{"id":"ba74eb01-cf4a-497a-819c-dafc225628da","arxiv_id":"2608.06156","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Resignation monotonicity is incompatible with justified representation, but the new Maximum Payment Rule and Maximum-Cardinality Affordable Rule recover the PJR+ proportionality guarantee after resignations.","lead":"Approval-based committee elections choose a fixed-size winner set, and this paper asks what should happen when some winners resign. It formalizes resignation monotonicity, shows that most proportional rules fail it, and introduces two new rules that can refill empty seats while keeping a fairness guarantee.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the central impossibility and MPR construction are internally consistent within the stated model.","rationale":"The reader's verdict is ACCEPT with high confidence, and the stress-test finds no internal mathematical error in the central claims. The impossibility theorem (3.4) and the positive construction (Theorem 5.2) are supported by detailed, internally consistent proofs. The main limitation is the modeling assumption (fixed k, static preferences, and existential recovery), which is explicitly scoped and does not invalidate the theorems. The abstract's omission of 'independent of losers' in the strategic candidacy claim is a minor presentation issue, not a load-bearing concern. The recommended verdict remains ACCEPT.","tokens_in":38358,"tokens_out":43014,"duration_ms":407734,"concrete_test":"Independently verify Theorem 3.4 by encoding the complete-graph K12 instance (n=12, k=6, candidates as edges) and exhaustively checking, for every initial JR committee and every branch of the case analysis (including all possible replacement candidates and all selected resignations), that the described sequence leads to a subinstance where no committee extending the survivors satisfies JR. If any branch admits a JR-satisfying extension, the impossibility proof has a gap.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I found no load-bearing flaw in the paper's central contributions. Theorem 3.4's proof is a careful adversarial case analysis over all possible replacement choices in the K12 instance; each branch terminates in the 'two-pair obstruction', and the argument only relies on the explicit existential definition of resignation monotonicity. Theorem 5.2's proof that MPR is PJR+-resignation monotone is sound: the augmenting-path step preserves flow to surviving winners, the 'some c' in T' with f'(c,t)<1' argument is valid, and the final improvement uses the same unused-capacity reasoning as Lemma 5.1. The supporting results (Theorem 5.8, MCAR, computational hardness) also check out. The reader's weakest assumption—static preferences and fixed committee size k after resignations—is a genuine scope limitation but is explicitly stated in Section 3 and does not contradict any theorem. The only minor issue is that the abstract omits the 'independent of losers' qualification in Proposition 7.1, but this does not affect the main impossibility or recovery claims.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces resignation monotonicity for approval-based committee elections: after a subset T of the winning candidates resigns, re-running the rule on the reduced instance must keep all non-resigning winners in some output committee. Section 3 shows that among Thiele and sequential Thiele rules only AV is resignation monotone (Theorem 3.1), and proves the central impossibility that no rule can satisfy both resignation monotonicity and justified representation (Theorem 3.4). Section 4 shows that in the fractional committee world the maximum-flow-based GRP rule is resignation monotone (Theorem 4.2). Section 5 introduces the Maximum Payment Rule (MPR), proves it satisfies the relaxed PJR+-resignation monotonicity (Theorem 5.2), establishes NP-hardness in general with tractable structured and bounded-resignation cases, and characterizes MPR as the unique homogeneous extension of Monroe when k divides n (Theorem 5.8). Section 6 introduces a second rule, MCAR, also PJR+-resignation monotone, with a greedy algorithm for voter-interval instances. Section 7 connects resignation monotonicity to strategic candidacy, showing that resignation monotone rules that are independent of losers are candidacy strategyproof while PAV is manipulable. The appendix contains full proofs and experiments measuring how often PAV and MES fail resignation monotonicity.","tokens_in":38554,"tokens_out":37734,"duration_ms":370600,"significance":"If the results hold, the paper settles a natural open question: JR is incompatible with retaining all surviving winners after resignations, and this is tight because the weaker PJR+ can be recovered by the newly designed MPR and MCAR. The fractional result (Theorem 4.2) and the characterization of MPR as the homogeneous Monroe extension (Theorem 5.8) are elegant and give the rules independent interest beyond the resignation setting. The proofs are thorough: Theorem 3.4's adaptive adversarial case analysis and the flow-augmentation arguments in Theorems 