Delegation with Costly Inspection

Mohammad Mahdavi; Mohammad T. Hajiaghayi; Piotr Krysta; Suho Shin

REVIEW 3 major objections 3 minor 35 references

Delegation with Costly Inspection

T0 review · 3 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper proves that when delegation is free, the better of an inspected single proposal and the highest expected-value alternative approximates the optimal delegated-choice mechanism within a factor of 3, and that this factor is tight…

desk verdict Solid new model and a clever 3-approx for costless delegation, but the guarantee rests on a tie-breaking convention that the paper understates, and Theorem 4.12 has a real proof gap. read the letter →

arxiv 2506.07162 v2 pith:DFROKGZJ submitted 2025-06-08 cs.GT cs.DSecon.TH

Mohammad T. Hajiaghayi , Piotr Krysta , Mohammad Mahdavi , Suho Shin This is my paper

classification cs.GTcs.DSecon.TH MSC 68W2591B03

keywords delegatedchoiceinspectioncostPandora'sboxwithnonobligatoryapproximationmechanismprincipal-agentproblemNP-hardnessstrategicmisreportingsingle-proposal

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

Delegated choice with inspection cost (DCIC) is a principal-agent problem in which the agent may lie about the value of the alternative he proposes, and the principal can verify by paying an inspection cost. The paper's main result is that, when delegation itself is free, a simple maximal mechanism—run the better of a single-proposal mechanism that inspects the proposed alternative and the no-inspection rule that picks the alternative with the largest expected value—approximates the optimal mechanism within a factor of 3. The factor is tight for this mechanism. The same ratio holds when inspection costs are given by any monotone set function, and it improves to 2 when all inspection costs are equal. These are the first constant-factor guarantees for this problem, and they show that a single inspection is enough to make delegation substantially robust to strategic misreporting.

What carries the argument

The load-bearing device is the notion of a solution accepted without overinspection: alternative $i$ is selected without inspecting any alternative whose inspection cost is at least $c_i$. The paper proves that, for the worst-case agent under favorable tie-breaking, the event that $i$ is accepted without overinspection is independent of $X_i$, because the agent's signal depends on his own utility ranking rather than on the principal's utility. This independence bounds the expected utility from such selections by $\max_i \mathbb{E}[X_i]$, producing the upper bound on the optimal mechanism. On the lower side, the key working identity is Lemma 4.6: a shifted single-proposal mechanism with inspection recovers $\tfrac{1}{2}\mathbb{E}[\max_i (X_i-c_i)^+]$ by accepting a proposed alternative exactly when its net value $X_i-c_i$ clears a threshold. The maximal mechanism that chooses the better of these two components is the 3-approximation.

What would settle it

Drop the assumption that an indifferent agent sends the principal-favorable signal. Take 10 alternatives with $X_i=1$ with probability $0.01$ and $0$ otherwise, and inspection cost $0$, plus a decoy alternative with $X=0$ and inspection cost $0.2$, where the agent's utility makes him propose the decoy whenever no alternative clears the threshold. The Theorem 4.8 mechanism then earns the maximum of $0.01$ and a negative single-proposal term, i.e., $0.01$, whereas the PNOI policy that inspects the 10 cheap alternatives and otherwise selects an uninspected one earns about $0.1046$; the ratio exceeds 10, so the claimed 3-approximation guarantee does not survive adversarial tie-breaking.

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Extended reading notes

Core claim

The paper shows that the optimal mechanism's value is governed by two quantities: the largest ex-ante expected alternative value, $\max_i \mathbb{E}[X_i]$, and the expected surplus of the best inspected alternative, $\mathbb{E}[\max_i (X_i-c_i)^+]$. For any mechanism, the principal's utility is at most the sum of these two quantities. On the constructive side, the better of two simple mechanisms—a single-proposal mechanism that inspects the proposed alternative and accepts it only when $X_i-c_i$ passes a threshold, and the no-inspection choice of the maximum expected value—attains the maximum of $\max_i \mathbb{E}[X_i]$ and $\tfrac{1}{2}\mathbb{E}[\max_i (X_i-c_i)^+]$. Comparing the two bounds yields the 3-approximation, and a matching lower-bound instance shows the factor is tight for this mechanism. The paper thus establishes that combining one round of inspection with a fallback on ex-ante expected values is sufficient to counteract strategic misreporting up to a constant factor.

