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Social choice and electoral systems |
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Multiwinner [1] or committee [2] [3] voting refers to electoral systems that elect several candidates at once. Such methods can be used to elect parliaments or committees.
There are many scenarios in which multiwinner voting is useful. They can be broadly classified into three classes, based on the main objective in electing the committee: [4]
A major challenge in the study of multiwinner voting is finding reasonable adaptations of concepts from single-winner voting. These can be classified based on the voting type—ranked voting as used in instant-runoff voting and single transferable voting vs. approval voting.
With multiwinner voting, there are many ways to decide which candidates should be elected. In some, each voter ranks the candidates; in others they cast X votes. Furthermore, depending on the system, each voter may cast single or multiple votes.
Some election systems elect multiple members by competition held among individual candidates. Each voter votes directly for one or more individual candidates. These systems include Plurality block voting and single non-transferable voting, adaptations of first-past-the-post voting to a multiwinner contest. Under SNTV, each voter casts only one vote, and that means no one party can take all the seats; although, because plurality system is used to allocate seats, parties are not guaranteed to take their proportional share of seats. A ranked-vote version of SNTV, Single transferable voting, elects a mixed, balanced group of members in a single contest in almost all cases.
In other systems, candidates are grouped in committees (slates or party lists) and voters cast votes for the committees (or slates). Sometimes only one slate or party takes all the seats, and sometimes members of various slates are elected.
Single transferable voting elects a mixed, balanced group of members in a single contest. It does this partly by allowing votes cast on unelectable candidates to be transferred to more-popular candidates. The quota used in STV ensures minority representation - no one group can take all the seats unless the district magnitude is small, or one party takes a great proportion of the votes cast.
Approval voting is a common method for single-winner elections and sometimes for multiwinner elections. In single-winner elections, voters mark their approved candidates, and the candidate with the most votes wins.
Already in 1895, Thiele suggested a family of weight-based rules called Thiele's voting rules. [2] [5] Each rule in the family is defined by a sequence of k weakly positive weights, w1,...,wk (where k is the committee size). Each voter assigns, to each committee containing p candidates approved by the voter, a score equal to w1+...+wp. The committee with the highest total score is elected. Some common voting rules in Thiele's family are:
There are rules based on other principles, such as minimax approval voting [6] and its generalisations, [7] as well as Phragmen's voting rules [8] and the method of equal shares. [9] [10]
The complexity of determining the winners vary: MNTV winners can be found in polynomial time, while Chamberlin-Courant [11] and PAV are both NP-hard.
Positional scoring rules are common in rank-based single-winner voting. Each voter ranks the candidates from best to worst, a pre-specified function assigns a score to each candidate based on his rank, and the candidate with the highest total score is elected.
In multiwinner voting held using these systems, we need to assign scores to committees rather than to individual candidates. There are various ways to do this, for example: [1]
In single-winner voting, a Condorcet winner is a candidate who wins in every head-to-head election against each of the other candidates. A Condorcet method is a method that selects a Condorcet winner whenever it exists. There are several ways to adapt Condorcet's criterion to multiwinner voting:
Excellence means that the elected committee should contain the "best" candidates. Excellence-based voting rules are often called screening rules. [18] They are often used as a first step in a selection of a single best candidate, that is, a method for creating a shortlist. A basic property that should be satisfied by such a rule is committee monotonicity (also called house monotonicity, a variant of resource monotonicity): if some k candidates are elected by a rule, and then the committee size increases to k+1 and the rule is re-applied, then the first k candidates should still be elected. Some families of committee-monotone rules are:
The property of committee monotonicity is incompatible with the property of stability (a particular adaptation of Condorcet's criterion): there exists a single voting profile that admits a unique Condorcet set of size 2, and a unique Condorcet set of size 3, and they are disjoint (the set of size 2 is not contained in the set of size 3). [18]
On the other hand, there exists a family of positional scoring rules - the separable positional scoring rules - that are committee-monotone. These rules are also computable in polynomial time (if their underlying single-winner scoring functions are). [1] For example, k-Borda is separable while multiple non-transferable vote is not.
Diversity means that the elected committee should contain the most-preferred candidates of as many voters as possible. Formally, the following axioms are reasonable for diversity-centred applications:
Proportionality means that each cohesive group of voters (that is: a group of voters with similar preferences) should be represented by a number of winners proportional to its size (the number of votes it receives). Formally, if the committee is of size k, there are n voters, and some L*n/k voters rank the same L candidates at the top (or give approval to the same L candidates), then these L candidates should be elected. This principle is easy to implement when the voters vote for parties (in party-list systems), but it can also be adapted to approval voting or ranked voting; see justified representation and proportionality for solid coalitions.
Proportionality may be measured just on the one usable preference that determines the vote's placement. In fact, in STV, only one preference is considered for each vote (unless fractional transfers are used as under a Gregory method). Voting blocs stay intact if back-up preferences are marked along party lines, but that is not always the case - in STV voters have the liberty to mark their preferences as they desire. Under STV the elected committee is composed of diverse representatives. Each substantial (quota-sized) group, as determined by the placement of the vote according to the top usable marked preference, elects its preferred candidate.
Haret, Klumper, Maly and Schafer [23] define, for each multiwinner voting instance, a corresponding budgeting game. In their game, each player has a fixed budget, and can choose a distribution of this budget among the candidates (so the strategy space of each player is a simplex). Each issue is active if its total allotment is least 1 and inactive otherwise. The utility of a player is the number of active issues that the player approves.
They prove that this game is a potential game and thus always has a pure-strategy Nash equilibrium. They also present several variants of equilibria in this game, and prove that they are equivalent (under certain assumptions) to various notions of justified representation in multiwinner voting.
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