Extended Justified Representation (EJR)

Updated 27 November 2025

Extended Justified Representation (EJR) is an axiomatic standard that guarantees proportional representation by requiring each sufficiently large, cohesive voter group to secure a fair number of approved candidates.
It refines basic representation axioms by ensuring that at least one voter in every ℓ-cohesive group receives at least ℓ approved representatives, thereby aligning group size with representation strength.
Algorithmic solutions such as Proportional Approval Voting (PAV) and the Method of Equal Shares (MES) provide practical means to achieve EJR despite inherent NP-hard challenges.

Extended Justified Representation (EJR) is an axiomatic standard capturing proportional representation strength in approval-based multiwinner voting, apportionment, participatory budgeting, and related settings. EJR generalizes the basic notion of justified representation to account not only for the existence of a representative for large cohesive groups, but also for the proportionality of representation aligned with the group’s size and its collective approvals. EJR is central to the axiomatic analysis of multiwinner voting protocols and provides a fairness yardstick for both algorithmic design and impossibility results across a wide variety of domains.

1. Formal Definition and Axiomatic Hierarchy

Given a set of voters $N = \{1,\dots,n\}$ , a candidate set $C$ , a target committee size $k \leq |C|$ , and each voter $i$ submitting an approval ballot $A_i \subseteq C$ , a committee $W \subseteq C$ of size $|W|=k$ is selected. A group $S \subseteq N$ is called $\ell$ -cohesive (for $\ell \in \mathbb{N}$ ) if (i) $|S| \geq \ell\cdot(n/k)$ and (ii) $|\cap_{i\in S} A_i| \geq \ell$ (Aziz et al., 2014, Sánchez-Fernández et al., 2016, Aziz et al., 2023, Tao et al., 2024).

Extended Justified Representation (EJR): A committee $W$ satisfies EJR if, for every $\ell \in [k]$ and every $\ell$ -cohesive group $S \subseteq N$ , there exists at least one voter $i \in S$ with $|A_i \cap W| \geq \ell$ (Aziz et al., 2014, Sánchez-Fernández et al., 2016, Aziz et al., 2023). That is, each sufficiently large and cohesive group is guaranteed that a member has at least as many approved representatives as their group proportion would suggest.

EJR refines the hierarchy of proportionality axioms: $\text{EJR} \implies \text{Proportional JR (PJR)} \implies \text{Justified Representation (JR)}$ where PJR weakens the demand so that $W$ must contain at least $\ell$ distinct candidates from the union of approvals of an $\ell$ -cohesive group, but not necessarily that a single voter from the group receives all $\ell$ (Sánchez-Fernández et al., 2016).

2. Theoretical Implications and Core Stability

EJR imposes strong proportionality guarantees. Specifically, if a group of at least $\ell\cdot(n/k)$ voters universally approves $\ell$ candidates, some member must receive all $\ell$ in the final committee. This is a strict strengthening over JR and PJR, which can permit significant under-representation of large cohesive subpopulations if they only ensure minimal, diffuse, or fractional coverage (Aziz et al., 2014, Sánchez-Fernández et al., 2016, Aziz et al., 2023).

EJR aligns with the notion of weak core stability in non-transferable utility (NTU) cooperative games associated to the approval voting setting. A committee with EJR ensures that no "cohesive" coalition (with size and common approvals satisfying the axiom conditions) can profitably deviate, although this does not amount to full core stability—non-cohesive coalitions can still block, and the full core can be empty even as EJR-committees exist in all instances (Aziz et al., 2014).

Extensions to more complex domains such as party-list apportionment and participatory budgeting preserve the conceptual core: EJR is adapted to ensure that, under (possibly vectorial) quota and budget constraints, a large, internally cohesive group always secures the appropriate threshold of representation (number of seats, funded projects, or utility) (Peters et al., 2020, Brill et al., 2019).

3. Algorithmic Solutions and Computational Complexity

EJR’s satisfaction by the Proportional Approval Voting (PAV) rule is a foundational result: PAV, using a harmonic utility function $H(j) = 1 + \frac12 + \cdots + \frac1j$ , always outputs an EJR committee (Aziz et al., 2014, Aziz et al., 2023). This property uniquely characterizes PAV among Thiele methods (weighted PAV with non-increasing weights) (Aziz et al., 2014, Sánchez-Fernández et al., 2016).

However, exact PAV is NP-hard to compute. EJR-committee existence is guaranteed in all profiles (Aziz et al., 2017, Skowron et al., 2017), and the problem of verifying whether a committee satisfies EJR is coNP-complete (Aziz et al., 2014, Aziz et al., 2017). Recent algorithmic breakthroughs present polynomial-time algorithms for constructing EJR-committees, notably via swap-based local search that optimizes (approximately) the PAV objective (Aziz et al., 2017, Skowron et al., 2017). Such algorithms iteratively replace committee members to maximize the PAV score, stopping once no swap yields a sufficiently large improvement, and can output a locally optimal committee that provably satisfies EJR in time polynomial in $n, m, k$ .