Load-bearing premise

The argument relies on the agent sending the principal-favorable signal whenever he is indifferent, especially the null signal when no proposed alternative clears the threshold; if indifference is resolved adversarially, the wasted inspection can break the half-surplus guarantee and the ratio 3.

Editorial extensions

If this is right

With costless delegation, one inspection suffices to make delegation robust to misreporting up to a constant factor, so the principal need not pre-commit to blind trust or exhaustive inspection.
The same 3-approximation holds when inspection costs are given by an arbitrary monotone set function, extending the result to settings where inspecting a group costs more or less than the sum of its parts.
If every alternative has the same inspection cost, the approximation improves to 2.
When delegation itself is costly, a constant-factor approximation survives in two regimes—$c_{\mathrm{Del}} < \tfrac12\mathbb{E}[\max_i (X_i-c_i)^+]$ or $c_{\mathrm{Del}} \ge \mathbb{E}[\max_i (X_i-c_i)^+] - 2\mathbb{E}[\mathrm{Mec}_P]$—leaving the intermediate regime open.
Since the problem generalizes Pandora's box with nonobligatory inspection, a known NP-hard problem, no exact polynomial-time optimal mechanism is expected unless P = NP.

Reading between the lines

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

The half-factor in Lemma 4.6 mirrors the classical threshold-mechanism half-guarantee for delegated search and prophet inequalities, which suggests that any improvement beyond 3 would need a richer signal structure than a single inspected proposal rather than a better threshold.
The overinspection-independence argument looks portable: any principal-agent search problem where the agent's signal is the only source of information and the principal can verify at a cost should admit a similar bound in terms of the best ex-ante value plus the best expected net surplus.
A testable extension is to randomize between the single-proposal mechanism with inspection and the PNOI policy when the delegation cost is moderate; the paper notes that randomization changes the agent's best-response and leaves this open, but the two-regime theorem suggests the failure region is exactly where neither component alone dominates.
If the favorable tie-breaking assumption is dropped, the SPMI's null-signal guarantee fails, so robustness to adversarial tie-breaking is the natural next stress test for any delegation mechanism.

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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

Here's my read. The DCIC model is a legitimate extension of both delegated choice and PNOI, and the overinspection decomposition (Lemmas 4.4-4.5) is a genuine new technique. NP-hardness via PNOI is clean. The 3-approximation for costless delegation is proved carefully, and Theorem 4.11 shows tightness for that mechanism. The combinatorial-cost extension and identical-cost 2-approximation are nice bonuses. I checked the key steps; Lemmas 4.4-4.7 hold up under the paper's stated assumptions.

The soft spot is the tie-breaking convention in Section 3.2. The paper calls it 'innocuous' and leans on reputation, but it is doing real work in Lemma 4.6. When no eligible solution exists, the agent is indifferent between the null signal and proposing an ineligible alternative; the proof assumes he sends null. An agent who breaks ties adversarially (e.g., all Y_i = 0) can force an expensive inspection and destroy the SPMI's guarantee. The stress-test's example makes this concrete: with 1000 boxes that pay 100 w.p. 0.01 (cost 0.1) and one box with cost 1000, favorable tie-breaking gives the claimed SPMI utility, while pessimistic tie-breaking leaves the mechanism with the max-expected-value fallback, and the ratio to the optimal inspection policy becomes unbounded. This doesn't contradict the theorem as stated — the model includes the convention — but it means the guarantee is contingent on that convention, and the paper's 'innocuous' characterization is misleading. The authors should either prove a robust version or openly frame the result as conditional on favorable tie-breaking.

