# Approval voting

Approval voting is a single-winner rated voting system where voters can approve of all the candidates as they like instead of choosing one. The method is designed to eliminate vote-splitting while keeping election administration simple and easy-to-count (requiring only a single score for each candidate). Approval voting has been used in both organizational and political elections to improve representativeness and voter satisfaction.
Critics of approval voting have argued the simple ballot format is a disadvantage, as it forces a binary choice for each candidate (instead of the expressive grades of other rated voting rules).

## 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. 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.
One study showed that approval would not have chosen the same two winners as plurality voting (Jacques Chirac and Jean-Marie Le Pen) in the first round of the 2002 French presidential election; it instead would have chosen Chirac and Lionel 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%. 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.

### Operational impacts

* Simple to tally—Approval ballots can be counted by some existing machines designed for plurality elections, as ballots are cast, so that final tallies are immediately available after the election, with relatively few if any upgrades to equipment.
* Just one round—Approval can remove the need for multiple rounds of voting, such as a primary or a run-off, simplifying the election process.
* Avoids overvotes—Approval voting does not have the notion of overvotes, where voting for one more than allowed will cancel the entire opportunity to vote. In plurality elections, overvotes have to be reviewed and resolved if possible while in approval voting, no time is wasted on this activity.

## Use
### 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.

#### 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. 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.

##### 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. 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.
The first election was held June 9, 2020, selecting two city commissioners, from seven candidates on the ballot. 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. A poll by opponents of approval was conducted to test whether voters had in fact voted strategically according to the Burr dilemma. They found that 30% of voters who bullet voted did so for strategic reasons, while 57% did so because it was their sincere opinion. 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.
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.

### 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. The party switched to using STAR voting in 2020.
It is also used in internal elections by the American Solidarity Party; the Green Parties of Texas and Ohio; the Libertarian National Committee; the Libertarian parties of Texas, Colorado, Arizona, and New York; Alliance 90/The Greens in Germany; and the Czech and German Pirate Party.
Approval has been adopted by several societies: the Society for Social Choice and Welfare (1992), Mathematical Association of America (1986), the American Mathematical Society, the Institute of Management Sciences (1987) (now the Institute for Operations Research and the Management Sciences), the American Statistical Association (1987), and the Institute of Electrical and Electronics Engineers (1987).
Steven Brams' analysis of the 5-candidate 1987 Mathematical Association of America presidential election shows that 79% of voters cast a ballot for one candidate, 16% for 2 candidates, 5% for 3, and 1% for 4, with the winner earning the approval of 1,267 (32%) of 3,924 voters. 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."
Approval voting was used for Dartmouth Alumni Association elections for seats on the College Board of Trustees, but after some controversy it was replaced with traditional runoff elections by an alumni vote of 82% to 18% in 2009. 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. In 2012, Suril Kantaria won with the support of 32% of the voters. In 2013, 2014 and 2016, the winners also earned the support of under 40% of the voters. Results reported in The Dartmouth show that in the 2014 and 2016 elections, more than 80 percent of voters approved of only one candidate. Students replaced approval voting with plurality voting before the 2017 elections.

### Historical

Robert J. Weber coined the term "Approval Voting" in 1971. It was more fully published in 1978 by political scientist Steven Brams and mathematician Peter Fishburn.
Historically, several voting methods that incorporate aspects of approval have been used:
* Approval 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.
* 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 that allowed approval of multiple candidates.
* According to Steven J. Brams, approval was used for unspecified elections in 19th century England.
* The Secretary-General of the United Nations is elected in a multi-round straw poll process where, in each round, members of the Security Council may approve or disapprove of candidates, or decide to express no opinion. Disapproval by permanent members of the Security Council is similar to a veto. A candidate with no vetoes, at least nine votes, and more votes than any other candidate is considered to be likely to be supported by the Security Council in its formal recommendation vote.
* Approval was used in Greek legislative elections from 1864 to 1923, after which it was replaced with party-list proportional representation.
* Sequential proportional approval voting was used in Swedish elections in the early 20th century, prior to being replaced by party-list proportional representation.
The idea of approval was adopted by X. Hu and Lloyd Shapley in 2003 in studying authority distribution in organizations.

## 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.'" 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. 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. In one sense, conditions where this can happen are robust and are not isolated cases. 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.

#### 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. When all voters have dichotomous preferences and vote the sincere, strategy-proof vote, approval is guaranteed to elect a Condorcet winner. 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.
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. 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.

#### 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.
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.

### 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. 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, e.g. the third and fourth choices are correlated to gain or lose decisive votes together; however, such situations are inherently unstable, suggesting such strategy should be rare.
Other strategies are also available and 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.
* 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. This strategy, if used by all voters, implies at equilibrium the election of the Condorcet winner whenever it exists.
* Vote for the most preferred candidate only. This strategy coincides with the optimal strategy when the best candidate is either much better than all others (i.e. is the only one with a positive expected value).
* If all voters are rational and cast a strategically optimal vote based on a common knowledge of how all other voters vote except for small-probability, statistically independent errors, then the winner will be the Condorcet winner, if one exists.

### 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) | Candidates | Candidates | Candidates | Candidates | Average |
| 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 :
| Strategy scenario | Winner | Runner-up | Candidate vote totals | Candidate vote totals | Candidate vote totals | 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 | 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 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

Modeling 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, 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.
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 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: | Majority | Monotone and Participation | Condorcet and Smith | IIA | Clone independence | Reversal symmetry | Sincere favorite | Strategyproof |
| Zero information | No | Yes | No | No | No | Yes | Yes | No |
| Leader rule | Yes | Yes | Yes | No | | | Yes | No |
| Trembling ballots | Yes | Yes | Yes | No | | | Yes | No |
| Binary preferences | Yes | Yes | Yes | Yes | Yes | Yes | Yes | Yes |