{"id":"e48194de-11b9-4920-9d67-64e2cf06622c","arxiv_id":"2607.26053","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For divisible public bads, proportional fairness and Lindahl equilibrium coincide; a flipped Nash-welfare rule satisfies the completion core on all instances.","lead":"The paper builds a theory of fairly choosing among shared bad outcomes, like landfill siting or power outages, where everyone is hurt but differently. It introduces two fairness definitions, connects market-like equilibria to proportional fairness, and shows a \"flip costs and maximize Nash welfare\" rule always meets one of them.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified","rationale":"The reader identified linear expected utility and the axes-cutting condition as the weak assumptions. I agree these are the least secure parts of the modelling framework, but they are explicit assumptions rather than errors. The linearity assumption is standard in the fair-division literature and is used consistently; the axes-cutting condition is a clearly delimited structural restriction whose failure the paper itself demonstrates and discusses. The central theorems hold under the stated assumptions, so the ACCEPT verdict stands. I found no internal inconsistency or hidden step in Lemmas 4.4, 4.5, Theorems 4.9, 4.10, 4.11, or the axes-cutting existence proof. A computational re-check of the equivalence on small instances would still be a useful, cheap sanity check of the paper's central claim.","tokens_in":40558,"tokens_out":22092,"duration_ms":197284,"concrete_test":"Brute-force verification of Theorem 4.3 on all small public-bads instances (e.g., n=3, m=4, integer costs bounded by 3). For each instance, enumerate all lotteries over supports, compute the set of PF allocations via the linear inequalities, compute Lindahl equilibria via Definition 4.1, and compute critical points of the Nash product in the upward closure (Definition 4.2). Assert that the three sets coincide exactly for every instance. This tests the equivalence independently of the paper's proof.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I read the paper in good faith and checked the central equivalence theorems and their proofs in detail. The claims are internally consistent and the proofs appear sound. The linear expected-cost model is explicitly stated at the outset, and the PF/Lindahl/critical-point equivalence (Theorem 4.3) is proven directly from that model; the zero-respecting refinement (Theorem 4.9) and the CEEI equivalence for private-induced instances (Theorem 4.11) also check out. The axes-cutting condition is transparently scoped: outside it, the paper itself provides counterexamples (Examples 4.12, 4.13) and impossibility results, so this is a defined limitation rather than a hidden assumption. The two core definitions are coherent, and the BE core proof (Theorem 4.10) and the completion core guarantee for Flipped-MNW (Theorem 5.7) are valid within the stated model. I did not find a load-bearing gap that would invalidate the central claims.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a fairness theory for allocating divisible public bads under linear expected costs. It introduces two core notions (bounded-externality and completion cores), proposes a Lindahl equilibrium definition for public bads, and proves an exact equivalence (Theorem 4.3) between proportional fairness, Lindahl equilibrium, and critical points of a truncated Nash product on the upward closure of the feasible cost set. It then characterizes strictly positive PF solutions via zero-respecting Lindahl equilibria (Theorem 4.9), shows they lie in the BE core (Theorem 4.10), and for private-induced instances coincide with CEEI for chores (Theorem 4.11). Under an axes-cutting condition, existence and strong fairness guarantees are proven (Theorems 4.15, 4.22); outside that condition, negative examples and an impossibility (Theorem 5.5) are given, and the Flipped-MNW rule is shown to satisfy the completion core (Theorem 5.7). The computational section gives enumeration results, a KKT characterization, and a Frank–Wolfe convergence bound.","tokens_in":40751,"tokens_out":27886,"duration_ms":239473,"significance":"The paper opens a coherent research direction in a neglected cell of the goods/bads—private/public classification. The main theorems are proved in detail, with full appendices and a self-contained re-proof of the Mariotti–Villar existence result; counterexamples are concrete and scoped, and the paper is explicit about its limitations (non-existence of strictly positive PF, IFS failure outside axes-cutting). If the results stand, they provide a market-based theory of proportional fairness for public bads and clarify how CEEI for chores embeds. The algorithmic section is a useful bonus. I found no numerical fitting or circularity; dependence on Bogomolnaia et al. (2017) and Mariotti–Villar (2005) is clearly identified.","major_comments":[],"minor_comments":[{"comment":"The statement that KKT points (β,x) are in 'one-to-one correspondence' with strictly positive PF allocations is too strong when distinct lotteries induce the same positive cost vector. In that case the same β corresponds to multiple dual solutions x. The intended content is a bijection between KKT points and strictly positive PF cost vectors (or an equivalence between KKT conditions and such allocations); please rephrase.","section":"§6, Proposition 6.3"},{"comment":"Several uniqueness/optimality claims are asserted with 'it can be checked' rather than proved. In particular, Example 4.13 does not show why the unique strictly positive PF allocation is (2/3,1/3) and why the unique completion-core allocation is (1/3,2/3). Short derivations, perhaps in an