Approval voting

Current electoral use

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.^[10][11][12] The party switched to using STAR voting in 2020.^[13][14]

It is also used in internal elections by the American Solidarity Party;^[15] the Green Parties of Texas^[16][17] and Ohio;^[18] the Libertarian National Committee;^[19] the Libertarian parties of Texas,^[20] Colorado,^[21][22] Arizona,^[23] and New York;^[24] Alliance 90/The Greens in Germany;^[25] and the Czech^[26] and German Pirate Party.^{[27][28][29][non-primary source needed]}

Approval has been adopted by several societies: the Society for Social Choice and Welfare (1992),^[30] Mathematical Association of America (1986),^[31] the American Mathematical Society,^[32] the Institute of Management Sciences (1987) (now the Institute for Operations Research and the Management Sciences),^[33] the American Statistical Association (1987),^[34] and the Institute of Electrical and Electronics Engineers (1987-2002).^[5]

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.^[5][35]

Approval voting was used for Dartmouth Alumni Association elections for seats on the College Board of Trustees until it was replaced with traditional runoff elections by an alumni vote of 82% to 18% in 2009.^[36][37] Dartmouth students used approval voting to elect their student body president from 2011-2016.^[38] The winner secured the support of 41% of voters (2011),^[39] 32% (2012),^[40] and under 40% (2013, 2014 and 2016).^[41][42][43] In the 2014 and 2016 elections, over 80 percent of voters selected only one candidate.^[42][43]

Historical

Robert J. Weber coined the term "Approval Voting" in 1971.^[44] It was more fully published in 1978 by political scientist Steven Brams and mathematician Peter Fishburn.^[45]

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.^[46]
• 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.^[47][48]
• According to Steven J. Brams, approval was used for unspecified elections in 19th century England.^[49]
• 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.^[50][51]
• Approval was used in Greek legislative elections from 1864 to 1923, after which it was replaced with party-list proportional representation.^[52]
• 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.^[53]

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.'"^[64] 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.

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.^[65] 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.^[66] In one sense, conditions where this can happen are robust and are not isolated cases.^[67] 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.^[68]

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.^[73] 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.^[74]

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.^[75]
• 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.^[76]
• 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).^[77]
• 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.^[78]