# Multiwinner approval voting

Multiwinner approval voting, sometimes also called approval-based committee (ABC) voting, 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

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, sequential proportional approval voting, Phragmen's voting rules and the method of equal shares. 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) 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, excess method, and minimax approval. 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:
* Inclusion-strategyproofness means that no manipulation can result in electing a strict superset of the manipulator's approved candidates (as in the PAV example above).
* Cardinality-strategyproofness is a stronger property: it means that no manipulation can result in electing a larger number of the manipulator's approved candidates.
Strategyproofness properties can also be classified by the type of potential manipulations:
* Independence of irrelevant alternatives means that the relative merit of two committees is not influenced by candidates outside these two committees. This prevents a certain form of strategic voting: altering one’s vote with respect to irrelevant candidates to manipulate the outcome.
* Monotonicity means that a voter never loses from revealing his true set of approved candidates. This prevents another form of strategic voting: hiding some approved candidates.
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. 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:
* Subset-inclusion-strategyproofness: if an agent i with approved-candidates A$_i$ reports a subset of A$_i$ (and all other reports are the same), then no previously-unelected candidate from A$_i$ is elected. This property is weaker than inclusion-strategyproofness, as it considers only one type of manipulation: reporting a subset of one’s truthful approval set.
* Party-list-proportionality: We define a party-list profile as a profile characteristic of party-list voting, that is: there is a partition of the voters into k groups and a partition of the projects into k subsets, such that each voter from group i votes only and for all projects in group i. Party-list proportionality means that, in a party-list profile, if some singleton ballot {x} appears at least B/n times, then x is elected. This property is weaker than the property of lower quota from apportionment, and weaker than the justified representation property.
** An alternative property, for which the impossibility holds, is disjoint diversity. It means that, in a party-list profile with at most k different parties, the rule selects at least one member from each party.
* Weak efficiency: if a candidate x is not supported by anyone, and there are at least k candidates that are supported, then x is not elected.
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 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 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 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.
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.

## 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.

### Divisible committees

Bei, Lu and Suksompong 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 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

* Korean villages used block approval voting for competitive elections following the surrender of Japan, according to observations made by journalist Anna Louise Strong in 1946: "In one village there were twelve candidates, of whom five were to be chosen for the Village Committee. Each voter was given twelve cards, bearing the names of the candidates. He then cast his chosen ones into the white box and the rejected ones into the black."
* Several Swiss cantons elect their government using such methods and so do French cities with population below 1000.
* In 1963, the proportional representation system in East Germany was replaced by a procedure in which the candidates had to receive more than 50% of the votes. Had more candidates than seats in this constituency won the majority, the order of the list would determine who would join the Volkskammer.