Auctions with Contract Design

Biaoshuai Tao; Chunxue Yang; Haikuo Yang; Jiarong Jin; Junzhu Ke; Pinyan Lu; Xiang Yan; Xiaolin Bu; Zhihua Zhu

Auctions with Contract Design

T0 review · 0 major / 5 minor · reviewed 2026-08-02 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper argues that in auctions where bidders make sunk quality investments before bidding, the revenue-maximizing quality reward converges to the auctioneer's full marginal benefit from quality as the number of bidders grows.

desk verdict Solid theory paper: new model, clean asymptotic result, transparent about its equilibrium-selection assumption; deserves a serious referee. read the letter →

arxiv 2607.13795 v1 pith:DNXFZQ65 submitted 2026-07-15 cs.GT

Xiaolin Bu , Jiarong Jin , Junzhu Ke , Pinyan Lu , Biaoshuai Tao , Xiang Yan , Chunxue Yang , Haikuo Yang

show 1 more author

Zhihua Zhu

This is my paper · ORCID

classification cs.GT MSC 91B2691A1091B41

keywords auctiontheorycontractdesignmoralhazardqualityinvestmentBayes-Nashequilibriumrevenueequivalencesecond-pricefirst-price

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

The paper introduces an auction model in which bidders invest in quality before the auction, so a losing bidder's investment is wasted—a moral hazard problem. The auctioneer commits to a linear contract T(q)=tq that rewards the winner based on realized quality. The central result: with identically distributed bidder types and bidders playing the unique symmetric 'natural' equilibrium, as n→∞ the optimal reward factor t converges to ω, the auctioneer's marginal benefit from quality. A revenue-equivalence theorem extends this conclusion to any auction rule satisfying the paper's assumptions, including first-price auctions, as long as a symmetric strictly increasing equilibrium exists. The paper also shows that using such contracts strictly improves revenue over no-contract auctions, with gains nearly proportional to the potential quality gain.

What carries the argument

The central object is the 'natural strategy'—bidding one's expected value plus expected reward—combined with the Key Monotonicity Lemma, which shows that the utility difference between a higher-cost action and a lower-cost action is increasing in a bidder's type. This yields threshold strategies in equilibrium. The threshold equation (D(θ*))^(n−1)(θ*(ν2−ν1)+t(μ2−μ1))=c2−c1 pins down the unique symmetric equilibrium in the second-price auction and is used to compute revenue in the large-n limit. The revenue-equivalence theorem relies on an envelope/payment-identity argument: for any symmetric strictly increasing equilibrium, a bidder's expected payment as a function of type equals the expecte

What would settle it

Compute the revenues in the paper's own two-bidder uniform example with parameters c1=0.5, c2=2, μ1=ν1=9, μ2=ν2=11, t=1, under the asymmetric equilibrium thresholds (3/5, 2/5) and under the symmetric natural equilibrium. If the asymmetric equilibrium yields higher revenue for some t, that would confirm the optimal reward depends on equilibrium selection. A sharper falsifier for the asymptotic claim would be to find a sequence of asymmetric equilibria as n→∞ whose revenue differs from the symmetric-threshold formula; the paper's revenue-equivalence theorem only covers symmetric strictly increas

Watch

Extended reading notes

Core claim

The paper's main discovery is that, in a large market, the auctioneer should set the linear quality reward equal to her own marginal value of quality, t≈ω. This is established for a two-action second-price auction with identical type distributions, where a unique symmetric Bayes-Nash equilibrium exists and is characterized by a threshold θ* solving (D(θ*))^(n−1)(θ*(ν2−ν1)+t(μ2−μ1))=c2−c1. The revenue-maximization analysis shows that, as n→∞, the optimal t approaches ω in the cases where high quality is worth incentivizing. The paper then proves a revenue-equivalence theorem: for any auction rule satisfying Assumption 2.2, if a symmetric strictly increasing Bayes-Nash equilibrium exists, the

Load-bearing premise

The results assume bidders actually play the symmetric 'natural' equilibrium (or, more generally, a symmetric strictly increasing equilibrium); if bidders coordinate on an asymmetric equilibrium instead—which the paper shows can exist—the revenue-maximizing reward can differ from the one characterized here.

Editorial extensions

If this is right

For second-price and first-price auctions, and any auction rule satisfying Assumption 2.2, the optimal linear reward factor converges to ω as the number of bidders grows, provided bidders play a symmetric strictly increasing equilibrium.
Revenue is invariant across auction rules under the same contract when a symmetric strictly increasing equilibrium exists, so the optimal contract design does not depend on the specific auction format.
Adding a quality-contingent contract strictly improves the auctioneer's expected revenue compared to the same auction without a reward, and the improvement is approximately linear in the potential quality gain ω(μ2−μ1).
In the small-ω regime (below the threshold K), the auctioneer gains nothing from using a contract, and the optimal reward can be any value that keeps all bidders on the low-quality action.
The revenue gain from contracts is always positive, even when bidders already have some intrinsic incentive to invest in quality, and it grows roughly linearly with the auctioneer's quality valuation.

