{"id":"0c8acd0c-5556-4853-8fe5-788cf4c12eb7","arxiv_id":"2606.23320","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Mixed voting rules sequentially combine PB rules with budget shares, and a Value-Based pre-allocation for MES yields an α_M-budget EJR+ guarantee that exceeds the baseline.","lead":"This paper introduces mixed voting rules for participatory budgeting, in which a sequence of rules each spend a share of the budget and later rules add to earlier selections. It shows that a Value-Based adaptation of the Method of Equal Shares can improve proportionality guarantees beyond a natural baseline while preserving good welfare trade-offs in real-world data.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified to the central claim (Theorem 2); proof appears sound.","rationale":"The reader's weakest assumption concerned Theorem 5's reliance on an unpublished result, which is a valid concern for that generalization but does not undermine the paper's central Theorem 2. My own review of Theorem 2's proof found no gap: the two cases cover the threshold dichotomy, the per-unit spending bounds are correct, and the construction in Case 2 is feasible because the upper-contour property ensures the Value-Based threshold cannot exceed v(p), making the payment comparison exact and the budget-shift feasible. The ex-post nature of the α_Value-Based guarantee is clearly acknowledged and is the intended workaround to Proposition 1's impossibility. The empirical weaknesses (no code, no error bars) are addressable and were already factored into the reader's CONDITIONAL verdict. Thus, no new load-bearing objection arises; the verdict remains CONDITIONAL for the reasons the reader stated, but the central theoretical claim stands.","tokens_in":39294,"tokens_out":20899,"duration_ms":169851,"concrete_test":"Formally verify Theorem 2 by encoding the proof in a proof assistant (e.g., Lean/Coq) for the cost-satisfaction model, specifically checking the Case 2 inequality α'_v ≥ α_v and the affordability step; alternatively, run a brute-force search over small PB instances (n≤6, m≤8) to test whether any MES_Value-Based outcome violates α_Value-Based-budget EJR+ up to any project.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I examined the proof of Theorem 2, the main theoretical contribution. The two-case argument is internally consistent: Case 1 bounds pre-allocation per-unit payments by 1/v* and transfers the standard MES affordability argument; Case 2 constructs P'_0=U(p), verifies the four numbered inequalities. The potentially delicate inequality (1) α'_v ≥ α_v is justified because for P'_0 an upper contour set, the Value-Based threshold cannot exceed v(p), so payments on P'_0 are identical to the original run; hence Σ(π_i-π'_i) equals the original payments on P0\\P'_0, bounded by c(P0\\P'_0), making the proposed budget shift feasible. The ex-post parameterization by α_Value-Based is legitimate given Proposition 1; it does not claim a universal α'>α guarantee. The only notable weakness is the dependence of Theorem 5 on the unpublished Skowron (2026) Strong EJR+ result, but this does not affect Theorem 2. The empirical section lacks code and error bars, but that is not a correctness issue for the central claim.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper introduces a framework for mixed participatory budgeting (PB) voting rules, in which a sequence of rules is executed with increasing rule budgets and each rule may spend the leftover budget of earlier rules. The main theoretical contribution is the adaptation of the Method of Equal Shares (MES) to a pre-selected set of projects P0 via four pre-allocation methods — Null, MES-Style, Equal-Split, and Value-Based — and a parameterized proportionality notion, α-budget EJR+ up to any project. Theorem 2 shows that the Value-Based method yields an α_Value-Based-budget EJR+ guarantee, which can exceed the universal α-budget baseline of Observation 1 when α_Value-Based > α. The paper also proves the baseline guarantee for Null (Theorem 1), a two-project guarantee for Equal-Split after Greedy (Theorem 3), negative results for Equal-Split and MES-Style (Theorem 4), an extension to general additive satisfaction functions (Theorem 5), and an optimality claim for Value-Based (Proposition 4). The experimental section evaluates mixed rules on 313 Pabulib instances, measuring utilitarian welfare and EJR+-based proportionality, with and without a budget-increase heuristic.","tokens_in":39576,"tokens_out":13593,"duration_ms":116489,"significance":"If Theorem 2 is correct, it is a meaningful contribution: it provides a PB completion method whose proportionality guarantee is strictly stronger than the universal α-budget baseline whenever the ex-post minimum