Condorcet methods
Positional voting
Cardinal voting
Quota-remainder methods
Approval-based committees
Fractional social choice
By ballot type
Strategy-resistance
Rational response
Positive results
Tournament sets
Justified representation (JR) is a criterion of fairness in multiwinner approval voting. It can be seen as an adaptation of the proportional representation criterion to approval voting.
Proportional representation (PR) is an important consideration in designing electoral systems. It means that the various groups and sectors in the population should be represented in the parliament in proportion to their size. The most common system for ensuring proportional representation is the party-list system. In this system, the candidates are partitioned into parties, and each citizen votes for a single party. Each party receives a number of seats proportional to the number of citizens who voted for it. For example, for a parliament with 10 seats, if exactly 50% of the citizens vote for party A, exactly 30% vote for party B, and exactly 20% vote for party C, then proportional representation requires that the parliament contains exactly 5 candidates from party A, exactly 3 candidates from party B, and exactly 2 candidates from party C. In reality, the fractions are usually not exact, so some rounding method should be used, and this can be done by various apportionment methods.
In recent years, there is a growing dissatisfaction with the party system.[1] A viable alternative to party-list systems is letting citizens vote directly for candidates, using approval ballots. This raises a new challenge: how can we define proportional representation, when there are no pre-specified groups (parties) that can deserve proportional representation? For example, suppose one voter approves candidate 1,2,3; another voter approves candidates 2,4,5; a third voter approves candidates 1,4. What is a reasonable definition of "proportional representation" in this case?[2] Several answers have been suggested; they are collectively known as justified representation.
Below, we denote the number of seats by k and the number of voters by n. The Hare quota is n/k - the minimum number of supporters that justifies a single seat. In PR party-list systems, each voter-group of at least L quotas, who vote for the same party, is entitled to L representatives of that party.
A natural generalization of this idea is an L-cohesive group, defined as a group of voters with at least L quotas, who approve at least L candidates in common.
Ideally, we would like to require that, for every L-cohesive group, every member must have at least L representatives. This condition, called Strong Justified Representation (SJR), might be unattainable, as shown by the following example.[3]
Example 1. There k=3 seats and 4 candidates {a,b,c,d}. There are n=12 voters with approval sets: ab, b, b, bc, c, c, cd, d, d, da, a, a. Note that the Hare quota is 4. The group {ab,b,b,bc} is 1-cohesive, as it contains 1 quota and all members approve candidate b. Strong-JR implies that candidate b must be elected. Similarly, The group {bc,c c,cd} is 1-cohesive, which requires to elect candidate c. Similarly, the group {cd,d,d,da} requires to elect d, and the group {da,a,a,ab} requires to elect a. So we need to elect 4 candidates, but the committee size is only 3. Therefore, no committee satisfies strong JR.
There are several ways to relax the notion of strong-JR.
One way is to guarantee representation only to an L-unanimous group, defined as a voter group with at least L quotas, who approve exactly the same set of at least L candidates. This condition is called Unanimous Justified Representation (UJR). However, L-unanimous groups are quite rare in approval voting systems, so Unanimous-JR would not be a very useful guarantee.
Remaining with L-cohesive groups, we can relax the representation guarantee as follows. Define the satisfaction of a voter as the number of winners approved by that voter. Strong-JR requires that, in every L-cohesive group, the minimum satisfaction of a group member is at least L. Instead, we can require that the average satisfaction of the group members is at least L. This weaker condition is called Average Justified Representation (AJR).[4] Unfortunately, this condition may still be unattainable. In Example 1 above, just like Strong-JR, Average-JR requires to elect all 4 candidates, but there are only 3 seats. In every committee of size 3, the average satisfaction of some 1-cohesive group is only 1/2.
We can weaken the requirement further by requiring that the maximum satisfaction of a group member is at least L. In other words, in every L-cohesive group, at least one member must have L approved representatives. This condition is called Extended Justified Representation (EJR); it was introduced and analyzed by Aziz, Brill, Conitzer, Elkind, Freeman, and Walsh.[3] There is an even weaker condition, that requires EJR to hold only for L=1 (only for 1-cohesive groups); it is called Justified Representation.[3] Several known methods satisfy EJR:
A further weakening of EJR is proportional justified representation (PJR). It means that, for every L ≥ 1, in every L-cohesive voter group, the union of their approval sets contains some L winners. It was introduced and analyzed by Sanchez-Fernandez, Elkind, Lackner, Fernandez, Fisteus, Val, and Skowron.[4]
The above conditions have bite only for L-cohesive groups. But L-cohesive groups may be quite rare in practice.[12] The above conditions guarantee nothing at all to groups that are "almost" cohesive. This motivates the search for more robust notions of JR, that guarantee something also for partially-cohesive group.
