faction 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 will choose their optimal strategic votes based on what they think the various pivot probababilities 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 will vote for any candidate that is more preferred than the leading candidate and will also vote for the leading candidate if that candidate is more preferred than the expected runner-up. Each of the remaining scenarios 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.

[edit] 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. Depending on how that 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 some criteria that are accepted and considered to be desirable by many voting theorists:

• Majority criterion—If there exists a majority that ranks (or rates) a single candidate higher than all other candidates, does that candidate always win?
• Monotonicity criterion—Is it impossible to cause a winning candidate to lose by ranking him higher, or to cause a losing candidate to win by ranking him lower?
• Consistency criterion—If the electorate is divided in two and a choice wins in both parts, does it always win overall?
• Participation criterion—Is voting honestly always better than not voting at all? (This is grouped with the distinct but similar Consistency Criterion in the table below.^[35])
• Condorcet criterion—If a candidate beats every other candidate in pairwise comparison, does that candidate always win? (This implies the majority criterion, above)
• Condorcet loser criterion—If a candidate loses to every other candidate in pairwise comparison, does that candidate always lose?
• Independence of irrelevant alternatives—Is the outcome the same after adding or removing non-winning candidates?
• Independence of clone candidates—Is the outcome the same if candidates identical to existing candidates are added?
• Reversal symmetry—If individual preferences of each voter are inverted, does the original winner never win?

	Majority	Monotone	Consistency & Participation	Condorcet	Condorcet loser	IIA	Clone independence	Reversal symmetry
Inherently dichotomous preferences	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes
Arbitrary cutoff	No	Yes	Yes	No	No	No	No	Yes
Nash equilibrium (Perfect information, rational voters, and perfect strategy)	Yes	Yes	No	Yes	No[36]	No	Yes	Yes

[edit] Other issues and comparisons

• Approval voting can allow voters to cast a compromise vote without abandoning their favorite candidate. Plurality voting, in contrast, sometimes leads to voters abandoning their first choice in order to help a "lesser of evils" to win. Thus, approval voting reduces but does not eliminate the need to form parties or listen to polls. While candidates who both have a chance to win might be tempted to compete instead of cooperate, voters at least can choose to do either. With plurality, voters wanting to compromise might not know which candidate to compromise vote for.
• Approval vote counting is faster than some other alternative voting methods, such as ordinal systems. On the other hands, ballots could be vulnerable to backers of one candidate attempting to add additional votes for their preferred candidate, making ballot security very important.
• If voters are sincere, approval voting would elect centrists at least as often as moderates of each extreme. If backers of a relatively extreme candidate are insincere and "bullet vote," they could help that candidate elect a candidate who would have been a more consensus choice.
• If voters are sincere, candidates trying to win an approval voting election might need to get as much as 100% approval to beat a strong competitor, and would have to find solutions that are fair to everyone in order to do so, whereas a candidate may win a plurality race by promising many perks to a simple majority or even a plurality of voters at the expense of the smaller voting groups.
• Approval voting fails the majority criterion, because it is possible that the candidate most preferred by the majority of voters, for example, winning 60% in a plurality election, will lose, if 65% indicate another candidate is at least acceptable to them. If 40% strongly dislike candidate A but like candidate B, and 60% mildly prefer candidate A over candidate B, approval voting might elect candidate B, whereas plurality would elect candidate A in a two candidate race.
• Approval voting without write-ins is easily reversed as disapproval voting where a choice is disavowed, as is already required in other measures in politics (e.g. representative recall).
• Approval voting makes it much easier for voters to vote against an candidate by voting for several others instead of just one other, increasing the probability that some other candidate will win and thus that the first will not.
• In contentious elections with large groups of organized voters who prefer their favorite candidate vastly over all others, approval voting may revert to plurality voting. Some voters will support only their single favored candidate when they perceive the other candidates more as competitors to their preferred candidate than as compromise choices. Range voting allows these voters to give intermediate approval ratings, but at the cost of longer ballot counts.

[edit] Multiple winners

Approval voting can be extended to multiple winner elections. The naive 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.

[edit] 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 are 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.)

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.

[edit] See also

• Borda count
• Bucklin voting
• Burr dilemma
• Condorcet method
• First Past the Post electoral system (also called Single-Member Plurality or Relative Majority)
• Instant-runoff voting
• Range voting
• Schulze method
• Voting system - many other ways of voting

[edit] Notes

[edit] External links

Look up approval in Wiktionary, the free dictionary.

• Citizens for Approval Voting
• Americans for Approval Voting
• Approval Voting Free Association Wiki
• Approval Voting: A Better Way to Select a Winner Article by Steven J. Brams.
• Approval Voting on Dichotomous Preferences Article by Marc Vorsatz.
• Scoring Rules on Dichotomous Preferences Article by Marc Vorsatz.
• Approval Voting: An Experiment during the French 2002 Presidential Election Article by Jean-François Laslier and Karine Vander Straeten.
• The Arithmetic of Voting article by Guy Ottewell
• Critical Strategies Under Approval Voting: Who Gets Ruled In And Ruled Out Article by Steven J. Brams and M. Remzi Sanver.
• Going from Theory to Practice:The Mixed Success of Approval Voting Article by Steven J. Brams and Peter C. Fishburn.
• Strategic approval voting in a large electorate Article by Jean-François Laslier.
• Spatial approval voting Article by Jean-François Laslier, published in Political Analysis (2006).
• Approval Voting with Endogenous Candidates An article by Arnaud Dellis and Mandor P. Oak.
• Approval Voting and Parochialism Article by Jonathan Baron, Nicole Altman and Stephan Kroll.