voter budget share α_Value-Based exceeds α. The mixed-rule framework is novel and the empirical finding that a small MES component can restore proportionality after a Greedy first stage is practically relevant. The proof of Theorem 2 is carefully constructed and I found no gap in its two-case argument. The negative results in Theorem 4 are also useful for delineating the limits of the other pre-allocation methods. However, the paper’s advertised extension to additive utilities rests on an unpublished result (Skowron 2026, personal communication), and the experimental section omits key parameter values and dispersion information; these issues currently prevent full verification of the paper’s broader claims.","major_comments":[{"comment":"Theorem 5 is stated as a theorem, but its proof relies on the unpublished claim that MES satisfies Strong EJR+ up to any project (Skowron 2026, personal communication). The proof in Appendix A says 'following Skowron [2026], we can also show this bound' without giving the argument. Since the abstract advertises this additive-utility extension, the result is not self-contained or publicly verifiable. Please either include a complete proof of the Strong EJR+ property for MES in this setting, or explicitly state Theorem 5 as conditional on that external result and adjust the paper’s claims accordingly. This is load-bearing for the additive-utility contribution.","section":"§7.2, Theorem 5 and Appendix A"},{"comment":"The budget-increase heuristic MESM+ is central to the experimental section, but the increment β is never specified. The text says only 'increase the budget available to MES by some β' and never reports the value used in Figures 4, 5, 11, and 15. Without β and the exact stopping rule, the empirical results are not reproducible. In addition, all reported metrics are averages over 313 instances with no error bars, confidence intervals, or per-instance distributions, which makes it hard to assess the robustness of claims such as 'dominates the curve without budget increase' and 'best for greedy shares in 0.6–0.9'. Please report β, release code/data or per-instance results, and include dispersion measures.","section":"§6.2 and §D"},{"comment":"The optimality claim for the Value-Based method is stated in terms of 'α_v-Strong EJR+', but the only formal definition in the paper is 'α-budget Strong EJR+ up to any project' (Definition 11). The relation between these notions is not established. Moreover, the proof uses Strong EJR+ violations in the cost-satisfaction setting, which is asserted to be a strengthening of EJR+ but depends on the same unpublished Skowron result as Theorem 5. Thus Proposition 4 is not currently verifiable independently of the issue raised in the first major comment, and the statement should be made conditional or supplied with a self-contained proof.","section":"§7.3, Proposition 4"}],"minor_comments":[{"comment":"The pre-allocation method is named 'Eqal-Split' in some places and 'Equal-Split' in others (e.g., Proposition 3, Theorem 3). Please standardize the spelling.","section":"§4, Method 3 and throughout"},{"comment":"In the definition of EJR+ up to any project, the group N' should be explicitly required to be nonempty, and the quantifier over p should be read as ranging over projects in the intersection of approvals of N'. This is minor but would improve precision.","section":"§2, Definition 3"},{"comment":"The comparison of ex-ante, intermediate, and ex-post variants is dense; a small table of α values for each variant in the example would make the argument much easier to follow.","section":"§D, Proposition 10"},{"comment":"The text notes that evaluating SeqPhragmén and GreedyCC with EJR+X may be unfair; this caveat is good. Consider adding the representation-based measure (Definition 16) to the main figures, since it is only shown in Appendix F.5.","section":"§6.4, Figure 6"}],"recommendation":"major_revision","confidential_remarks":"The core theoretical result, Theorem 2, appears sound and is a genuine contribution. My main concern is that the paper advertises an additive-utility extension (Theorem 5) and an optimality result (Proposition 4) that depend on an unpublished personal communication; this should be fixed before publication, either by including a complete proof or by explicitly marking those results as conditional. The experimental section also needs parameter disclosure and better statistical reporting. These issues are fixable within the manuscript's scope, so I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The headline is simple: this paper gives a reasoning framework for mixed PB rules that I had not seen before, and the main theoretical result (Theorem 2) appears sound on close reading. The Value-Based pre-allocation and the α_M-parameterized EJR+ guarantee are new and do not feel like repackaged existing tools. I checked the two-case proof in Theorem 2 carefully and the stress-test note is right: no circular dependency, and the α_M parameterization is legitimate because it is a post-hoc statement about the produced outcome, not a fitted bound.