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.
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 than one candidate.
The system was described in 1976 by Guy Ottewell 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. Approval voting is used by some professional societies. Voting systems which incorporated aspects of approval voting have been used historically.
Approval voting has been adopted by the Mathematical Association of America (1986) The Institute of Management Sciences (1987) (now the Institute for Operations Research and the Management Sciences), the American Statistical Association (1987), 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 According to Steven J. Brams and Peter C. Fishburn, only the IEEE has rescinded the decision.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." Unlike the situation with MAA and ASA, Approval Voting was implemented by the IEEE board and rescinded by the board.
Historically, several voting methods which incorporate aspects of approval voting have been used:
Supposing that voters voted for their two favorite candidates and that Tennessee has 100 residents, the results would be as follows (a more sophisticated approach to voting is discussed below):
The effect of this system as an electoral reform measure is not without critics. Instant-runoff voting advocates like the Center for Voting and Democracy argue that approval voting would lead to the election of "lowest common denominator" candidates disliked by few, and liked by few, but this could also be seen as an inherent strength against demagoguery. In an editorial, 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.
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. This seems a more reasonable result 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.
According to Brams, Approval voting usually elects Condorcet winners in practice.
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.' " 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.
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:
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. 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. 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 voting, representing the flexibility and responsiveness of approval voting, not just to voter ordinal preferences, but cardinal utilities as well.
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). 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. 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.
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 voting, the strategy-proof vote, if it exists, is a sincere vote.
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 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.
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. 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.
Other strategies are also available and will 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 has no knowledge of how other voters will vote, 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.
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.
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.
| Faction of Voters|
(living close to)
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.
|Candidate Vote Totals|
|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 without any knowledge of, or at least without any regard for, how other voters might vote. 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 this scenario, the winner is also the Condorcet winner.
Advocates of approval voting often note that a single simple ballot can serve for single, multiple, or negative choices. It requires the voter to think carefully about whom or what they really accept, rather than trusting a system of tallying or compromising by formal ranking or counting. Compromises happen but they are explicit, and chosen by the voter, not by the ballot counting. Some features of approval voting include:
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.
Approval ballots can be of at least four semi-distinct forms. The simplest form is a blank ballot where the names of supported candidates is written in by hand. A more structured ballot will list all the candidates and allow a mark or word to be made by each supported candidate. A more explicit structured ballot can list the candidates and give two choices by each. (Candidate list ballots can include spaces for write-in candidates as well.)
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.
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