Unanimous groups

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.

Cohesive groups

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.

• Proportional Approval Voting guarantees each L-cohesive group, 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.^{[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.^{[5]: Prop.2}
• AJR can be satisfied by fractional committees - committees in which members can serve for a fraction of a term. In particular, the Nash rule satisfies AJR.^[6] See also fractional social choice.

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:

• Every committee with average-satisfaction larger than L-1 satisfies EJR (by the pigeonhole principle).^[5] Hence, PAV and Local-Search-PAV satisfy EJR. PAV is the only one of Thiele's voting rules that satisfies EJR.^[3]
• The Method of Equal Shares^[7] is another polynomial-time computable rule that satisfies EJR.
• Another polytime algorithm that guarantees EJR is EJR-Exact.^[5]
• A simple algorithm that finds an EJR allocation is called "Greedy EJR". Looping L from k downwards to 1, this algorithm checks whether there is an L-cohesive subset of voters. If so, it chooses a largest L-cohesive subset, and adds some L candidates that are approved by all of them.^{[8]: Algorithm 1}
• Sequential-PAV satisfies EJR only for 1-cohesive groups, and only for k ≤ 5. For k ≥ 6, it fails EJR even for 1-cohesive groups.
• Monroe's rule fails EJR
• It is co-NP-complete to check whether a given committee satisfies EJR.

A further weakening of EJR is called 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]

• EJR implies PJR, but not vice-versa. For example,^[9]: Sec.4 consider a setting with 2k candidates and k voters. Voter i approves candidate i, as well as candidates k+1,...,2k. Note that the quota is one voter, and every L voters are an L-cohesive group. The committee 1,...,k satisfies PJR, since for every L voters, the union of their approval sets contains L winners. But it does not satisfy EJR, since every voter has only 1 approved winner. In contrast, the committee k+1,...,2k satisfies EJR.
• PAV satisfies EJR, so it satisfies PJR too; and it is also the only weight-based approval voting rule that satisfies PJR. However, Sequential-PAV violates PJR.
• Some of Phragmen's voting rules satisfy PJR, namely: the Leximax Phragmen - which is NP-hard to compute, and the Sequential Phragmen - which is polytime computable, and in addition, satisfies committee monotonicity. ^[10]
• When k divides n, Monroe and Greedy Monroe satisfy PJR. However, when k does not divide n, both Monroe and Greedy Monroe might violate PJR, except when L=1.^[4]
• Another rule that is both PJR and polytime computable is the maximin-support rule.^[11]
• It is co-NP-complete to check whether a given committee satisfies PJR.^[5]

Partially cohesive groups

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.

• As of 2023, it is an open question whether a CS committee always exists.

Peters, Pierczynski 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 at 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 every in every (L,B)-weak-cohesive group, there is at least one members who approves some B winners. Clearly, FJR implies EJR.

• FJR can always be satisfied by the Greedy Cohesive Rule (which is not polytime); it is open whether there are polytime algorithms satisfying FJR.

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.

• PAV, local-search-PAV and MES satisfy EJR+; the proofs are the same as the original proofs, as the original proofs do not use cohesivity - they only use the fact that one candidate approved by all group members is not elected.
• There is also a polytime greedy algorithm that finds an EJR+ committee: the Greedy Justified Candidate Rule.
• PJR+ can be verified in polynomial time by reduction to submodular optimization - in contrast to PJR which is coNP-hard to verify.
• EJR+ can be verified in polynomial time by the following simple algorithm: For every L between 1 and k, and for every unelected candidate c: count the number of voters who approve c, and approve fewer than L winners. If the number of such voters is at least L quotas, then the committee violates EJR+.
• EJR+ satisfies a weak form of Committee monotonicity: for all k, there is an EJR+ committee W of size k, and an unelected candidate c, such that adding c to W yields an EJR+ committee (of size k+1).

Perfect representation

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]

• PER is compatible with PJR and JR: for every instance that admits perfect representation, there exists a committee that satisfies PJR. However, PER is not compatible with EJR: there are instances in which perfect representations exist, but none of them satisfies EJR. PER is satisfied by the Monroe rule, and by the leximax-Phragmen's rule;^[10] but it is violated by Greedy Monroe, Sequential PAV, and PAV.^[4]

See also: Fully proportional representation.

