Justified Representation (JR) Axioms

• Justified Representation axioms are criteria ensuring that any sufficiently large, cohesive voter group secures at least one candidate they commonly approve.
• The axioms, including EJR, PJR, FPJR, and Droop-based variants, offer a hierarchical framework for fairness with specific mathematical definitions and computational properties.
• These principles guide the design of voting rules like PAV and MES, providing actionable, algorithmic approaches for achieving proportional representation.

Justified Representation (JR) axioms are central criteria in approval-based committee voting, intended to formalize proportional representation for large, cohesive groups of voters. These axioms, together with their extensions (EJR, PJR, and more recently robust, full, and quota-based variants), organize a rich landscape of fairness guarantees, each with specific mathematical formulations, computational properties, and implications for mechanism design.

1. Formal Definitions and Hierarchical Structure

At their core, JR axioms define how groups of voters whose approval sets overlap (“cohesive groups”) ought to be represented in any winning committee of fixed size $k$, given $n$ voters. The baseline is the classical Hare quota $n/k$, which divides the electorate into $k$ “deserving” groups; an alternative is the more demanding Droop quota $D(n,k) = \lfloor n/(k+1)\rfloor + 1$ (Casey et al., 1 Aug 2025).

Justified Representation (JR):

Given a profile $(A_1, \dots, A_n)$ and committee $W$ of size $k$, $W$ provides JR if there does not exist a group $N^* \subseteq N$ with $|N^*| \geq n/k$ such that $\bigcap_{i\in N^*} A_i \ne \varnothing$ and $A_i \cap W = \varnothing$ for all $i \in N^*$ (Aziz et al., 2014). Generalizing, a group is called $\ell$-cohesive if $|N^*| \geq \ell (n/k)$ and $|\bigcap_{i\in N^*} A_i| \geq \ell$.

Extended Justified Representation (EJR):

$W$ provides EJR if, for every $1 \leq \ell \leq k$ and every $\ell$-cohesive group $N^*$, there exists at least one $i\in N^*$ with $|A_i \cap W| \geq \ell$ (Aziz et al., 2014).

Proportional Justified Representation (PJR):

Requires that, for every $\ell$-cohesive group $N^*$, the committee $W$ covers at least $\ell$ members from the union of their approvals: $|W \cap (\bigcup_{i\in N^*} A_i)| \geq \ell$ (Sánchez-Fernández et al., 2016).

Full Proportional Justified Representation (FPJR) and Full JR (FJR):

Defined by a weaker notion of cohesiveness—each member must approve enough candidates from a witness set—and either a groupwise (FPJR) or individual (FJR) satisfaction guarantee (Kalayci et al., 21 Jan 2025, Phillips et al., 28 May 2025). For instance, FJR requires that, for every “weakly $\ell$-cohesive” group $S$ (i.e., $|S|/n \geq |T|/k$ for some $T$ with $|T| \geq \ell$ and $|A_i \cap T| \geq \ell$ for all $i \in S$), there is $i \in S$ with $|A_i \cap W| \geq \ell$. FPJR uses a collective utility measure instead.

Droop-style JR axioms: Use the Droop quota in place of Hare, requiring, e.g., $|S| > \ell \cdot n/(k+1)$ for $\ell$-cohesiveness, making the axiom strictly more demanding (Casey et al., 1 Aug 2025).

The table below shows the relationships between cohesiveness (C1, C2) and representation (R1, R2) definitions (cf. (Kalayci et al., 21 Jan 2025)):

Cohesiveness	Representation	Notion
(C1) unanimous, $|N^*|\geq\ell n/k$	(R1) group utility $|W\cap (\cup_{i \in N^*} A_i)|\geq\ell$	PJR
(C1)	(R2) individual, $\exists i\in N^*, |A_i\cap W|\geq\ell$	EJR
(C2) weak, $|S|/n\geq|T|/k$, $|A_i\cap T|\geq\ell$	(R1)	FPJR
(C2)	(R2)	FJR

2. Significance and Interpretations

JR guarantees “minimal proportionality”: any sufficiently large group sharing a candidate receives some representation. EJR, PJR, FPJR, and FJR strengthen and generalize this principle, shifting the focus from minimal to more robust or richer forms of representation (Aziz et al., 2014, Sánchez-Fernández et al., 2016, Kalayci et al., 21 Jan 2025, Casey et al., 1 Aug 2025).

