Multiwinner approval voting explained

Multiwinner approval voting,[1] sometimes also called approval-based committee (ABC) voting,[2] refers to a family of multi-winner electoral systems that use approval ballots. Each voter may select ("approve") any number of candidates, and multiple candidates are elected.

Multiwinner approval voting is an adaptation of approval voting to multiwinner elections. In a single-winner approval voting system, it is easy to determine the winner: it is the candidate approved by the largest number of voters. In multiwinner approval voting, there are many different ways to decide which candidates will be elected.

Approval block voting

See main article: Approval block voting.

In approval block voting (also called unlimited voting), each voter either approves or disapproves of each candidate, and the k candidates with the most approval votes win (where k is the predetermined committee size). It does not provide proportional representation.

Proportional approval voting

Proportional approval voting refers to voting methods which aim to guarantee proportional representation in case all supporters of a party approve all candidates of that party. Such methods include proportional approval voting,[3] [4] sequential proportional approval voting, Phragmen's voting rules and the method of equal shares.[5] [6] In the general case, proportional representation is replaced by a more general requirement called justified representation.

In these methods, the voters fill out a standard approval-type ballot, but the ballots are counted in a specific way that produces proportional representation. The exact procedure depends on which method is being used.

Party-approval voting

Party-approval voting (also called approval-based apportionment)[7] is a method in which each voter can approve one or more parties, rather than approving individual candidates. It is a combination of multiwinner approval voting with party-list voting.

Other methods

Other ways of extending approval voting to multiple winner elections are satisfaction approval voting,[8] excess method,[9] and minimax approval.[10] These methods use approval ballots but count them in different ways.

Strategic voting

Many multiwinner voting rules can be manipulated: voters can increase their satisfaction by reporting false preferences.

Example

The most common form of manipulation is subset-manipulation, in which a voter reports only a strict subset of his approved candidates. This manipulation is called Hylland free riding: the manipulator free-rides on others approving a candidate, and pretends to be worse off than they actually are. Then, the rule is induced to "compensate" the manipulator by electing more of their approved candidates.

As an example, suppose we use the PAV rule with k=3, there are 4 candidates (a,b,c,d), and 5 voters, of whom three support a,b,c and two support a,b,d. Then, PAV selects a,b,c. But if the last voter reports only d, then PAV selects a,b,d, which is strictly better for him.

Strategyproofness properties

A multiwinner voting rule is called strategyproof if no voter can increase his satisfaction by reporting false preferences. There are several variants of this property, depending on the potential outcome of the manipulation:

Strategyproofness properties can also be classified by the type of potential manipulations:[11]

Lackner and Skowron focus on the class of ABC-counting rules (an extension of positional scoring rules to multiwinner voting). Among these rules, Thiele's rules are the only ones satisfying IIA, and dissatisfaction-counting-rules are the only ones satisfying monotonicity. Utilitarian approval voting is the only non-trivial ABC counting rule satisfying both axioms. It is also the only non-trivial ABC counting rule satisfying SD-strategyproofness—an extension of cardinality-strategyproofness to irresolute rules. If utilitarian approval voting is made resolute by a bad tie-breaking rule, it might become non-strategyproof.

Strategyproofness and proportionality

Cardinality-strategyproofness and inclusion-strategyproofness are satisfied by utilitarian approval voting (majoritarian approval voting rule with unlimited ballots), but not by any other known rule satisfying proportionality.

This raises the question of whether there is any rule that is both strategyproof and proportional. The answer is no: Dominik Peters proved that no multiwinner voting rule can simultaneously satisfy a weak form of proportionality, a weak form of strategyproofness, and a weak form of efficiency.[12] Specifically, the following three properties are incompatible whenever k ≥ 3, n is a multiple of k, and the number of candidates is at least k+1:

The proof is by induction; the base case (k=3) was found by a SAT solver. For k=2, the impossibility holds with a slightly stronger strategyproofness axiom.

Degree of manipulability

Lackner and Skowron quantified the trade-off between strategyproofness and proportionality by empirically measuring the fraction of random-generated profiles for which some voter can gain by misreporting. Example results, when each voter approves 2 candidates, are: Phragmen's sequential rule is manipulable in 66% of the profiles; Sequential PAV - 68%; PAV - 71%; Satisfaction AV and Maximin AV - 86%; Approval Monroe - 92%; Chamberlin-Courant - 95%. They also checked manipulability of Thiele's rules with p-geometric score function (where the scores are powers of 1/p, for some fixed p). Note that p=1 yields utilitarian AV, whereas p→∞ yields Chamberlin-Courant. They found out that increasing p results in increasing manipulability: rules which are more similar to utilitarian AV are less manipulable than rules that are more similar to CC, and the proportional rules are in-between.