The second issue is in Theorem 4.12. The theorem's second branch states a (2−ε) approximation when c_Del ≥ M − 2E[Mec_P], but the proof uses the stronger condition c_Del ≥ M − E[Mec_P]. Recomputing with the stated assumption gives roughly a 3-approximation, so the proof doesn't establish the claim. This is secondary to the main costless result, but it needs fixing.

Who is this for? People working on delegated choice, Pandora's box, and optimal stopping with verification. It deserves a serious referee. The model is new, the technique is non-obvious, and the paper is honest about what it doesn't get. I'd recommend conditional acceptance: patch Theorem 4.12 and address the tie-breaking head-on, either by strengthening the mechanism or by clearly labeling the result as being under a specific tie-breaking rule.

Referee Report

3 major / 3 minor

Summary. The paper introduces Delegated Choice with Inspection Cost (DCIC), a principal–agent model in which the agent may misreport the utility of a proposed alternative and the principal can inspect alternatives at an exogenous cost, with an additional fixed delegation cost c_Del. It observes that DCIC generalizes Pandora's box with nonobligatory inspection (PNOI), implies NP-hardness, and shows that neither pure inspection nor a single-proposal mechanism with inspection (SPMI) alone can be constant-factor approximable. Its main positive result is a 3-approximation for costless delegation: a maximal mechanism that takes the better of (i) an SPMI with a single threshold (capturing half of E[max_i (X_i - c_i)^+]) and (ii) the no-inspection choice of the alternative with largest E[X_i]. The 3 bound is shown tight for this mechanism and the analysis extends to monotone combinatorial costs, with an improved 2-approximation for equal inspection costs. For the costly-delegation setting, Theorem 4.12 claims constant-factor approximations in two regimes of c_Del. The paper also gives several impossibility and tightness examples.

Significance. If the main theorem holds, the paper makes a useful conceptual connection between delegated choice and nonobligatory-inspection Pandora's box, and the overinspection-based upper bound (Lemmas 4.4 and 4.5) is an elegant technique that could be reused. The proofs for the costless case are carefully derived, the mechanism is simple, and the tightness example for the proposed mechanism is instructive. However, the significance is substantially tempered by two issues: the 3-approximation depends on a favorable tie-breaking convention that the paper calls 'innocuous', and the costly-delegation theorem contains a mismatch between statement and proof plus an algebra error that turns the claimed 2 - epsilon into 2 + epsilon. These concerns must be fixed before the results can be relied upon.

major comments (3)

[3.2, Lemma 4.6] The proof of Lemma 4.6 relies on the assumption that when no eligible solution exists, the agent sends the null signal because he is indifferent and behaves in favor of the principal. This assumption is not innocuous: if ties are broken adversarially, the agent can propose an ineligible alternative, the principal inspects it and pays its cost, and the claimed lower bound 1/2 E[max_i (X_i - c_i)^+] fails. For example, take 1000 alternatives with X_i = 100 w.p. 0.01 (else 0) and c_i = 0.1, plus one alternative with X = 0 and c = 1000, and all Y_i = 0. With threshold tau = 50, favorable tie-breaking gives SPMI utility about 99.9, while a pessimistic tie-breaking agent proposes the high-cost box whenever it exists, giving utility about -1000. The maximal mechanism then falls back to the no-inspection rule (utility 1), while a feasible inspection policy obtains about 90, so the approximation ratio is at least 90 rather than 3. Thus Theorem 4.8 is contingent on the tie-breaking convention; the authors should either adopt it as an explicit modeling assumption or modify the mechanism and analysis to be robust to pessimistic tie-breaking.
[Theorem 4.12] The statement of Theorem 4.12 requires c_Del >= E[max_i (X_i - c_i)^+] - 2 E[Mec_P], but the proof (line 'Now consider the second case where c_Del >= E[max] - E[Mec_P]') uses the strictly stronger condition c_Del >= E[max] - E[Mec_P]. Since 2 E[Mec_P] >= E[Mec_P], the stated condition does not imply the proof's condition, so the theorem as stated is unproved. The condition in the theorem should be corrected, or the proof should be modified to work under the weaker stated assumption.
[Theorem 4.12, second branch] Even under the proof's corrected condition, the displayed chain gives E[Opt] <= (1 + 1/(1 - epsilon)) E[Mec] = (2 - epsilon)/(1 - epsilon) E[Mec], which is greater than 2 for every epsilon in (0,1). The conclusion '2 - epsilon-approximation for any epsilon' is therefore false; the argument yields a 2 + epsilon-approximation. This quantitative claim appears in the abstract and in Section 2.4 and should be corrected accordingly.

