Strengthening Full Justified Representation: Efficient Verification and Computation

Nicholas Teh

Strengthening Full Justified Representation: Efficient Verification and Computation

T0 review · 0 major / 2 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A fractional strengthening of full justified representation is both checkable and satisfiable in polynomial time.

desk verdict FJR+ is a genuinely new, verifiable strengthening of FJR and EJR+ with a real RBG completion theorem; the sub-core part rests on an unproved imported lemma, but the stress-test counterexample to that lemma does not survive contact with the paper's definitions. read the letter →

arxiv 2608.11500 v1 pith:M2UMKRP6 submitted 2026-08-11 cs.GT cs.AIecon.TH

Nicholas Teh This is my paper · ORCID

classification cs.GTcs.AIecon.TH

keywords approval-basedcommitteeelectionsfulljustifiedrepresentationFJR+EJR+residual-budgetgreedypriceabilitysub-coreparticipatorybudgeting

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

Approval-based committee elections ask how to convert voter approvals into a size-$k$ committee that is proportionally representative. The strongest satisfiable justification axiom, full justified representation (FJR), can be met in polynomial time, but checking a proposed committee is coNP-complete. This paper defines FJR+, a strict strengthening of FJR and EJR+ stated with fractional voter and candidate weights, and shows it can be verified by linear programming and satisfied by a greedy rule. The main theorem proves that Residual-Budget Greedy (RBG) returns a partial committee $P$ such that every size-$k$ completion satisfies FJR+, which leaves the remaining seats free to be chosen for priceability; the completed committee lies in the sub-core, and a cost-based version works for participatory budgeting. If true, this removes the verification-versus-computation gap while making the proportionality requirement stronger, not weaker.

What carries the argument

The load-bearing object is Residual-Budget Greedy (RBG) with candidate price $\rho$. It walks levels $h=k,k-1,\dots,1$; a candidate $c$ not yet chosen is affordable when $\mathrm{score}_h(c;P,b)=\sum_{i \in N_c: u_i(P)<h} b_i/(h-u_i(P)) \geq \rho$, and when chosen its payments satisfy the same individual caps and sum to $\rho$. The proof tracks $F(P,b)-\rho Y(P)$, where $F$ sums $b_i t_i/a_i$ over weighted voters with gap $a_i=\ell-u_i(P)$ and remaining assignment $t_i$; the identity showing that this quantity never decreases is the engine that rules out every fractional violation for every completion. Verification of FJR+ rests on normalizing total candidate weight to $1$ and maximizing total voter weight by linear programming for each level.

What would settle it

Enumerate all small profiles (say $n \le 5$, $k \le 3$) and check whether every priceable committee lies in the sub-core; a single priceable committee outside the sub-core refutes the external lemma on which Theorem 4.5 rests. Alternatively, run the linear program $\mathrm{LP}_\ell(W)$ on every size-$k$ completion of the set returned by $\mathrm{RBG}(q_H)$ on the profile of Proposition 3.8 and look for an optimum at least $q_H$.

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Extended reading notes

Core claim

FJR+ is defined by fractional violations. At a representation level $\ell$, a committee $W$ has a violation when there are nonnegative voter weights $z_i$, candidate weights $y_c$, and assignments $x_{ic}$ such that only voters with $u_i(W)<\ell$ receive weight, each such voter is assigned exactly $\ell z_i$ units from approved candidates, assignments to candidate $c$ never exceed $y_c$, assignments to an elected candidate $w$ never exceed $z_i$, and total voter weight $Z$ is at least $q_H = n/k$ times total candidate weight $Y$. Binary variables recover exactly FJR violations; a single positive candidate recovers EJR+ violations; and an example shows FJR+ is strictly stronger than FJR plus EJR+ together. The paper proves that RBG run with price $q_H$ returns a set $P$ such that every size-$k$ committee $W \supseteq P$ satisfies FJR+, and that the Droop version with strict inequality satisfies Droop-FJR+. Sequential Phragm\'en then completes $P$ to a priceable committee, which, by the priceability-to-sub-core implication, lies in the sub-core; this is stated as Theorem 4.5, with a participatory-budgeting analogue in Theorem 6.7.

Load-bearing premise

The sub-core guarantee is imported from an unproved external lemma that every priceable committee lies in the sub-core; if that implication is false or needs extra conditions, the sub-core parts of Theorems 4.5 and 6.7 collapse.