\n\nWhat is actually good: the framework defines mixed rules cleanly, and the Observation 1 baseline is the right benchmark. Proposition 1 shows you cannot do better than the baseline for all instances, which makes the α_M parameterization a sensible escape. Theorem 2, showing that Value-Based beats the baseline exactly when it creates a higher minimum voter budget share, is the core contribution, and it is coherent. The empirical study is extensive and the observation that mixing small amounts of MES fixes proportionality of Greedy is practically useful. The paper also gives credit for completion methods and positions itself correctly against prior work.\n\nThe soft spots are proportionate. Theorem 5 relies on an unpublished Strong EJR+ theorem (Skowron 2026, personal communication). That is a real gap: the general additive-utility generalization stands on an external result that the reader cannot check. The experiments report no code, no error bars, and the budget-increase increment β is not specified—these are reproducibility issues that matter for a paper whose empirical claims are part of its contribution. The negative results for Equal-Split and MES-Style are convincing but rest on relatively artificial worst-case instances, which is fine for impossibility statements. I would also mention the empirical claim that a 60-90% Greedy share is \"best\" is likely dataset-dependent; the paper is careful enough to hedge this, but it could be misinterpreted.\n\nWho is this for: computational social choice people working on PB, especially anyone designing or evaluating proportional rules. The reading group would get a good discussion from the framework and from the parameterized-axiom idea. For peer review, the answer is yes—send it out. The referee needs to check Theorem 2, verify the Strong EJR+ dependency, and demand replication details for the experiments, but the core is solid. It is not a desk-reject; with minor work it could be a strong paper.","headline":"A genuinely new framework for composing PB rules sequentially, with a sound main theorem and a few addressable loose ends; worth serious refereeing.","tokens_in":39991,"tokens_out":1194,"would_cite":true,"duration_ms":15698,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["91B14","91B12"],"pacs":[],"model":"deepseek-v4-flash","headline":"Mixed rules that let a later proportional rule rebalance earlier choices can exceed the baseline guarantee; the paper proves this for a value-based MES variant and shows the trade-off on 313 real elections.","keywords":["participatory budgeting","mixed voting rules","Method of Equal Shares","EJR+","Value-Based pre-allocation","proportionality axioms","greedy rule","additive satisfaction functions"],"falsifier":"Search small approval-budgeting instances for an outcome of MES with Value-Based pre-allocation that violates α_Value-Based-budget EJR+ up to any project; one such instance would disprove Theorem 2. A second, cheaper check is to audit the unpublished Strong EJR+ claim used in Theorem 5: if MES can fail Strong EJR+ up to any project under additive utilities, that extension dissolves.","tokens_in":39237,"feed_emoji":"🗳️","tokens_out":9618,"duration_ms":92836,"temperature":0.7,"pith_summary":"This paper introduces mixed voting rules for participatory budgeting: a sequence of rules, each spending an assigned slice of the budget, with later rules only allowed to add projects to what earlier rules chose. Its central theoretical claim is that when the Method of Equal Shares (MES) is used as a later rule with a 'Value-Based' pre-allocation, the final outcome satisfies a proportionality guarantee that is strictly stronger than the universal baseline for any pre-selected set. The paper also extends this guarantee to general additive satisfaction functions, and reports experiments on 313 real-world PB elections suggesting that mixing a utilitarian rule with MES can improve welfare without sacrificing representation. A sympathetic reader would care because the paper shows a carefully designed combination of two rules can do something no single rule can guarantee.","feed_headline":"Value-based budget reset beats the fairness baseline","feed_subtitle":"A value-aware second rule lifts guaranteed representation; tests on 313 real elections show welfare stays high.","key_machinery":"The key object is the Value-Based pre-allocation method inside an MES stage of a mixed rule. It