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Full Proportional Justified Representation

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Pith's one-line read This paper introduces Full Proportional Justified Representation, a mid-strength proportionality axiom that priceable rules and the Monroe rule satisfy and that PAV can violate; verifying a committee satisfies it is coNP-complete.

desk verdict FPJR is a genuine completion of the taxonomy, but the unqualified Greedy Monroe guarantee in the abstract is false for k∤n. read the letter →

arxiv 2501.12015 v1 pith:2I7LKCZT submitted 2025-01-21 cs.GT cs.AI

Yusuf Hakan Kalayci , Jiasen Liu , David Kempe This is my paper · ORCID
classification cs.GTcs.AI MSC 91B1291B1468Q17
keywords approval-basedmultiwinnervotingproportionaljustifiedrepresentationfullpriceabilityMonroerulemethodofequalsharescoNP-completenesscorestability
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper introduces and studies Full Proportional Justified Representation (FPJR), the missing fourth combination in the standard taxonomy of approval-based proportionality axioms: it uses the weak-cohesiveness condition from Full Justified Representation and the collective-utility representation condition from Proportional Justified Representation. The central claim is that FPJR is a viable middle ground: it is strictly stronger than PJR, strictly weaker than FJR, and incomparable with EJR, yet it is satisfied by priceable committees and, when $k$ divides $n$, by the Monroe rule and its greedy variant. The paper also establishes that Proportional Approval Voting can fail FPJR, showing that EJR does not imply FPJR, and that verifying FPJR, FJR, or core stability for a given committee is coNP-complete. A sympathetic reader would care because FPJR names a fairness guarantee that efficient and commonly discussed rules actually deliver, while still capturing the harder weakly cohesive notion of group solidarity.

What carries the argument

The load-bearing object is the weakly $\ell$-cohesive coalition with witness set $T$, defined by the size constraint $|S|/n \geq |T|/k$ and the per-voter approval constraint $|A_v \cap T| \geq \ell$ for every $v \in S$. The engine of the argument is Lemma 3.2, an averaging step: summing $|A_v \cap T|$ over $S$ shows that some candidate $c \in T$ is approved by at least $\ell \cdot n/k$ voters of $S$. That high-approval candidate is what converts a supposed FPJR violation into a contradiction: in the priceability proof it exposes leftover voter budget that could buy the unelected candidate, and in the Monroe proof it yields a committee-and-assignment swap or a greedy-move improvement that increases the Monroe score. The hardness results are carried by the same Balanced Biclique reductions used for PJR and EJR, with more involved analyses of the weakly cohesive coalition that emerges.

What would settle it

Exhibit an election with committee size $k$ not dividing the number of voters $n$ on which the Monroe rule or Greedy Monroe rule returns a committee containing a weakly $\ell$-cohesive coalition whose members collectively approve fewer than $\ell$ winners; a single such instance would refute the abstract's unqualified claim that these rules satisfy FPJR.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that the fourth cell of the cohesiveness-versus-representation table is occupied by a meaningful axiom rather than an incidental variant. A committee satisfies FPJR exactly when every weakly $\ell$-cohesive coalition $S$—one large enough to be entitled to a candidate set $T$ of proportional size, with each member approving at least $\ell$ candidates of $T$—collectively approves at least $\ell$ members of the committee. The paper proves that every priceable committee has this property, which carries the Method of Equal Shares and Phragmén-type rules along with it, and that when the committee size divides the electorate, the Monroe rule and the efficient Greedy Monroe rule satisfy it as well. It then shows the boundary is real: PAV can violate FPJR, so neither EJR nor FPJR implies the other, and some FPJR committees are not priceable. Finally, the same Balanced Biclique reductions used for earlier axioms yield coNP-completeness for verifying FPJR, FJR, and core stability.

Load-bearing premise

The boundary of the main algorithmic result is the divisibility premise: Theorem 3.11 proves FPJR for Monroe and Greedy Monroe only when $k$ divides $n$, because the swap argument relies on every committee candidate having exactly $n/k$ assigned voters; without that equality the proof breaks and the paper supplies no replacement argument or counterexample.