Editorial extensions

If this is right

Any rule that returns an FJR+ committee automatically satisfies FJR and EJR+, so the stronger axiom comes with both classical guarantees at no extra computational cost.
Because every size-$k$ completion of the RBG partial committee is FJR+, the completion step can be optimised for priceability, utilitarian welfare, or other objectives without sacrificing FJR+.
When at least $k$ candidates receive an approval, the completed committee is priceable, and priceability implies the sub-core, so the same outcome satisfies FJR+ and the sub-core simultaneously.
At the Droop quota $n/(k+1)$ with a strict group-size inequality, the same algorithm satisfies Droop-FJR+, and the equality case cannot be ruled out in general.
In approval-based participatory budgeting with arbitrary project costs, PB-RBG returns a feasible partial outcome whose every feasible superset is PB-FJR+, and the natural continuation is priceable and lies in the PB sub-core.

Reading between the lines

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

The LP-based verification defines a quantitative FJR+ gap, the maximum ratio $Z/Y$ across levels; that number could rank nearly proportional committees or guide search when exact FJR+ is too demanding.
FJR+ is not monotone: adding approvals for already elected candidates can create a fractional violation, so dynamic or perpetual settings would need rerunning the rule or a monotone variant rather than incremental updates.
The single-voter knapsack hardness for additive utilities suggests that exact polynomial FJR+ computation is limited to approval utilities, making approximation guarantees or restricted utility classes the natural next target in participatory budgeting.
The completion freedom invites a testable extension: among the priceable completions of a fixed RBG partial committee, one can optimise utilitarian welfare and measure whether priceability and FJR+ are preserved in practice.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The core result is real and worth engaging with. The paper introduces FJR+, a fractional-weight strengthening of FJR that also implies EJR+, gives a polynomial-size LP verification, and proves that every execution of the broader RBG rule returns a partial committee whose every size-k completion satisfies FJR+. That completion freedom is the key move, and using it to run sequential Phragmén to get priceability is a nice piece of work. The PB extension with project-specific costs is coherent and not bolted on. I spot-checked the main examples and the potential argument in Theorem 3.5; the proofs in the appendix are detailed and the identity in (9) does the work it claims.

The soft spots are real but not load-bearing. The sub-core conclusion depends entirely on Proposition 4.4, an imported lemma attributed to Dong and Peters without proof. For a paper that otherwise proves its central claims carefully, that is a genuine presentation gap; I would want the proof in an appendix or a precise pointer to a theorem statement. I also checked the stress-test counterexample to Proposition 4.4. It does not work: in that example q_H = 2, so for T = {a,b,d}, q_H|T| = 6, not 4, and the group S = {1,2,3,4} does not satisfy the sub-core size condition. The alternative S = {5,6}, T = {c} has equality rather than strict inclusion. So the stress-test concern does not land, and priceability implying the sub-core remains plausible and consistent with the literature. The other caveat is minor: the Table 1 citation for coNP-completeness of sub-core verification looks like it points to a paper about auditing core stability rather than the sub-core as defined here; worth fixing but not threatening.

This paper is for researchers in computational social choice who care about verifiable proportionality axioms. It delivers a new definition that strictly dominates the previous pair, with polynomial-time verification and computation, and it is honest about what is new at the FJR level versus what it credits to Ai and to Frank-Peters. I would send it to a serious referee. It deserves a careful reading, and the only required revision is to nail down the priceability-to-sub-core implication rather than importing it silently.

Recommendation: accept into the review process, with a request to prove or precisely cite Proposition 4.4 and to re-check the Table 1 attribution.

