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Proportional Fairness for Harmful Decisions

T0 review · 0 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read For public bads, proportional fairness, Lindahl equilibrium, and a Nash critical-point condition are the same thing.

desk verdict 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. read the letter →

arxiv 2607.26053 v1 pith:ESJTPAGR submitted 2026-07-28 cs.GT

Benjamin Cookson , Soroush Ebadian , Dominik Peters , Nisarg Shah This is my paper · ORCID
classification cs.GT MSC 91B3291B50
keywords publicbadsproportionalfairnessLindahlequilibriumcorechoresNashproductcompetitivefromequalincomesfairdivision
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 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.

What carries the argument

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

What would settle it

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.

Watch

Extended reading notes

Core claim

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

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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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

0 major / 4 minor

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.

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.

minor comments (4)
  1. [§6, Proposition 6.3] 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.
  2. [§4.2, Examples 4.12–4.13] 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.
  3. [Definition 4.2] 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.
  4. [Lemma 4.4] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central PF/Lindahl/critical-point equivalences are proven directly from the stated linear-cost model.

full rationale

I walked the main derivation chain and found no step where a claimed result reduces by construction to its own input. Theorem 4.3 is established by direct proofs: Lemma 4.4 constructs Lindahl prices from PF and derives PF from Lindahl using the definitions of the two concepts, and Lemma 4.5 proves the equivalence between PF and critical points of the Nash product in the upward closure C^S_≥ using only the linear-cost structure and the PF inequality. Theorem 4.9 is a corollary: the proof shows that, within Lindahl equilibria, the zero-respecting condition is equivalent to strictly positive costs, so the claimed equivalence follows from Theorem 4.3 rather than being assumed. Theorem 4.10 is a nontrivial derivation from the definition of the bounded-externality core: the BE core is strictly weaker than PF, as can be seen from the paper's own examples, so the implication is not a restatement. Theorem 4.11 is proven constructively from the CEEI definition and the private-bads characterization of Bogomolnaia et al. (2017). The completion-core guarantee for Flipped-MNW (Theorem 5.7) is a valid reduction to the public-goods core theorem via the dual valuation v_i(a)=max c_i - c_i(a); the completion core was defined separately for fairness reasons, and the equivalence with the dual core is derived, not presupposed. No fitted parameters are presented as predictions, and no load-bearing uniqueness theorem is imported from author self-citations. The citations to Kroer and Peters (2025) and Mariotti and Villar (2005) are contextual or are accompanied by self-contained proofs in the appendices; they do not carry the central argument. The axes-cutting restriction is explicitly scoped, with the paper itself providing counterexamples outside it, so this is a stated limitation rather than a hidden assumption. I therefore find no circularity.

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

The central results are theorems over a stated model: linear expected costs, nonnegative costs with 0 not in C, and no indifferent agents. Standard convex analysis is used in proofs. No parameters are fitted to data, and no new unobserved entities are postulated. The axes-cutting condition is an explicit structural assumption for some existence and algorithmic results, not a global axiom.

assumptions (4)
  • domain assumption Linear expected costs: agent i's cost of lottery x is sum_a x(a) c_i(a), so C = conv{c(a)}.
    Used throughout Section 3 and in the PF/Lindahl/critical-point equivalences. If costs are non-linear or non-expected, the first-order characterizations fail.
  • domain assumption 0 not in C and no agent is indifferent between all alternatives.
    Stated at the start of Section 3. Ensures S is nonempty in critical-point arguments and that Flipped-MNW's dual goods instance is non-degenerate.
  • standard math Standard convex analysis toolkit: separating hyperplane theorem, KKT conditions, AM-GM inequality, concavity of log.
    Used in Lemmas 4.16-4.19, Theorem 4.3, and Proposition 6.3. This is unremarkable background mathematics.
  • domain assumption Axes-cutting condition: for every agent i there is an alternative a_i with positive cost to i and zero cost to all others.
    Definition 4.14. Needed for existence of strictly positive PF allocations and for the Frank-Wolfe convergence theorem (Theorem 6.4); it is not a global assumption of the paper.

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Cite this review

Pith. "Pith review of Proportional Fairness for Harmful Decisions." pith.science (2026). https://pith.science/paper/ESJTPAGR

@misc{pith2026260726053,
  author       = {Pith},
  title        = {Pith review of: Proportional Fairness for Harmful Decisions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ESJTPAGR}},
  note         = {Machine review of arXiv:2607.26053}
}
read the original abstract

We study allocation of (divisible) public bads, where agents incur costs for alternatives and the goal is to pick a lottery over the alternatives. We show that the traditional definitions of the core, a central criterion of proportional representation for allocation of public goods, private goods, and private bads (chores), do not make sense for allocation of public bads. We introduce two formalizations of the core tailored to public bads. Under a structural condition which subsumes allocation of private bads, we show that zero-respecting Lindahl equilibria satisfy both formalizations, exhibit additional fairness guarantees, and strictly generalize competitive equilibria from equal incomes (CEEI) for allocation of private bads. Without this structural condition, we show that Lindahl equilibria exhibit undesirable behaviors, prove sharp impossibility results separating public bads from public goods, but show that a rule using a reduction to public goods recovers one of our formalizations of the core. Our results lay the groundwork for studying fair allocation of public bads, an overlooked yet fundamental problem, and highlight several structural and algorithmic directions that remain open.

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.