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Single winner electoral system

Approval voting is an electoral system in which voters can select any number of candidates instead of selecting only one.

Approval voting ballots show a list of all the candidates running and each voter indicates support for as many candidates as they see fit. Final tallies show how many votes each candidate received, and the winner is the candidate with the most support.

Approval voting advocates Steven Brams and Dudley R. Herschbach predict that Approval should increase voter participation, prevent minor party candidates from being spoilers, and reduce negative campaigning. One study showed that Approval would not have chosen the same two winners as plurality voting (Chirac and Le Pen) in the first round of the 2002 French presidential election; it instead would have chosen Chirac and 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. A generalized version of the Burr dilemma applies to Approval when two candidates are appealing to the same subset of voters. Although Approval differs from the voting system used in the Burr dilemma, Approval can still leave candidates and voters with the generalized dilemma of whether to compete or cooperate. But, Approval satisfies the favorite betrayal criterion, which means that it is always safe for a voter to give their true favorite maximum support. While in the modern era there have been relatively few competitive Approval elections where tactical voting is more likely, Brams argues that Approval usually elects Condorcet winners in practice. In 2018, Fargo, North Dakota, passed a local ballot initiative adopting Approval for the city's local elections, and it was used to elect officials in June 2020, becoming the first United States city and jurisdiction to adopt Approval. In November 2020, St. Louis, Missouri, passed Proposition D to authorize a variant of Approval (as unified primary) for municipal offices. It passed in the House on Apr 15, 2024.

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. In 2018, Fargo, North Dakota passed a ballot initiative adopting Approval for local elections, becoming the first U. S. 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 2020, St. Louis, Missouri passed an initiative to adopt Approval followed by a top two runoff (see Unified primary), thus becoming the second U. S. city to adopt Approval and the first to use a variant of it. The first such primary was held in March 2021, with voters expressing 1.1 to 1.6 approvals per ballot in races with more than two candidates.

The idea of approval was adopted by X. Hu and Lloyd Shapley in 2003 in studying authority distribution in organizations. 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). 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." (It is worth noting that most IEEE elections had only two candidates listed on the ballot.) 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 also true of other associations; 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. Approval also can be used in social scenarios as a fairer system compared to a first past the post equivalent, being able to avoid a spoiler effect while still being very quick to calculate.

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 might 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 allows for 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. 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.

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.

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.

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

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. However, an optimal vote can require voting for a candidate and not voting for a more preferred candidate if there 4 candidates or more. 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 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 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. Optimal strategic Approval fails to satisfy the Condorcet criterion and can elect a Condorcet loser. Strategic Approval 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.