Referee Report

0 major / 2 minor

Summary. The paper introduces full justified representation+ (FJR+), a fractional strengthening of FJR and EJR+ for approval-based committee elections. A level-ℓ fractional violation is defined by voter weights z_i, candidate weights y_c, and assignment variables x_ic subject to (C1)-(C5) and Z ≥ q_H Y; the Droop version uses Z > q_D Y. The authors show that FJR+ implies FJR and EJR+ and is strictly stronger than their conjunction (Proposition 3.8), characterize FJR+ on party-list profiles by lower quota (Proposition 3.9), and give polynomial-time verification via one LP per level (Theorem 3.4). The main computational result is Theorem 3.5: Residual-Budget Greedy with price q_H returns a partial committee P such that every size-k completion satisfies FJR+, and with price q_D gives Droop-FJR+. The paper then proves that the payments make P affordable (Proposition 4.2), completes P by sequential Phragmén to a priceable committee (Theorem 4.3), and, using the known implication from priceability to the sub-core (Proposition 4.4), obtains a polynomial-time committee rule satisfying FJR+ and the sub-core, priceable whenever at least k candidates are approved (Theorem 4.5). Section 5 gives separation and monotonicity results, and Section 6 extends the framework to approval-based participatory budgeting with arbitrary project costs: PB-FJR+, PB-RBG, and a priceable PB-sub-core completion (Theorems 6.3, 6.4, 6.7).

Significance. If the results hold, which I believe they do, the paper makes a substantial contribution: it identifies a natural proportionality axiom that is strictly stronger than FJR and EJR+ yet simultaneously admits polynomial-time verification and polynomial-time construction, and it carries the same guarantee over to participatory budgeting with arbitrary project costs. The proof strategy is convincing: the RBG potential F(P,b)-ρY(P) with identity (9), the nondecreasing-threshold argument for Phragmén completion (Lemma C.1), and the budget-accounting proofs in the PB section are all coherent. I specifically examined the alleged counterexample to Proposition 4.4 raised in review; it is arithmetically invalid, since with n=6, k=3, q_H=2 and |T|=3 the required coalition size is q_H|T|=6, while the proposed S has size 4; the alternative S={5,6}, T={c} does not give a strict inclusion. Hence the sub-core conclusion of Theorem 4.5 is not undermined. The paper ships complete appendices, explicit constructions for strictness, and polynomial-time algorithms with stated running times.

minor comments (2)

[Section 4.3, Proposition 4.4] The sub-core implication in Proposition 4.4 is imported without proof from Dong and Peters [2026]; because it is load-bearing for Theorem 4.5, a one-paragraph proof or a precise theorem reference would improve self-containedness.
[Appendix B.4, Proposition 3.9] In the proof of Proposition 3.9, the displayed inequality appears garbled as 'jknj n k'; it should read 'k n_j/n < ℓ', and the surrounding derivation should be typeset more clearly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: FJR+ verification and RBG guarantees are proved from definitions by invariants; the only imported step, Proposition 4.4, is external and a correctness risk, not a circular one.

full rationale

The manuscript's derivation chain is self-contained at every central step. FJR+ is defined directly by the fractional system (C1)-(C5), so the LP verification in Theorem 3.4 is a restatement of the definition rather than a fitted or imported prediction; the RBG guarantee in Theorem 3.5 is proved by the invariant F(P,b)-rho*Y(P), with no parameter fit to FJR+ outcomes; the priceable completion in Theorem 4.3 follows from the nondecreasing-load property of Algorithm 2 proved in Appendix C; and the PB sub-core proof in Appendix E.1 derives the budget inequality directly without assuming the conclusion. The only externally imported load-bearing fact is Proposition 4.4, 'Every priceable committee lies in the sub-core,' attributed to Dong and Peters [2026] and stated without proof; if that implication is false or requires unstated conditions, the sub-core conclusions of Theorems 4.5 and 6.7 would fail. That is a correctness risk, not a circularity: the paper does not define priceability in terms of the sub-core, and it does not use its own theorem as the premise. Self-citations in the related-work and discussion sections are contextual and not load-bearing. No free parameter is fitted to the target axioms, and no equation reduces to its own input by construction beyond the intended definitional character of the LP check.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted to data; the only numeric inputs are the Hare and Droop quotas fixed by the model. The axioms section lists the LP-solvability assumption, the imported priceability-to-sub-core lemma, the integer-utility level reduction in PB, and the Droop strict-inequality convention. The fractional weights in the FJR+ definition are existential variables in a linear system, not new postulated entities.