Editorial extensions

If this is right

  • Any committee produced by a priceable rule—the Method of Equal Shares, Phragmén's method, and similar rules—automatically satisfies FPJR, strengthening the proportionality guarantee those rules were already known to give.
  • When the desired committee size divides the number of voters, the Monroe rule and its greedy variant satisfy FPJR, so FPJR is achievable in polynomial time even though verifying it is coNP-hard.
  • PAV and sequential PAV can elect committees that violate FPJR, which means the popular EJR guarantee does not protect against weak-coalition underrepresentation; the Monroe rule provides the reverse separation.
  • Core stability implies FJR, FJR implies FPJR, and FPJR implies PJR; the new axiom therefore slots between PJR and FJR in the implication hierarchy while remaining incomparable with EJR.
  • Verifying FPJR, FJR, or core stability for a proposed committee is coNP-complete, closing a gap in the complexity landscape of proportionality axioms.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: the divisibility assumption in the Monroe result is likely not a proof artifact but a genuine boundary; searching for a counterexample with $k \nmid n$, or proving a modified statement with floor and ceiling assignments, is the most direct next test.
  • Editorial inference: because priceability implies FPJR but not conversely, FPJR offers a way to certify a class of non-priceable committees as proportionally sound; whether every FPJR committee can be realized by a mild perturbation of a price system is a natural open question.
  • Editorial inference: the coNP-completeness of verifying FPJR suggests that a verifiable FPJR+ analogue in the style of PJR+ and EJR+ would require a different cohesiveness condition, and the paper's open question about efficiently verifiable stronger axioms points exactly there.
  • Editorial inference: in participatory budgeting, generalizing FPJR with candidate costs and voter budgets would likely give a proportionality axiom that the Method of Equal Shares satisfies, extending the paper's priceability result to the budgeted setting.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper proposes a new proportionality axiom, Full Proportional Justified Representation (FPJR), defined as requiring every weakly ell-cohesive coalition to have collective approval utility at least ell. It studies the relations of FPJR to PJR, EJR, FJR, priceability, perfect representation, and PAV, proves that priceable committees and perfectly representative committees satisfy FPJR, and claims that the Monroe rule and Greedy Monroe rule satisfy FPJR when k divides n. It proves coNP-completeness of verifying FPJR, FJR, and core stability via balanced-biclique reductions. The central claims are mostly coherent, but the unqualified Greedy Monroe statement in the abstract is false for k not dividing n, and the proof of Theorem 3.6 has a gap.

Significance. The paper fills a natural gap in the justified-representation taxonomy and gives the first systematic study of the (C2)-(R1) cell. The implication from priceability to FPJR, the PAV counterexample, and the hardness results for FPJR, FJR, and core stability are useful additions; the latter two indeed appear to be missing from the literature. The proofs are constructive, and the hardness reductions are modeled carefully on the existing EJR and PJR constructions. The main caveat is that the headline Monroe and Greedy Monroe claim is false as stated when k does not divide n, and the proof of Theorem 3.6 contains a localized but load-bearing gap. Both issues are repairable within the scope of the manuscript, and with those corrections the paper would be a solid contribution.

major comments (2)
  1. [Abstract, Section 1, Theorem 3.11] The abstract and introduction state without the divisibility condition that the Monroe rule and Greedy Monroe rule satisfy FPJR. Theorem 3.11 is proved only for k dividing n, and the restriction is essential. For n=6, k=4, let voters v1,v2,v3 approve {a,b}, v4 approve {d}, v5 approve {e}, and v6 approve {f}. Under the Greedy Monroe procedure in Section 2.2.2, a legitimate tie-breaking run selects a first, assigning v1 and v2; then d, assigning v4 and one arbitrary remaining voter, say v3; then e, assigning v5; then f, assigning v6. The output is W={a,d,e,f}. The coalition S={v1,v2,v3} is weakly 2-cohesive with witness T={a,b}, because |S|/6=1/2=|T|/4 and every voter in S approves both a and b. Yet the union of approvals of S intersects W in {a}, of size 1, which is less than 2, violating FPJR. Thus the unqualified claim is false; every FPJR statement about Monroe and Greedy Monroe must be restricted to k dividing n.
  2. [Section 3.1, proof of Theorem 3.6, Eq. (2)] The upper bound in Eq. (2) is applied to all candidates in T \ O on the ground that none of them are in W, but T \ O can contain elected candidates that are not approved by any voter in S; for such candidates the priceability condition for unelected candidates does not apply. The proof can be repaired by summing only over N_c ∩ S in both Eq. (1) and Eq. (2): for elected candidates in T \ O, the set N_c ∩ S is empty, while for unelected candidates priceability gives the needed bound. As written, however, the argument contains a genuine gap in the main implication from priceability to FPJR.
minor comments (4)
  1. [Proof of Theorem 3.11] The line saying that c in T prime and c not in W_S implies c not in W is false as stated, because c could be in W but not approved by any voter in S. The contradiction is restored because Lemma 3.2 gives a nonempty N_c ∩ S prime; the proof should say this explicitly.
  2. [Example 3.4] The text says that Sequential PA V also selects this committee, but the related discussion concerns LS-PAV; please use the established name of the sequential variant.
  3. [Theorem 4.4] The theorem statement says that the problem is coNP-hard, while the proof establishes coNP-completeness; uniform phrasing with Theorems 4.2 and 4.3 would be clearer.
  4. [Definition 3.1] The expression W ∩ S v∈S A_v in Definition 3.1 appears to be a rendering artifact and should be W ∩ (union over v in S of A_v).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: FPJR results are derived forward from definitions and external theorems; no fitted input or self-citation chain is load-bearing.