minor comments (3)

[Theorem 4.11] The example gives a ratio of (3 - 3 epsilon + epsilon^2)/1, which is below 3 for all epsilon in (0,1); this does not support the statement that the mechanism 'cannot have approximation factor better than 3+epsilon'. The intended tightness claim should be rephrased, e.g., as a family of instances whose approximation ratio approaches 3.
[Section 2.4 and Theorem 4.12] The 'rich class of instances' is only the union of the two closed-form regimes for c_Del; the paper should state explicitly that intermediate delegation costs are left open.
[Proposition 4.10] In the proof, the sentence 'If for any signal sigma, inspection policy Sig(sigma) selects a solution i without inspecting any solution...' appears to require that the agent's best response indeed sends such a signal; the argument would benefit from making the worst-case agent explicit.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the 3-approximation arises from an upper/lower-bound sandwich using external SPM and PNOI PTAS results, not from fitted constants or self-referential definitions.

full rationale

The paper's derivation chain is self-contained against external benchmarks. The central upper bound (Lemma 4.5) is proved by constructing a worst-case agent response and bounding the utility from selections with and without overinspection; the lower bound (Lemma 4.6) reduces the SPMI to the Kleinberg-Kleinberg single-threshold SPM on shifted variables Z_i=(X_i-c_i)^+, an external theorem whose stated assumptions do not include the target 3-approximation. The factor 3 in Theorem 4.8 then follows by comparing the two bounds, with no fitted constant smuggled in to match the claimed ratio. Theorem 4.12 invokes the PNOI PTAS of Beyhaghi-Cai and Fu-Liu-Liu as a black box, again external and parameter-free. Self-citations in the related-work section and acknowledgements are survey-style and not used to justify any theorem. The tie-breaking convention in Section 3.2 is a stated modeling assumption; even if it is load-bearing for Lemma 4.6, it is not a circular definition of the result. The algebra slip in Theorem 4.12's second branch (the stated condition uses 2E[Mec_P] while the proof uses E[Mec_P]) is a correctness defect, not a circularity, because no equation is defined in terms of the result it is meant to prove.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters are fitted; thresholds and approximation parameters come from prior theorems. No new physical or model entities are introduced. The adversarial agents in the proofs are devices, not additions to the model.

assumptions (6)

domain assumption The principal knows the marginal distributions D^X_i and costs c_i but not the agent's utility distribution D^Y; the mechanism must be agent-oblivious and is evaluated against the worst-case D^Y.
Defines the benchmark in Eq. (3.1)-(3.2) and justifies the adversarial agent constructions in Lemmas 4.4 and 4.5.
domain assumption The agent observes all realized utilities, best-responds, and breaks ties in favor of the principal.
Explicitly imposed in Section 3.2; used in Lemma 4.6 to ensure an indifferent agent sends the null signal, and in Proposition 4.10.
domain assumption Alternative utilities X_i and Y_i are nonnegative and independent across alternatives.
Section 3 setup; independence across i is used for product distributions and in the independence step of Lemma 4.4.
domain assumption The principal can commit to any signaling mechanism and inspection policy; the game is sequential with commitment.
Standard delegation assumption; without commitment, the analysis in Section 4 would not hold.
standard math Kleinberg-Kleinberg single-proposal single-threshold theorem (Theorem 3.2) holds for zero-cost inspection.
External theorem used in Lemma 4.6 to obtain the 1/2 E[max(X_i - c_i)^+] guarantee.
standard math PNOI is NP-hard and admits a PTAS (Fu et al. 2023; Beyhaghi and Cai 2023).
Used for Proposition 3.8 (NP-hardness of DCIC) and for the PTAS-based mechanism in Theorem 4.12.