assumptions (4)

standard math Linear programs can be solved exactly in polynomial time in the real-RAM model.
Invoked in Theorems 3.4 and 6.3, where FJR+ verification is reduced to solving k or m polynomial-size LPs. The paper does not discuss exact rational solvers or strict inequality comparisons for the Droop case.
domain assumption Priceability implies the sub-core.
Imported from Dong and Peters [2026] as Proposition 4.4. The sub-core conclusions of Theorem 4.5 and Theorem 6.7 depend on this cited lemma, which is not proved in the text.
domain assumption Approval utilities are integer-valued, so a PB-FJR violation at a real level beta also produces a violation at level ceil(beta).
Used in Section 6 before Definition 6.1 to restrict PB-FJR levels to [m]. The argument is standard for the recalled PB-FJR definition with actual voter groups and candidate sets, but it is stated without proof.
domain assumption Droop proportionality axioms use the strict group-size inequality |S| > q_D |T|.
Adopted from Casey and Elkind [2025] in Section 2.4. Proposition 3.7 shows the strict form is essential for the Droop-FJR+ guarantee, so this convention is load-bearing rather than cosmetic.

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Pith. "Pith review of Strengthening Full Justified Representation: Efficient Verification and Computation." pith.science (2026). https://pith.science/paper/M2UMKRP6

@misc{pith2026260811500,
  author       = {Pith},
  title        = {Pith review of: Strengthening Full Justified Representation: Efficient Verification and Computation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M2UMKRP6}},
  note         = {Machine review of arXiv:2608.11500}
}

abstract

Full justified representation (FJR) is among the strongest known satisfiable proportionality axioms for approval-based committee elections. Recent work has shown that an FJR committee can be found in polynomial time, but verifying whether a given committee satisfies FJR remains coNP-complete. We introduce FJR+, a strict strengthening of FJR and EJR+ that can be verified and satisfied in polynomial time. We then analyze the Residual-Budget Greedy (RBG) algorithm and prove that it selects a partial committee such that every size-$k$ completion satisfies FJR+. This freedom allows us to use sequential Phragm\'en to obtain a priceable completion. The resulting rule always satisfies FJR+ and the sub-core, and it is priceable whenever at least $k$ candidates receive an approval. We also obtain a Droop-quota version of FJR+. Finally, we extend FJR+ to approval-based participatory budgeting with arbitrary project costs. A project-specific version of RBG computes this property in polynomial time and can be continued to a priceable outcome satisfying a cost-based version of the sub-core.

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Works this paper leans on

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Full Justified Representation under Hare and Droop Quotas in Polynomial Time

Yizhou Ai. Full justified representation under hare and droop quotas in polynomial time. arXiv preprint, arXiv:2608.05417, 2026

work page Pith review arXiv 2026

[2] [2]

Justified representation in approval-based committee voting

Haris Aziz, Markus Brill, Vincent Conitzer, Edith Elkind, Rupert Freeman, and Toby Walsh. Justified representation in approval-based committee voting. Social Choice and Welfare, 48: 0 461--485, 2017

2017

[3] [3]

On the complexity of extended and proportional justified representation

Haris Aziz, Edith Elkind, Shenwei Huang, Martin Lackner, Luis S\' a nchez-Fern\' a ndez, and Piotr Skowron. On the complexity of extended and proportional justified representation. In Proceedings of the 32nd AAAI Conference on Artificial Intelligence (AAAI), pages 902--909, 2018

2018

[4] [4]

Best-of-both-worlds fairness in committee voting

Haris Aziz, Xinhang Lu, Mashbat Suzuki, Jeremy Vollen, and Toby Walsh. Best-of-both-worlds fairness in committee voting. arXiv preprint, arXiv:2303.03642, 2023

arXiv 2023

[5] [5]

Fair lotteries for participatory budgeting

Haris Aziz, Xinhang Lu, Mashbat Suzuki, Jeremy Vollen, and Toby Walsh. Fair lotteries for participatory budgeting. In Proceedings of the 38th AAAI Conference on Artificial Intelligence (AAAI), pages 9469--9476, 2024

work page 2024

[6] [6]

Robust and verifiable proportionality axioms for multiwinner voting

Markus Brill and Jannik Peters. Robust and verifiable proportionality axioms for multiwinner voting. In Proceedings of the 24th ACM Conference on Economics and Computation (EC), page 301, 2023

2023

[7] [7]

Completing priceable committees: Utilitarian and representation guarantees for proportional multiwinner voting

Markus Brill and Jannik Peters. Completing priceable committees: Utilitarian and representation guarantees for proportional multiwinner voting. In Proceedings of the 38th AAAI Conference on Artificial Intelligence (AAAI), pages 9528--9536, 2024

work page 2024

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Phragm \'e n's voting methods and justified representation

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