full rationale

The paper's central claims are forward derivations from its definitions. FPJR is defined in Definition 3.1, and the main positive results show that priceable committees satisfy FPJR (Theorem 3.6) and that Monroe and Greedy Monroe satisfy FPJR when k divides n (Theorem 3.11). These proofs use only the paper's own definitions, standard counting arguments, and cited external results, notably Proposition 1 of Peters and Skowron 2020b for the priceability-to-PJR implication. The hardness results in Theorems 4.2-4.4 explicitly build on reductions due to Aziz et al. 2017 and 2018a, with the paper supplying the additional verification arguments; those reductions are external benchmarks rather than self-referential assumptions. No parameter is fitted to any subset of data and then renamed as a prediction; the paper is purely axiomatic. The only self-citation, Kalayci et al. 2024 in the Related Work section, is not load-bearing for the main results. The known restriction that Theorem 3.11 is proved only when k divides n is stated in the theorem and in Section 2.2.2, and the abstract's unqualified wording that greedy Monroe satisfies FPJR is a correctness/scoping concern about an overclaim, not a circularity: it does not reduce the derivation to its own inputs. Under the requested circularity standard, no step in the derivation chain is equivalent to its inputs by construction, so the appropriate finding is no significant circularity with score 0.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The paper's central claims rest only on standard combinatorial definitions and on cited results from the approval-voting literature. There are no fitted parameter values, no ad hoc constants, and no invented entities. The analysis is new; the ingredients are standard. The only external results that do real work are priceability-to-PJR, the Aziz et al. reductions, and Balanced Biclique NP-completeness.

assumptions (8)
  • standard math Balanced Biclique is NP-complete (Garey and Johnson).
    Used as the source problem for all coNP-hardness reductions in Section 4.
  • domain assumption The Aziz et al. [2018a] reduction from Balanced Biclique proves a PJR violation for the constructed instance.
    Invoked in Theorem 4.2 forward direction: if an ℓ×ℓ biclique exists, PJR is violated by R'∪V2.
  • domain assumption The Aziz et al. [2017] reduction from Balanced Biclique proves an EJR violation for the Algorithm 2 instance.
    Invoked in Theorems 4.3 and 4.4 forward directions.
  • domain assumption Priceability implies PJR (Proposition 1 of Peters and Skowron 2020b).
    Used in the |T|=ℓ case of Theorem 3.6.
  • domain assumption The Method of Equal Shares and Phragmen's rule always output priceable committees (Peters and Skowron 2020b).
    Used for Corollary 3.9 to conclude these rules satisfy FPJR.
  • domain assumption Priceable committees may violate EJR (Peters and Skowron 2020b).
    Used with Theorem 3.6 to derive Corollary 3.7 that FPJR does not imply EJR.
  • domain assumption Monroe's rule can violate EJR, and EJR+ is stronger than EJR (Aziz et al. 2017, Brill and Peters 2023).
    Used to argue that FPJR and EJR+ are incomparable in the Introduction.
  • domain assumption Core stability implies FJR, FJR implies EJR, and EJR implies PJR.
    Standard known implication chain used in Corollary 3.3 and in the hardness implications.

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Pith. "Pith review of Full Proportional Justified Representation." pith.science (2026). https://pith.science/paper/2I7LKCZT

@misc{pith2026250112015,
  author       = {Pith},
  title        = {Pith review of: Full Proportional Justified Representation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2I7LKCZT}},
  note         = {Machine review of arXiv:2501.12015}
}
read the original abstract

In multiwinner approval voting, forming a committee that proportionally represents voters' approval ballots is an essential task. The notion of justified representation (JR) demands that any large "cohesive" group of voters should be proportionally "represented". The "cohesiveness" is defined in different ways; two common ways are the following: (C1) demands that the group unanimously approves a set of candidates proportional to its size, while (C2) requires each member to approve at least a fixed fraction of such a set. Similarly, "representation" have been considered in different ways: (R1) the coalition's collective utility from the winning set exceeds that of any proportionally sized alternative, and (R2) for any proportionally sized alternative, at least one member of the coalition derives less utility from it than from the winning set. Three of the four possible combinations have been extensively studied: (C1)-(R1) defines Proportional Justified Representation (PJR), (C1)-(R2) defines Extended Justified Representation (EJR), (C2)-(R2) defines Full Justified Representation (FJR). All three have merits, but also drawbacks. PJR is the weakest notion, and perhaps not sufficiently demanding; EJR may not be compatible with perfect representation; and it is open whether a committee satisfying FJR can be found efficiently. We study the combination (C2)-(R1), which we call Full Proportional Justified Representation (FPJR). We investigate FPJR's properties and find that it shares PJR's advantages over EJR: several proportionality axioms (e.g. priceability, perfect representation) imply FPJR and PJR but not EJR. We also find that efficient rules like the greedy Monroe rule and the method of equal shares satisfy FPJR, matching a key advantage of EJR over FJR. However, the Proportional Approval Voting (PAV) rule may violate FPJR, so neither of EJR and FPJR implies the other.

Figures

Figures reproduced from arXiv: 2501.12015 by the authors.

Figure 1
Figure 1. In the diagram, we illustrate how proportionality axioms in approval-based committee selection [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗

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