defines the threshold value v* as the value (number of supporters, or welfare per cost) of the most valuable unselected project that could still be added once all pre-selected projects of at least its value are paid for, and it charges each voter at most 1/v* per unit satisfaction for pre-selected projects. This payment cap is what keeps the minimum voter budget share α_Value-Based high and makes the proof's per-unit-satisfaction argument go through. The paper also uses the parameterized axiom α-budget EJR+ up to any project to measure the resulting proportionality, and the monotone-property basel","core_discovery":"On the paper's own terms, the central claim is that MES with Value-Based pre-allocation achieves α_Value-Based-budget EJR+ up to any project whenever it is run with available budget share α and its minimum voter budget share α_Value-Based is at least α. Here α_Value-Based is the smallest total endowment (payments for pre-selected projects plus remaining MES budget) among all voters, measured as a fraction of the fair per-voter share B/n; it is always at least α and strictly larger whenever pre-selected projects exist. Because no rule can exceed the plain α-budget baseline for all instances, this is a genuine improvement in the class of instances the parameter identifies. The proof works by b","pith_inferences":["Editorial extension: α_Value-Based can be computed after a run, so the guarantee is usable as a per-outcome certificate—an administrator could report 'this result is α-budget EJR+ for α=0.92' rather than relying on a worst-case statement.","Editorial extension: the pre-allocation idea transfers to any repeated or inherited-choice setting, where earlier commitments play the role of pre-selected projects; the threshold v* then measures the opportunity cost of past decisions.","Editorial extension: replacing Greedy with other first-stage rules is a directly testable next step; the paper's experiments with other rules suggest the Value-Based mechanism, not the identity of the first rule, is what drives the improved trade-off."],"forward_implications":["Any mixed rule containing MES with Value-Based pre-allocation inherits the α_Value-Based-budget EJR+ up to any project guarantee, and adding later completion rules such as Greedy preserves the guarantee while making the outcome exhaustive.","In the unit-cost multiwinner special case, the same arguments yield plain α_Value-Based-budget EJR+, a cleaner bound because the 'up to any project' caveat disappears.","When MES is run with the budget-increase heuristic, the guarantee holds with respect to the minimum voter budget share of the final iteration, which can exceed 1, so the practical heuristic used in elections does not void the theoretical result.","If the unpublished Strong EJR+ theorem for MES is accepted, the Value-Based guarantee extends to all additive satisfaction functions, which covers settings where voters care about the number of approved projects rather than their cost."],"fun_headline_variants":["Value-annotated MES enforces better proportionality in PB","Hybrid PB rules achieve strong fairness without welfare sacrifice","Rewiring MES budgets by prior picks upgrades EJR+ guarantee","Mixed-member style PB rules balance utility and fairness"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The paper's broadest positive result depends on an unpublished theorem that MES satisfies Strong EJR+ up to any project; if that theorem is wrong, the additive-utility generalization collapses, though the core cost-satisfaction result does not depend on it.","fun_headline_variants_meta":{"raw":{"variants":["Value-annotated MES enforces better proportionality in PB","Hybrid PB rules achieve strong fairness without welfare sacrifice","Rewiring MES budgets by prior picks upgrades EJR+ guarantee","Mixed-member style PB rules balance utility and fairness"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000533,"raw_usage":{"total_tokens":2435,"prompt_tokens":811,"completion_tokens":1624,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":555,"completion_tokens_details":{"reasoning_tokens":1566}},"tokens_in":555,"tokens_out":1624,"duration_ms":12476,"temperature":1.0,"reasoning_tokens":1566,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T10:25:02.428072+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Search small approval-budgeting instances for an outcome of MES with Value-Based pre-allocation that violates α_Value-Based-budget EJR+ up to any project; one such instance would disprove Theorem 2. A second, cheaper check is to audit the unpublished Strong EJR+ claim used in Theorem 5: if MES can fail Strong EJR+ up to any project under additive utilities, that extension dissolves.","supporting_citations":[],"review_version":2}