The Method of Equal Shares (MES), initially developed for participatory budgeting, also extends to approval-based committee voting, delivering EJR and supporting randomized (ex-post) and fractional (ex-ante) committee outcomes achieving EJR and stronger ex-ante proportionality criteria (Aziz et al., 2023, Peters et al., 2020).

The classical EJR property is binary, but its quantitative relaxation—EJR-degree—measures how many members of each $\ell$ -cohesive group are assigned their fair number of representatives (approving at least $\ell$ committee members) (Tao et al., 2024). For a given instance, the EJR-degree $c$ of a committee $W$ is the minimum over all $\ell$ -cohesive groups $S$ of $|\{i \in S : |A_i \cap W| \geq \ell\}|$ . Maximizing the EJR-degree is desirable for enhancing group satisfaction and group stability.

The associated optimization problem, MDEJR (Maximum Degree of Extended Justified Representation), is NP-hard and hard to approximate to within a factor of $(k/n)^{1-\varepsilon}$ . Nonetheless, any (polytime-computable) EJR-committee provides an $(k/n)$ -approximation by default. Furthermore, special parameterizations (e.g., fixing both committee size and desired degree) permit fixed-parameter tractable algorithms (Tao et al., 2024).

5. Extensions, Stronger Notions, and Temporal/Generalized Domains

EJR has been generalized and strengthened in numerous directions:

Temporal Voting: EJR, EJR $^+$ , and FJR have been adapted to temporal voting scenarios, where the notion of group cohesion and representation is distributed across multiple rounds. EJR $^+$ strictly strengthens EJR while remaining verifiable and computable in polynomial time. Full Justified Representation (FJR) strengthens further by requiring, in each group, that all members reach the collective threshold (Phillips et al., 28 May 2025).
Participatory Budgeting: EJR in PB protects budget shares of cohesive groups, requiring that for every group contributing more than the budget fraction of a bundle of projects, at least one member achieves full value. MES satisfies approval-based EJR for arbitrary costs and a relaxed “up to one project” EJR for additive utilities. Full Justified Representation (FJR) in PB settings admits greedy exponential-time rules with even stronger guarantees (Peters et al., 2020).
Apportionment: EJR under approval-based apportionment demands that any sufficiently large and cohesive group is directly allocated their proportional seat share by the committee. This strong guarantee can coexist with classical committee monotonicity (house monotonicity) via dedicated two-step apportionment rules (Brill et al., 2019).
Sub-Committee Voting: EJR extends to block-structured settings, imposing proportionality for cohesive groups across multiple blocks and quotas, with heightened computational intractability and existence limitations (Aziz et al., 2017).

EJR is strictly stronger than PJR and JR, providing greater guarantees for group welfare (average satisfaction) and resistance to group-based dissatisfaction (blocking coalitions). Comparative analysis indicates that PJR is compatible with Perfect Representation (PR) while EJR is not; EJR commits strongly to proportionality for cohesive blocks even if this excludes perfect balancing possible under PR (Sánchez-Fernández et al., 2016). The price of EJR’s stringency is seen in computational intractability for verification, partial incompatibility with some parliamentary apportionment desiderata, and (occasionally) a lack of existence in “vector” or multi-block generalizations.

Recent research situates EJR within a refined hierarchy of proportionality and stability axioms, such as EJR $^+$ , FJR, and core stability, with precise implications and strict separations between these concepts (Phillips et al., 28 May 2025, Aziz et al., 2023). Table 1 illustrates the logical relationships among major axioms (→ denotes implication):

Strongest	Middle	Weakest
Core, FJR, EJR⁺	EJR	PJR, JR

This structure reflects the static and dynamic settings discussed in (Phillips et al., 28 May 2025, Aziz et al., 2023).

7. Illustrative Examples

Several canonical examples clarify EJR’s operational meaning:

Single-winner case ( $k = 1$ ): For $n=4$ , committee $\{c_1\}$ may yield EJR-degree of 4 if every voter approves $c_1$ , but only 1 if only one voter does so (Tao et al., 2024).
Overlapping Cohesive Groups ( $k = 3, n = 9$ ): Committees maximizing average satisfaction may still violate EJR by leaving some cohesive groups without concentrated representation, while committees meeting EJR guarantee highly satisfied subgroups (Tao et al., 2024).
SCV/block scenarios: In sub-committee settings, SCV-EJR can fail to exist, and even when possible, is computationally hard to enforce (Aziz et al., 2017).