4.2 and 5.2 are careful and internally consistent, and the computational hardness/tractability statements are standard but complete. The experimental appendix provides useful quantitative context for how often common rules fail in practice. Overall this is a strong contribution to computational social choice.","major_comments":[],"minor_comments":[{"comment":"The metadata abstract states that all resignation monotone rules are immune to strategic candidacy, while Proposition 7.1 requires the additional independence-of-losers condition; the full-text abstract correctly includes the qualifier. Please reconcile the two abstracts.","section":"Abstract / Section 7"},{"comment":"In the case val(f') ≥ val(f), the sentence that W' is payment-maximizing for E and therefore also for E−T skips the needed argument that φ_{E−T}(W'') ≤ φ_{E−T}(W') for every committee W''; adding the one-line subnetwork argument would improve verifiability.","section":"Theorem 5.2"},{"comment":"The assertion that the proof of Theorem 5.2 goes through with a 1-locally payment-maximizing replacement is terse; a brief explanation that the contradiction in Theorem 5.2 uses only a single swap (replacing one c' ∈ T') would help the reader follow the reduction.","section":"Theorem 5.5"},{"comment":"For the MES case, the 'first time' argument should explicitly note that if a weak candidate c' ∈ T with N(c') ⊆ N(c) is taken before c, then c and c' have equal effective prices, so there exists a tie-breaking in which c is taken first; this is implicit in the current wording.","section":"Proposition 7.3"},{"comment":"The experimental plots do not report confidence intervals or standard errors for the estimated probabilities; given the parameter sweep, a brief statement about Monte Carlo uncertainty would be useful.","section":"Appendix A"}],"recommendation":"minor_revision","confidential_remarks":"This is a well-executed paper. I checked the central proofs (Theorem 3.4, Theorem 4.2, Theorem 5.2, Theorem 5.8, and the MCAR arguments) and found no load-bearing errors. The only issue worth the editor's attention is the abstract-level overstatement regarding independence of losers in the strategic candidacy claim, which is cosmetic and easily fixed. The paper fits the journal's scope and I support publication after a minor revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The reader's take matches mine: this is a serious theory paper and the central results hold up. What's genuinely new is the resignation monotonicity axiom, the impossibility with JR (Theorem 3.4), and the Maximum Payment Rule (MPR). The proof of Theorem 3.4 is a careful adversarial case analysis, and I don't see a gap. The characterization of MPR as the unique homogeneous extension of Monroe (Theorem 5.8) is elegant and makes the rule independently interesting, quite apart from the resignation story. The PJR+-recovery result for MPR (Theorem 5.2) is also sound; the flow-swapping arguments are correct. I checked the main proofs and the stress-test note is right: no load-bearing flaw.\n\nSoft spots are minor. The model fixes committee size k and assumes static preferences after resignations—this is explicitly stated in Section 3, but it's a real scope limitation. The sequential Thiele counterexamples rely on 'sufficiently large p' without explicit bounds, so those are existence proofs with implicit thresholds. The appendix is dense but complete. The auxiliary experiments have no code, but they are clearly labeled as exploratory. Also, the abstract omits the 'independent of losers' qualification in Proposition 7.1, which could mislead a casual reader, but the main claims are unaffected.\n\nThis paper deserves a serious referee. The new axiom is natural, the impossibility tightens a recent result, and MPR is a concrete, well-characterized rule that connects to apportionment. I'd send it to review. I'd also cite it in my own work on dynamic ABC elections. There's no circularity in the citation pattern; Dong and Peters is cited properly, and the proofs stand on their own. The paper is for computational social choice researchers, especially those working on proportionality axioms and dynamic candidate sets.","headline":"A clean, well-proved theory paper that introduces a natural new axiom, proves a strong incompatibility, and offers a new rule (MPR) that is likely to become a reference point.","tokens_in":39074,"tokens_out":1193,"would_cite":true,"duration_ms":13921,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["91B12","91B14","05C21","68Q17"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that no approval-based committee rule can satisfy both resignation monotonicity and justified representation, and constructs a maximum-flow rule that recovers the stronger PJR+ guarantee after resignations.","keywords":["approval-based committee elections","resignation monotonicity","justified representation","maximum flow","proportionality axioms","strategic candidacy","Monroe rule","NP-hardness"],"falsifier":"To falsify the central impossibility, exhibit any approval-based committee rule that outputs a committee satisfying