Barrot, Lang and Yokoo[13] present a similar study of another family of rules, based on ordered weighted averaging and the Hamming distance. Their family is also characterized by a parameter p, where p=0.5 yields utilitarian AV, whereas p=1 yields egalitarian AV. They arrive at a similar conclusion: increasing p results in a larger fraction of random profiles that can be manipulated.

Restricted preference domains

One way to overcome impossibility results is to consider restricted preference domains. Botan[14] consider party-list preferences, that is, profiles in which the voters are partitioned into disjoint subsets, each of which votes for a disjoint subset of candidates. She proves that Thiele's rules (such as PAV) resist some common forms of manipulations, and it is strategyproof for "optimistic" voters.

Irresolute rules

The strategyproofness properties can be extended to irresolute rules (rules that return several tied committees). Lackner and Skowron define a strong extension called stochastic-dominance-strategyproofness, and prove that it characterizes the utilitarian approval voting rule.

Kluiving, Vries, Vrijbergen, Boixel and Endriss provide a more thorough discussion of strategyproofness of irresolute rules; in particular, they extend the impossibility result of Peters to irresolute rules. Duddy[15] presents an impossibility result using a different set of axioms.

Non-dichotomous preferences

There is an even stronger variant of strategyproofness called non-dichotomous strategyproofness: it assumes that agents have an underlying non-dichotomous preference relation, and they use approvals only as an approximation. It means that no manipulation can result in electing a committee that is ranked higher by the manipulator. Non-dichotomous strategproofness is not satisfied by any non-trivial multiwinner voting rule.[16]

Scheuerman, Harman, Mattei and Venable present behavioral studies on how people with non-dichotomous preferences behave when they need to provide an approval ballot, when the outcome is decided using utilitarian approval voting.[17] [18]

Extensions

Variable number of winners

Freeman, Kahng and Pennock study multiwinner approval voting in which the number of winners is not fixed in advance, but determined by the votes. For example, when selecting candidates for interview, if there are many strong candidates, then the number of candidates selected for interview may be larger. They extend the notion of average satisfaction to this setting.[19]

Divisible committees

Bei, Lu and Suksompong[20] extend the committee election model to a setting in which there is a continuum of candidates, represented by a real interval [0, ''c''], as in fair cake-cutting. The goal is to select a subset of this interval, with total length at most k, where here k and c can be any real numbers with 0<k<c. They generalize the justified representation notion to this setting. Lu, Peters, Aziz, Bei and Suksompong[21] extend these definitions to settings with mixed divisible and indivisible candidates (see justified representation).

Usage

Multiwinner approval voting, while less common than standard approval voting, is used in several places.

Block approval voting

External links

References

  1. 1407.3247 . cs.GT . Haris . Aziz . Serge . Gaspers . Computational Aspects of Multi-Winner Approval Voting . 2014-07-11 . Gudmundsson . Joachim . Mackenzie . Simon . Mattei . Nicholas . Walsh . Toby.
  2. Aziz . Haris . Brill . Markus . Conitzer . Vincent . Elkind . Edith . Freeman . Rupert . Walsh . Toby . 2017 . Justified representation in approval-based committee voting . Social Choice and Welfare . en . 48 . 2 . 461–485 . 1407.8269 . 10.1007/s00355-016-1019-3 . 8564247.
  3. http://www2.math.uu.se/~svante/papers/sjV9.pdf
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  11. Lackner . Martin . Skowron . Piotr . 2018-07-13 . Approval-based multi-winner rules and strategic voting . Proceedings of the 27th International Joint Conference on Artificial Intelligence . IJCAI'18 . Stockholm, Sweden . AAAI Press . 340–346 . 978-0-9992411-2-7.
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  14. Botan . Sirin . 2021-05-03 . Manipulability of Thiele Methods on Party-List Profiles . Proceedings of the 20th International Conference on Autonomous Agents and MultiAgent Systems . AAMAS '21 . Richland, SC . International Foundation for Autonomous Agents and Multiagent Systems . 223–231 . 978-1-4503-8307-3.
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  19. Freeman . Rupert . Kahng . Anson . Pennock . David M. . 2021-01-07 . Proportionality in approval-based elections with a variable number of winners . Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence . IJCAI'20 . Yokohama, Yokohama, Japan . 132–138 . 978-0-9992411-6-5.
  20. Bei . Xiaohui . Lu . Xinhang . Suksompong . Warut . 2022-06-28 . Truthful Cake Sharing . Proceedings of the AAAI Conference on Artificial Intelligence . en . 36 . 5 . 4809–4817 . 2112.05632 . 10.1609/aaai.v36i5.20408 . 2374-3468.
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