Variable number of winners

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:

• Each voter gains satisfaction, not only from an elected candidate that he approves, but also from a non-elected candidate that he does not approve (this makes the problem similar to multi-issue voting, where each candidate is a binary issue).
• A group is L-large if it contains at least L*n/m voters (where m is the total number of candidates), and L-cohesive if in addition the group members agree on the placement of at least L candidates (that is: the intersection of A_i plus the intersection of C\A_i is at least L).
• A committee is r-AS (r-average-satisfaction) if for every L-cohesive group, the average of the members' satisfaction is at least r*L. The JR, PJR and EJR conditions are generalized in a similar way.
• The PAV rule chooses a committee that maximizes the sum of Harmonic(sat_i), where sat_i is the satisfaction of voter i. The sequential Phragmen rule and the Method of Equal Shares divide the load of each elected candidate among the voters who approve it, and the load of each unelected candidate among the voters who do not approve it. All these rules satisfy PJR. MES violates EJR; it is not known whether the other two satisfy it.
• A deterministic rule cannot guarantee r-AS for r = (m-1)/m+epsilon, for any epsilon>0. PAV, Phragmen and MES cannot guarantee r-AS for r = 1/2+epsilon. But there is a randomized rule that satisfies (29/32)-AS.

Price of justified representation

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]

Party-approval 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:

• PJR means that, for every L ≥ 1, in every L-cohesive voter group, the parties in the union of their approval sets are allocated at least L seats.
• EJR means that, for every integer L ≥ 1, in every L-cohesive voter group, the parties approved by at least one voter are allocated at least L seats.
• CS means that, for any voter group of size L*n/k voters (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 from parties he approves is not larger than in the original committee.

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.

Rank-based elections

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]

Trichotomous ballots

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 and regressive proportionality

Degressive proportionality means that small groups should be given above-proportional representation. It is common, for example, in the European parliament. For example, Penrose has suggested that each group should be represented in proportional 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 utilitarian approval voting (AV) rule aims to maximize 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 generalise 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.

Divisible goods

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}

• EJR means that, in any L-cohesive group, there is at least one agent that approves a subset of length at least L of the selected piece.
• PJR means that, for any L-cohesive group, the union of their approval-sets contains a subset of length at least L of the selected piece.

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(u_i), satisfies EJR and hence PJR. Note that the Nash rule can be seen as a continuous analoge of proportional approval voting, which maximizes the sum of Harmonic(u_i). 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:

• EJR-M guarantees to any L-cohesive group, when there is a set of resources of total size exactly L, that at least one group member receives utility at least L. EJR-M reduces to EJR both in settings with only indivislbe candidates and in settings with only a divisible candidate.
• EJR-b (for any real number b) guarantees to any L-cohesive group, that at least one group member receives utility larger tha L-b.

They prove that:

• For any b<1, EJR-b may be unattainable.
• The Nash rule does not satisfy EJR-b for any b.
• A rule called Greedy-EJR satisfies EJR-M, but runs in exponential time, and has proportionality degree ~L/2.
• A generalization of Equal Shares satisfies EJR-1 but not EJR-M, but satisfies EJR for divisible-only instances, and has proportionality degree ~L/2.
• A generalization of PAV, using an analytic extension to the harmonic series, satisfies EJR-1 but not EJR-M, does not satisfy EJR for divisible-only instances, and has proportionality degree larger than L-1.

Other adaptations

• Bulteau, Hazon, Page, Rosenfeld and Talmon^[25] adapted the JR axioms to multi-issue voting (also called: perpetual voting, public decision making, or sequential decision making). Their work was later extended by Chandak, Sashwat and Peters.^[26]
• Aziz, Lee and Talmon^[27] adapted the JR axioms to participatory budgeting.
• Brill, Laslier and Skowron^[28] adapted JR to degressive proportionality - assigning more weight to minorities.
• Mavrov, Munagala and Shen^[29] study the core and the JR axioms when there are constraints on the committee.
• Munagala, Shen, Wang and Wang^[30] study a multiplicative approximation of the core when agents may have non-additive satisfaction functions.