• JR prevents complete exclusion of large, homogeneous minorities, ensuring pluralism in committee selection.
• EJR/PJR and robust (EJR+, PJR+) variants (Brill et al., 2023) handle increasing group size or relax the need for perfect cohesiveness, providing finer guarantees and, often, stronger protection against underrepresentation.
• Droop-based axioms (vs. Hare) require that even smaller cohesive groups can secure representation, raising the bar for proportionality, but often at significant algorithmic and feasibility cost (Casey et al., 1 Aug 2025).

JR and its extensions also have a cooperative-game interpretation. Committees satisfying these axioms prevent “blocking coalitions” from being better off by deviating (as in core stability (Aziz et al., 2014, Peters et al., 2019, Xia, 5 Mar 2025)).

3. Voting Rules and Satisfiability

• Proportional Approval Voting (PAV): Satisfies both JR and EJR via its diminishing returns (harmonic weighting) model (Aziz et al., 2014, Sánchez-Fernández et al., 2016). In fact, canonical PAV is uniquely characterized by EJR among weighted PAV rules.
• Reweighted Approval Voting (RAV): Satisfies JR only for small $k$ (up to $k=5$) but fails for larger $k$ (Sánchez-Fernández et al., 2016).
• Monroe and Greedy Monroe rules: Satisfy PJR (and FPJR) “when $k|n$,” but not necessarily EJR (Sánchez-Fernández et al., 2016, Kalayci et al., 21 Jan 2025).
• Phragmén’s rules and Method of Equal Shares (MES): Satisfy stronger axioms like priceability and laminar proportionality, and guarantee EJR or FPJR when properly parameterized (Peters et al., 2019, Kalayci et al., 21 Jan 2025, Casey et al., 1 Aug 2025).
• Local-Search PAV (lsPAV): Can be adjusted (e.g., using $\epsilon$-lsPAV for $\epsilon=1/k^2$) to satisfy Droop-EJR$^+$ (Casey et al., 1 Aug 2025).

For each axiom—Hare or Droop, strong or robust—there exist specific rules that can be designed or tuned to achieve the corresponding representation guarantee.

4. Computational Aspects

The algorithmic complexity of finding or verifying committees with specific JR-like guarantees varies significantly:

• JR: Existence guaranteed for all profiles; committees can be found and checked in polynomial time by examining all groups of size at least $n/k$ for cohesive overlap (Aziz et al., 2014, Aziz et al., 2016).
• EJR: Checking is coNP-complete. Finding EJR committees is NP-hard (winner determination for rules like PAV is NP-hard) (Aziz et al., 2014, Aziz et al., 2016).
• PJR: Finding a PJR committee can be done in polynomial time in certain cases, but verifying whether a given committee satisfies PJR is coNP-complete (Aziz et al., 2016).
• FPJR, FJR, and Droop+ variants: Verification of FPJR/FJR is coNP-complete (Kalayci et al., 21 Jan 2025). For Droop-JR$^+$ and Droop-EJR$^+$, adjusted algorithms guarantee existence, but feasibility and verification may be computationally hard or require modified thresholds/budgeting (Casey et al., 1 Aug 2025).
• Robust EJR+ and PJR+: Can be checked in polynomial time and are thus especially practical for real-world verification (Brill et al., 2023).

Parameterization (number of voters, candidates, approval set size) sometimes admits FPT results, but in general, optimizing (E)JR degree is intractable (W[2]-hard) unless additional structure, such as bounded maximum degree, is imposed (Tao et al., 27 Dec 2024).