JR on every instance and, whenever an elected candidate resigns, always outputs a committee containing every other previous winner. To test the MPR recovery guarantee, enumerate small instances, compute an MPR committee $W$, delete each $T\\subseteq W$, and check whether any committee $(W\\setminus T)\\cup T'$ with $|T'|=|T|$ satisfies PJR+ in the resigned instance; a counterexample would disprove Theorem 5.2.","tokens_in":38146,"feed_emoji":"🗳️","tokens_out":8101,"duration_ms":81409,"temperature":0.7,"pith_summary":"Approval-based committee elections pick a fixed-size set of winners, and when an elected member resigns, the natural demand is that everyone else who won keeps their seat when the election is re-run. The paper formalizes this as resignation monotonicity and proves a stark incompatibility: no rule can be resignation monotone and satisfy justified representation (JR), the weakest standard proportionality axiom. The escape is to lower the demand from \"the rule itself must keep all survivors\" to \"some proportional committee containing all survivors exists after resignations.\" The paper constructs the Maximum Payment Rule (MPR), which maximizes the value of a maximum flow in the approval network, and proves it satisfies this relaxed property for the stronger axiom PJR+. If the paper is right, committees can be made resilient to resignations, but only by abandoning strict resignation monotonicity or by allowing fractional committees.","feed_headline":"No voting rule can be proportional and resignation-proof","feed_subtitle":"A maximum-flow rule keeps surviving winners and restores proportional representation after resignations.","key_machinery":"The load-bearing object is the network representation of an election: a flow network with source $s$, a node per voter with capacity $k/n$ from $s$, infinite-capacity edges from voters to candidates they approve, and capacity-1 edges from candidates to sink $t$. A committee $W$ is scored by $\\phi(W)$, the value of a maximum flow through the subnetwork restricted to $W$; fractional versions of the same network characterize GRP committees. The arguments run on flow augmentation: when winners resign, deleting their flow and augmenting along paths can only weakly increase flow to surviving winners, and if a replacement committee's total flow fell below the original, some replacement candidate has flow less than 1, which yields an improving swap that contradicts local payment-maximality and hence forces PJR+. The same network gives MPR its algorithmic profile: it is NP-hard to compute in general, polynomial-time on voter-interval and candidate-interval domains via the Monroe connection, and FPT in the number of voters.","core_discovery":"The central claim is Theorem 3.4: there is no resignation monotone ABC rule satisfying JR. The proof adapts a robust-decremental impossibility for PJR+ from the dynamic-candidate-set literature and forces any JR committee through an adaptive sequence of resignations that ends in a two-pair obstruction where a single replacement cannot cover all uncovered voter pairs whose common candidate has resigned. On the positive side, the paper shows that fractional committees escape the impossibility: the rule outputting all GRP committees is resignation monotone, because GRP committees are exactly those extending a maximum flow, and after deleting resigned candidates one can augment the flow without decreasing any remaining candidate's flow. For integral committees, the paper proposes MPR, which selects committees $W$ maximizing the maximum flow value $\\phi(W)$ in the network representation, and proves MPR is PJR+-resignation monotone: after any resignation $T$, some committee $(W\\setminus T)\\cup T'$ satisfies PJR+. MPR is the unique homogeneous rule that coincides with Monroe when $k$ divides $n$, induces the largest remainder method on party-list instances, and satisfies perfect representation, so it fails EJR+.","pith_inferences":["Since PJR+ is stronger than JR, the impossibility for JR suggests that no integral rule can satisfy both strict resignation monotonicity and any proportional axiom, so MPR's relaxed guarantee is essentially the best an integral rule can offer.","The Monroe characterization suggests a recipe: any homogeneous extension of a proportional apportionment method built from a flow-maximization network could inherit similar resignation recovery properties; testing this on other apportionment methods is a natural next step.","In practice, a rule like MPR would need to publish the flow certificate for the original committee, because finding replacement candidates is much easier with it than without it, as the comparison between MPR and MCAR indicates.","The strategic candidacy result suggests a new design criterion: if a rule is resignation monotone and independent of losers, it automatically resists candidate-controlled ballot stuffing, which may be a useful robustness check beyond the usual candidate-monotonicity axioms."],"forward_implications":["Any rule that offers even the weakest proportional guarantee (JR) will sometimes drop a