One such notion, which is very common in cooperative game theory, is core stability (CS).[3] It means that, for any voter group with L quotas (not necessarily cohesive), if this group deviates and constructs a smaller committee with L seats, then for at least one voter, the number of committee members he approves is not larger than in the original committee. EJR can be seen as a weak variant of CS, in which only L-cohesive groups are allowed to deviate. EJR requires that, for any L-cohesive group, at least one member does not want to deviate, as his current satisfaction is already L, which is the maximum satisfaction possible with L representatives.
Peters, Pierczyński and Skowron[13] present a different weakening of cohesivity. Given two integers L and B≤L, a group S of voters is called (L,B)-weak-cohesive if it contains at least L quotas, and there is a set C of L candidates, such that each member of S approves at least B candidates of C. Note that (L,L)-weak-cohesive is equivalent to L-cohesive. A committee satisfies Full Justified Representation (FJR) if in every (L,B)-weak-cohesive group, there is at least one members who approves some B winners. Clearly, FJR implies EJR.
Brill and Peters[14] present a different weakening of cohesivity. Given an elected committee, define a group as L-deprived if it contains at least L quotas, and in addition, at least one non-elected candidate is approved by all members. A committee satisfies EJR+ if for every L-deprived voter group, the maximum satisfaction is at least L (at least one group member approves at least L winners); a committee satisfies PJR+ if for every L-deprived group, the union of their approval sets contains some L winners. Clearly, EJR+ implies EJR and PJR+, and PJR+ implies PJR.
A different, unrelated property is Perfect representation (PER). It means that there is a mapping of each voter to a single winner approved by him, such that each winner represents exactly n/k voters. While a perfect representation may not exist, we expect that, if it exists, then it will be elected by the voting rule.[4]
See also: Fully proportional representation.
The following diagram illustrates the implication relations between the various conditions: SJR implies AJR implies EJR; CS implies FJR implies EJR; and EJR+ implies EJR and PJR+. EJR implies PJR, which implies both UJR and JR. UJR and JR do not imply each other.
EJR+ is incomparable to CS and to FJR.[14]: Rem.2
PER considers only instances in which a perfect representation exists. Therefore, PER does not imply, nor implied by, any of the other axioms.
Given the voters' preferences and a specific committee, can we efficiently check whether it satisfies any of these axioms?[5]
The satisfaction of a voter, given a certain committee, is defined as the number of committee members approved by that voter. The average satisfaction of a group of voters is the sum of their satisfaction levels, divided by the group size. If a voter-group is L-cohesive (that is, their size is at least L*n/k and they approve at least L candidates in common), then:
Proportional Approval Voting guarantees an average satisfaction larger than L-1. It has a variant called Local-Search-PAV, that runs in polynomial time, and also guarantees average satisfaction larger than L-1 (hence it is EJR).[5]: Thm.1,Prop.1 This guarantee is optimal: for every constant c>0, there is no rule that guarantees average satisfaction at least L-1+c (see Example 1 above).[5]: Prop.2
Skowron[15] studies the proportionality degree of multiwinner voting rules - a lower bound on the average satisfaction of all groups of a certain size.
Freeman, Kahng and Pennock[16] adapt the average-satisfaction concept to multiwinner voting with a variable number of winners. They argue that the other JR axioms are not attractive with a variable number of winners, whereas average-satisfaction is a more robust notion. The adaptation involves the following changes:
The price of justified representation is the loss in the average satisfaction due to the requirement to have a justified representation. It is analogous to the price of fairness.[8]
Bredereck, Faliszewski, Kaczmarczyk and Niedermeier[12] conducted an experimental study to check how many committees satisfy various justified representation axioms. They find that cohesive groups are rare, and therefore a large fraction of randomly selected JR committees, also satisfy PJR and EJR.
The justified-representation axioms have been adapted to various settings beyond simple committee voting.