5. Variants and Extensions

Robust and Verifiable Axioms

Robust axioms (EJR+, PJR+) replace exact group cohesiveness with “almost” solid groups, improving both applicability in real elections and enabling polynomial-time verification (Brill et al., 2023). Probabilistic settings and uncertainty models have extended these axioms further, formulating the probability that a random realization of voter preferences admits representation and examining the complexity thereof (Aziz et al., 28 Jul 2024).

Droop-Quota Axioms

Adapting JR, EJR, and related axioms to use the Droop quota (instead of Hare) strictly strengthens the criteria: cohesive groups can be smaller and still “deserve” representation. This leads to more challenging existence and algorithmic questions. The paper (Casey et al., 1 Aug 2025) formally proves that for every standard JR-style axiom, there exists a voting rule (possibly modified) that guarantees the Droop variant, expanding the frontier of achievable proportionality.

Quantitative Degree of Representation

Optimization variants that maximize the number of represented voters in every group (or the minimal representation degree—JR-degree, EJR-degree) have been introduced. Maximizing this degree is, however, provably hard to approximate (Tao et al., 27 Dec 2024). Such metrics serve as finer benchmarks for the equitable performance of voting rules.

6. Relationships to Other Fairness and Stability Concepts

• Perfect Representation (PR): Implies PJR (and FPJR), but is incompatible with EJR (Sánchez-Fernández et al., 2016, Sánchez-Fernández et al., 2017).
• Core Stability: Stronger than (E)JR; PAV achieves a factor-2 approximation to the core, while MES yields a (logarithmic) approximation (Peters et al., 2019).
• Priceability and Laminar Proportionality: Priceable committee outcomes (where voters “buy” representation) deliver FPJR and, when combined with MES-style rules, satisfy strong fairness properties (Peters et al., 2019, Kalayci et al., 21 Jan 2025, Casey et al., 1 Aug 2025).
• Monotonicity: Some monotonicity axioms (especially strong forms) are fundamentally incompatible with PR and certain representation requirements, framing a trade-off in mechanism design (Sánchez-Fernández et al., 2017).

7. Applications and Empirical Insights

• Participatory Budgeting: EJR and FJR have been adapted to handle arbitrary project costs and additive utilities, providing guarantees matched to budget share and enabling rules like MES to offer strong proportionality even under complex domain constraints (Peters et al., 2020).
• Deliberative Selection and Public Sphere Ranking: JR-inspired constraints in ranking or recommendation tasks ensure minority perspectives are always represented while optimizing for other system objectives (e.g., engagement or diversity) (Revel et al., 19 Mar 2025).
• Reconfiguration and Connectivity: The set of JR or EJR committees may not always be connected, but approximate variants (e.g., 2-JR or 4-EJR intermediates) bridge the gap, and rules like PAV, MES, and GJCR yield rich, connected solution spaces (Dong et al., 21 Apr 2025).
• Temporal and Constrained Domains: Strong extension of (E)JR, FPJR, and core stability to temporal voting or public decisions under constraints shows a hierarchy of proportionality concepts remains available, albeit sometimes only approximately (Phillips et al., 28 May 2025, Chingoma et al., 4 Sep 2024).
• Learning and Likelihood Perspective: A linear theory models the whole family of JR-axioms as linear thresholds, supporting sample complexity analysis (via the Natarajan dimension) and providing probabilistic guarantees that strong axioms like EJR+ or the core are satisfied with high probability in natural models (Xia, 5 Mar 2025).

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The family of justified representation axioms spans a spectrum from minimal group-wise fairness (JR, PJR) through robust, individual, and fully proportional variants (EJR, FJR, FPJR), and up to even stronger market and core-based criteria. Each comes with precise mathematical definitions, computational complexity characterizations, and proven connections to key mechanism-design desiderata (e.g., monotonicity, priceability, perfect representation). Recent work systematically extends the paradigm via linear theory, uncertainty models, variants adapted to quotas or constrained public decisions, and application to participatory budgeting and deliberative selection, enabling both theoretical analysis and practical rule design in modern social choice.