surviving winner after a resignation, so a strict \"never remove winners\" policy forces Approval Voting or another non-proportional rule.","MPR guarantees that after any resignation, a PJR+ committee extending all surviving winners exists, and such a replacement committee can be computed in polynomial time once the original committee is known.","In fractional committee voting, proportionality and resignation monotonicity are compatible: the GRP rule is resignation monotone, so the incompatibility is an artifact of integrality.","On party-list instances MPR reduces to the largest remainder apportionment method, so it inherits the usual quota properties of Hamilton's method while also satisfying PJR+.","Resignation monotone rules that are independent of losers are immune to a losing candidate introducing new weak candidates to get elected, while PAV is vulnerable to this manipulation."],"supporting_citations":[{"why":"supplies the robust-decremental impossibility that Theorem 3.4 adapts from PJR+ to JR, and the ideas behind MCAR.","marker":"Dong and J. Peters (2025)"},{"why":"introduces the network representation and the max-flow characterization of GRP used in the fractional positive result and in MPR's definition.","marker":"Suzuki and Vollen (2024)"},{"why":"defines justified representation, the axiom that Theorem 3.4 shows incompatible with resignation monotonicity.","marker":"Aziz et al. (2017)"},{"why":"establishes that a committee containing a maximal affordable set satisfies PJR+, the guarantee MCAR builds on.","marker":"Brill and J. Peters (2024)"},{"why":"provides the Monroe-rule algorithms and hardness results that the MPR-Monroe equality imports for tractable cases.","marker":"Betzler et al. (2013)"},{"why":"gives the polynomial algorithm for Monroe in voter-interval domains used to compute MPR there.","marker":"Chen et al. (2023)"},{"why":"supplies the fixed-dimension integer programming bound making MPR FPT in the number of voters.","marker":"Lenstra (1983)"},{"why":"gives the perfect-representation axiom and the PR/EJR+ incompatibility used to position MPR's proportionality direction.","marker":"Sánchez-Fernández et al. (2017)"}],"fun_headline_variants":["No proportional committee rule survives resignations","Resignation-proof rules fail proportionality","Flow-based rule restores winners after resignations","MPR: a flow rule that withstands resignations","Proportionality and resignation-proofness conflict"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The paper fixes the election model so that a resignation only removes the resigned candidates from the candidate set and from every ballot, keeps the committee size $k$ and the voter set unchanged, and requires all non-resigning winners to stay winners; if instead vacant seats may go unfilled or voters update their approvals after a resignation, the impossibility and the MPR recovery guarantee need not hold.","fun_headline_variants_meta":{"raw":{"variants":["No proportional committee rule survives resignations","Resignation-proof rules fail proportionality","Flow-based rule restores winners after resignations","MPR: a flow rule that withstands resignations","Proportionality and resignation-proofness conflict"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001296,"raw_usage":{"total_tokens":5328,"prompt_tokens":1022,"completion_tokens":4306,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":638,"completion_tokens_details":{"reasoning_tokens":4238}},"tokens_in":638,"tokens_out":4306,"duration_ms":32132,"temperature":1.0,"reasoning_tokens":4238,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T13:50:41.467750+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"To falsify the central impossibility, exhibit any approval-based committee rule that outputs a committee satisfying JR on every instance and, whenever an elected candidate resigns, always outputs a committee containing every other previous winner. To test the MPR recovery guarantee, enumerate small instances, compute an MPR committee $W$, delete each $T\\subseteq W$, and check whether any committee $(W\\setminus T)\\cup T'$ with $|T'|=|T|$ satisfies PJR+ in the resigned instance; a counterexample would disprove Theorem 5.2.","supporting_citations":[{"cited_title":"Proceedings of the 25th ACM Conference on Economics and Computation (EC) , pages =","cited_arxiv_id":null,"evidence_quote":"introduces the network representation and the max-flow characterization of GRP used in the fractional positive result and in MPR's definition."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"establishes that a committee containing a maximal affordable set satisfies PJR+, the guarantee MCAR builds on."},{"cited_title":"Proceedings of the 26th European Conference on Artificial Intelligence (ECAI) , series =","cited_arxiv_id":null,"evidence_quote":"gives the polynomial algorithm for Monroe in voter-interval domains used to compute MPR there."},{"cited_title":", title =","cited_arxiv_id":null,"evidence_quote":"supplies the fixed-dimension integer programming bound making MPR FPT in the number of voters."}],"review_version":1}