appendix, would improve verifiability.","section":"§4.2, Examples 4.12–4.13"},{"comment":"The notation C^S_≥ is not fully defined: C^S is mentioned before the projection operation is specified, and the expression '{z^S=(z_i)_{i∈S}: …}' introduces z^S without defining it. Please clarify that C^S is the projection of C onto coordinates S after setting N∖S to zero.","section":"Definition 4.2"},{"comment":"The proof asserts that for a Lindahl equilibrium (x,p), p_i·x>1/n is possible only if c_i(x)=0. This is true, but it deserves a one-sentence justification: if c_i(x)>0, any positive-cost alternative in supp(x) must have positive price, otherwise reducing its weight would lower cost without lowering spending; then an ε-reduction of such an alternative makes the spending constraint slack while strictly reducing cost. Adding this would make the argument easier to follow.","section":"Lemma 4.4"}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is technically strong and the AI-use disclosure is transparent; I have no additional concerns for the editor."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is the paper that actually covers the public-bads cell of the goods/bads taxonomy, and it does it carefully. The central claim—PF, Lindahl equilibrium, and Nash-product critical-point characterizations line up for public bads with linear costs—is proven in detail, and I checked the main theorems (4.3, 4.9, 4.10, 4.11, 4.22, 5.7) for internal consistency. The proofs are written out, not sketched, including a re-proof of the Mariotti-Villar existence result in the appendix. I think the central argument holds up.\n\nWhat is actually new: the BE core and completion core definitions, the strengthened Lindahl notion that fixes Tao-Zheng's weak version, the zero-respecting Lindahl equivalence to strictly positive PF, the exact CEEI correspondence for private-induced instances, the impossibility separating public bads from public goods, and Flipped-MNW with its completion-core guarantee. The paper also gives a clean axiomatic characterization and an efficient Frank-Wolfe algorithm for axes-cutting instances.\n\nThe paper is unusually honest about where things break. Strictly positive PF can fail to exist (Example 4.12) and can violate the completion core (Example 4.13); Theorem 5.5 shows no rule with weak symmetry and lower contraction consistency can even guarantee IFS. These are not hidden assumptions—the axes-cutting condition is defined precisely, and the failures outside it are demonstrated rather than ignored. The impossibility results are a feature, not a defect.\n\nSoft spots are minor and mostly about scope. The entire model assumes agents evaluate lotteries by expected cost, so risk-averse or correlation-sensitive preferences are out of scope—but that is stated at the outset, not smuggled in. Some worked examples say “it can be checked” instead of showing the arithmetic; a reader who wants to verify every last detail will have to do a little work. The AI-assisted proof of Lemma 4.5 is fully written out, so I don’t see a transparency problem there. No code artifacts, but the algorithmic results are standard iterative methods applied to a convex program, so this is not an obstacle.\n\nThis is a serious piece of work that deserves a careful referee. The citation pattern is appropriate: Mariotti and Villar and Tao and Zheng are acknowledged and genuinely extended, and the authors do not oversell the scope. I would cite this paper in the next year and I would bring it to a reading group. Send it to peer review—it will not waste a referee's time.","headline":"This is the first paper that gives a coherent fairness theory for public bads, and the main equivalences hold up; the scope limits are explicit and honestly handled.","tokens_in":41234,"tokens_out":1839,"would_cite":true,"duration_ms":19577,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["91B32","91B50"],"pacs":[],"model":"deepseek-v4-flash","headline":"For public bads, proportional fairness, Lindahl equilibrium, and a Nash critical-point condition are the same thing.","keywords":["public bads","proportional fairness","Lindahl equilibrium","core","chores","Nash product","competitive equilibrium from equal incomes","fair division"],"falsifier":"Take a private bads instance with two agents and two chores where the agents have different costs, induce the public-bads instance, compute the unique strictly positive proportional-fair allocation, and check whether the corresponding private allocation is a CEEI; any such instance that is PF but not CEEI would refute Theorem 4.11. Alternatively, find an axes-cutting instance whose unique strictly positive PF allocation lies outside the completion core, which would refute Theorem 4.22.","tokens_in":40441,"feed_emoji":"⚖️","tokens_out":6570,"duration_ms":67732,"temperature":0.7,"pith_summary":"This paper asks what proportional fairness should mean when the thing being allocated is a public bad—a single outcome, such as a landfill site or a blackout schedule, that imposes costs on everyone. It argues that the classical core, which lets a coalition walk away, is meaningless for public bads because a coalition's chosen outcome still harms outsiders, and it proposes two replacement core notions: the bounded-externality core and the completion core. Its central positive result is an exact equivalence: a lottery is proportionally fair exactly when it is a Lindahl equilibrium, exactly when the positive-cost part of its cost vector is a critical point of the Nash product in the upward closure of the feasible cost set. Under a structural condition that subsumes private chores, strictly positive fair lotteries coincide with competitive equilibrium from equal incomes (CEEI) and recover