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Cite this review

Pith. "Pith review of Delegation with Costly Inspection." pith.science (2026). https://pith.science/paper/DFROKGZJ

@misc{pith2026250607162,
  author       = {Pith},
  title        = {Pith review of: Delegation with Costly Inspection},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DFROKGZJ}},
  note         = {Machine review of arXiv:2506.07162}
}

abstract

We study the problem of delegated choice with inspection cost (DCIC), which is a variant of the delegated choice problem by Kleinberg and Kleinberg (EC'18) as well as an extension of the Pandora's box problem with nonobligatory inspection (PNOI) by Doval (JET'18). In our model, an agent may strategically misreport the proposed element's utility, unlike the standard delegated choice problem which assumes that the agent truthfully reports the utility for the proposed alternative. Thus, the principal needs to inspect the proposed element possibly along with other alternatives to maximize its own utility, given an exogenous cost of inspecting each element. Further, the delegation itself incurs a fixed cost, thus the principal can decide whether to delegate or not and inspect by herself. We show that DCIC indeed is a generalization of PNOI where the side information from a strategic agent is available at certain cost, implying its NP-hardness by Fu, Li, and Liu (STOC'23). We first consider a costless delegation setting in which the cost of delegation is free. We prove that the maximal mechanism over the pure delegation with a single inspection and an PNOI policy without delegation achieves a $3$-approximation for DCIC with costless delegation, which is further proven to be tight. These results hold even when the cost comes from an arbitrary monotone set function, and can be improved to a $2$-approximation if the cost of inspection is the same for every element. We extend these techniques by presenting a constant factor approximate mechanism for the general setting for rich class of instances.

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Optimal delegation

Ricardo Alonso and Niko Matouschek. Optimal delegation. The Review of Economic Studies, 75 0 (1): 0 259--293, 2008

work page 2008

[2] [2]

A model of delegated project choice

Mark Armstrong and John Vickers. A model of delegated project choice. Econometrica, 78 0 (1): 0 213--244, 2010

work page 2010

[3] [3]

Delegated Stochastic Probing

Curtis Bechtel and Shaddin Dughmi. Delegated stochastic probing. arXiv preprint arXiv:2010.14718, 2020

work page Pith review arXiv 2010

[4] [4]

Delegated pandora's box

Curtis Bechtel, Shaddin Dughmi, and Neel Patel. Delegated pandora's box. In Proceedings of the 23rd ACM Conference on Economics and Computation, pages 666--693, 2022

work page 2022

[5] [5]

Pandora's problem with combinatorial cost

Ben Berger, Tomer Ezra, Michal Feldman, and Federico Fusco. Pandora's problem with combinatorial cost. In Proceedings of the 24th ACM Conference on Economics and Computation, pages 273--292, 2023

work page 2023

[6] [6]

Pandora’s problem with nonobligatory inspection: Optimal structure and a ptas

Hedyeh Beyhaghi and Linda Cai. Pandora’s problem with nonobligatory inspection: Optimal structure and a ptas. In Proceedings of the 55th Annual ACM Symposium on Theory of Computing, pages 803--816, 2023

work page 2023

[7] [7]

Pandora's problem with nonobligatory inspection

Hedyeh Beyhaghi and Robert Kleinberg. Pandora's problem with nonobligatory inspection. In Proceedings of the 2019 ACM Conference on Economics and Computation, pages 131--132, 2019

work page 2019

[8] [8]

Matching with Nested and Bundled Pandora Boxes

Robin Bowers and Bo Waggoner. Matching with nested and bundled pandora boxes. arXiv preprint arXiv:2406.08711, 2024

work page Pith review arXiv 2024