Brill, Golz, Peters, Schmidt-Kraepelin and Wilker adapted the JR axioms to party-approval voting. In this setting, rather than approving individual candidates, the voters need to approve whole parties. This setting is a middle ground between party-list elections, in which voters must pick a single party, and standard approval voting, in which voters can pick any set of candidates. In party-approval voting, voters can pick any set of parties, but cannot pick individual candidates within a party. Some JR axioms are adapted to this setting as follows.[17]
A voter group is called L-cohesive if it is L-large, and all group members approve at least one party in common (in contrast to the previous setting, they need not approve L parties, since it is assumed that each party contains at least L candidates, and all voters who approve the party, automatically approve all these candidates). In other words, an L-cohesive group contains L quotas of voters who agree on at least one party:
The following example[17] illustrates the difference between CS and EJR. Suppose there are 5 parties {a, b, c, d, e}, k=16 seats, and n=16 voters with the following preferences: 4*ab, 3*bc, 1*c, 4*ad, 3*de, 1*e. Consider the committee with 8 seats to party a, 4 to party c, and 4 to party e. The numbers of representatives the voters are: 8, 4, 4, 8, 4, 4. It is not CS: consider the group of 14 voters who approve ab, bc, ad, de. They can form a committee with 4 seats to party a, 5 seats to party b, and 5 seats to party d. Now, numbers of representatives are: 9, 5, [0], 9, 5, [0], so all members of the deviating coalition are strictly happier. However, the original committee satisfies EJR. Note that the quota is 1. The largest L for which an L-cohesive group exists is L=8 (the ab and ad voters), and this group is allocated 8 seats.
The concept of JR originates from an earlier concept, introduced by Michael Dummett for rank-based elections. His condition is that, for every integer L ≥ 1, for every group of size at least L*n/k, if they rank the same L candidates at the top, then these L candidates must be elected.[18]
Talmon and Page[19] extend some JR axioms from approval ballots to trichotomous (three-choice) ballots, allowing each voter to express positive, negative or neutral feelings towards each candidate. They present two classes of generalizations: stronger ("Class I") and weaker ("Class II").
They propose some voting rules tailored for trichotomous ballots, and show by simulations the extent to which their rules satisfy the adapted JR axioms.
Degressive proportionality (sometimes progressive proportionality) accords smaller groups more representatives than they are proportionally entitled to and is used by the European Parliament. For example, Penrose has suggested that each group should be represented in proportion to the square root of its size.
The extreme of degressive proportionality is diversity, which means that the committee should represent as many voters as possible. The Chamberlin-Courant (CC) voting rule aims to maximize diversity. These ideas are particularly appealing for deliberative democracy, when it is important to hear as many diverse voices as possible.
On the other end, regressive proportionality means that large groups should be given above-proportional representation. The extreme of regressive proportionality is individual excellence, which means that the committee should contain members supported by the largest number of voters.[9]: Sec.4.5 The block approval voting (AV) rule maximizes individual excellence.
Lackner and Skowron[20] show that Thiele's voting rules can be used to interpolate between regressive and degressive proportionality: PAV is proportional; rules in which the slope of the score function is above that of PAV satisfy regressive proportionality; and rules in which the slope of the score function is below that of PAV satisfy degressive proportionality. Moreover,[21] If the satisfaction-score of the i-th approved candidate is (1/p)i, for various values of p, we get the entire spectrum between CC and AV.
Jaworski and Skowron[22] constructed a class of rules that generalize the sequential Phragmén’s voting rule. Intuitively, a degressive variant is obtained by assuming that the voters who already have more representatives earn money at a slower rate than those that have fewer. Regressive proportionality is implemented by assuming that the candidates who are approved by more voters cost less than those that garnered fewer approvals.
Bei, Lu and Suksompong[23] 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. To generalize the JR notions to this setting, they consider L-cohesive groups for any real number L (not necessarily an integer):[23]: App.A
They consider two solutions: the leximin solution satisfies neither PJR nor EJR, but it is truthful. In contrast, the Nash rule, which maximizes the sum of log(ui), satisfies EJR and hence PJR. Note that the Nash rule can be seen as a continuous analog of proportional approval voting, which maximizes the sum of Harmonic(ui). However, Nash is not truthful. The egalitarian ratio of both solutions is k/(n-nk+k).
Lu, Peters, Aziz, Bei and Suksompong[24] extend these definitions to settings with mixed divisible and indivisible candidates: there is a set of m indivisible candidates, as well as a cake [0,c]. The extended definition of EJR, which allows L-cohesive groups with non-integer L, may be unattainable. They define two relaxations:
They prove that:
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