strong core guarantees; outside that condition, impossibility results show no rule can combine symmetry, consistency, and even the weakest individual-fairness guarantee.","feed_headline":"For public bads, proportional fairness is exactly Lindahl equilibrium","feed_subtitle":"Shared harms like landfills get personalized prices; on chore instances it matches the CEEI benchmark.","key_machinery":"The load-bearing structure is the upward closure C^S_≥ of the feasible cost set: the set of cost vectors dominated by some feasible lottery. Replacing the Pareto-frontier condition used for private chores by a critical-point condition over this larger upward-closed set is what makes proportional fairness and Lindahl equilibrium coincide for public bads. The other named mechanism is the paper's Lindahl equilibrium for public bads (spending at least 1/n, cost-minimization under personalized prices, and price sums weakly above 1, equality on supported alternatives), which is equivalent to PF through pain-per-buck pricing. The two core notions—bounded-externality core, where deviators compensate","core_discovery":"On the paper's terms, the central discovery is that proportional fairness for public bads is not a broken version of the goods-world maximum-Nash-welfare story but a distinct market story. Theorem 4.3 shows that a lottery x is proportionally fair iff it is a Lindahl equilibrium under personalized prices and iff, letting S be the agents with positive cost, c_S(x) is a critical point of the product of costs in the upward-closed feasible set C^S_≥. Restricting to strictly positive cost vectors, Theorem 4.9 turns this into zero-respecting Lindahl equilibria and the usual full Nash-product critical-point condition. On private-induced instances this coincides exactly with CEEI for chores (Theorem","pith_inferences":["If the expected-cost assumption is relaxed to risk-averse preferences, the ratio inequality defining PF no longer rests on a convex cost set; a natural next test is whether a suitably reweighted Lindahl definition recovers an equivalence for concave disutility.","The bounded-externality core and completion core may be two ends of a spectrum parameterized by how much externality a deviating coalition may leave on outsiders; one could search for an intermediate core that keeps strong guarantees on axes-cutting instances and IFS everywhere.","Because Flipped-MNW's completion-core guarantee is proved through the public-goods core, any future strengthening of public-goods core rules would directly strengthen this rule; conversely, its failure on private-induced instances points to an inherent trade-off between protecting outsiders and honoring private-bads core.","The acknowledgment that Lemma 4.5 and Example 4.6 were AI-derived is not itself a mathematical claim, but it identifies a step where independent formal verification would harden the chain of results."],"forward_implications":["For private bads (chores), the strictly positive proportional-fairness allocation is exactly the CEEI allocation, so the two theories coincide in that domain.","In axes-cutting instances, a strictly positive PF allocation always exists, satisfies both core notions and strong individual fair share, and can be approximated in polynomial time by a greedy Frank–Wolfe algorithm.","In general instances, no rule can simultaneously be weakly symmetric and lower-contraction-consistent while returning only allocations that meet individual fair share.","The Flipped-MNW rule, which converts the public-bads instance into a public-goods instance by complementing costs, always returns completion-core allocations and satisfies participation.","There are at most 2^m − 1 proportionally fair cost vectors, so the solution set can be enumerated by checking each support via convex programming."],"fun_headline_variants":["Public bads: proportional fairness is Lindahl equilibrium","Fair public bads: personalized Lindahl prices","Chore fairness extends to public bads via Lindahl","Lindahl pricing achieves fairness for public bads"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The entire framework assumes that every agent evaluates a lottery by its expected cost, so the feasible cost set is the convex hull of the alternatives and every fairness ratio is linear; if agents are risk-averse about harm or harms are correlated across people, the equivalence between proportional fairness, Lindahl equilibrium, and Nash critical points does not follow from these proofs.","fun_headline_variants_meta":{"raw":{"variants":["Public bads: proportional fairness is Lindahl equilibrium","Fair public bads: personalized Lindahl prices","Chore fairness extends to public bads via Lindahl","Lindahl pricing achieves fairness for public bads"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000757,"raw_usage":{"total_tokens":3192,"prompt_tokens":723,"completion_tokens":2469,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":467,"completion_tokens_details":{"reasoning_tokens":2407}},"tokens_in":467,"tokens_out":2469,"duration_ms":17475,"temperature":1.0,"reasoning_tokens":2407,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T00:44:11.995824+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a private bads instance with two agents and two chores where the agents have different costs, induce the public-bads instance, compute the unique strictly positive proportional-fair allocation, and check whether the corresponding private allocation is a CEEI; any such instance that is PF but not CEEI would refute Theorem 4.11. Alternatively, find an axes-cutting instance whose unique strictly positive PF allocation lies outside the completion core, which would refute Theorem 4.22.","supporting_citations":[],"review_version":1}