Reading between the lines

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

The full-pass-through result is derived for a linear contract and linear quality benefit; a natural extension, left implicit by the paper, is that for nonlinear benefit Ω(q), the optimal contract may need to match the marginal benefit at each quality level rather than a single scalar t.
The optimality of t≈ω is conditional on equilibrium selection. Since the paper itself constructs asymmetric Bayes-Nash equilibria, if bidders coordinate on asymmetric thresholds—for example in the two-bidder uniform example—the revenue-maximizing reward could differ from ω.
A testable practical implication is that in large ad auctions, the platform's per-unit quality reward should be set equal to its internal marginal valuation of quality, rather than to a share of bidder surplus or a cost-reimbursement parameter.
The revenue-equivalence theorem suggests that adding quality contracts to auctions with reserve prices or other allocation rules may preserve revenue as long as symmetric monotone equilibria exist, but the paper's analysis stops short of optimizing the mechanism jointly with the contract.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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 is my take on arXiv:2607.13795. The paper is a careful theory contribution. The model is genuinely new: bidders make sunk quality investments before the auction, creating a moral hazard problem, and the auctioneer commits to an ex-post linear reward to incentivize effort. That is a natural bridge between auction theory and contract theory, and the paper delivers a clean headline result: as the number of bidders grows, the optimal reward factor converges to the auctioneer's marginal benefit of quality—full pass-through—and this holds across a broad class of auctions via a revenue equivalence theorem. The proof of the asymptotic result is explicit and checkable: revenue is written as a function of the equilibrium threshold, the first-order condition gives t=ω, and the substitution D(θ*)=1-x/n is handled properly. The revenue equivalence proof is a standard but correct envelope argument.

The paper also earns credit for disclosure. It states plainly that the main results rely on equilibrium selection—bidders must play the unique symmetric natural BNE in SPA or a symmetric strictly increasing equilibrium elsewhere—and it explicitly constructs an asymmetric BNE in Appendix A.2. That is a real limitation, not a hidden flaw. If bidders coordinate on asymmetric equilibria, the revenue-maximizing t need not be the characterized one. But the authors do not hide it.

Soft spots are minor. The existence proof for SPA (Theorem 3.5) uses Brouwer and some continuity arguments that are stated tersely; a referee may want more detail. The finite-n analysis in Appendix A.3 shows t is not exactly ω for uniform types, which is fine but means the headline is asymptotic. The paper restricts to linear contracts; whether linear contracts are optimal among all contracts remains open, and the authors say so. None of these undercut the central claim.

This is a paper that delivers what it promises. I would bring it to a reading group, and I would cite it if I were working on auctions or contracts. My bottom line: send it to a serious referee. It deserves careful review, and I would expect acceptance after minor revision.

Referee Report

0 major / 5 minor

Summary. The paper introduces a model in which n bidders first make a costly, sunk quality-enhancing investment and then compete in an auction; the winner receives a quality-contingent transfer T(q)=tq. The auctioneer's revenue is payment minus the transfer plus a linear quality benefit ωq. For the second-price auction, the paper characterizes natural-strategy threshold equilibria, proves existence of Bayes-Nash equilibria, and shows that for i.i.d. types there is a unique symmetric natural-strategy equilibrium. Its main result is that, as n→∞, a revenue-maximizing linear contract satisfies t→ω, i.e., full pass-through of the auctioneer's marginal quality benefit. A revenue-equivalence theorem extends this conclusion to any auction rule admitting a symmetric strictly increasing equilibrium, and a symmetric equilibrium is constructed for the first-price auction. The paper also quantifies the revenue gain over the no-contract benchmark. All proofs are provided in the text and appendices, and the main asymptotic argument is explicit.

Significance. If correct, the paper makes a substantial contribution: it connects auction theory with moral-hazard contract design in a clean, tractable framework, and it delivers a sharp, parameter-free conclusion (t→ω) that is robust across a broad class of auction formats via revenue equivalence. The derivation is internally consistent: the revenue is written in terms of the equilibrium threshold, the scaling D(θ*)=1-x/n is handled carefully, and the first-order condition genuinely yields t=ω rather than an identity. The paper also contains an explicit construction of the FPA symmetric equilibrium and a transparent envelope-style revenue-equivalence proof. The main caveat is equilibrium selection: the optimality statement is conditional on bidders coordinating on the unique symmetric natural BNE, and Appendix A.2 shows that asymmetric natural-strategy equilibria can exist. The paper is honest about this restriction in the theorem statements, but the abstract and introduction could more prominently carry the same qualification.

minor comments (5)

[§3.1, §A.2, abstract] The headline 'optimal reward factor converges to the marginal benefit' should be qualified in the abstract and introduction as holding under symmetric equilibrium selection. Appendix A.2 constructs an asymmetric natural-strategy BNE (thresholds 3/5 and 2/5 for n=2), so the 'natural' restriction alone does not select the equilibrium used in Theorems 3.7 and 4.1. The theorem statements are accurate, but the broader framing should not imply robustness to arbitrary Bayes-Nash play.
[§1.1 vs. discussion after Theorem 3.7] The introduction says the optimal contract 'always result[s] in strictly better revenue' compared with no contract, but the first case of Theorem 3.7 gives the same revenue Bν1+ωμ1 for t optimal and t=0. The discussion after the theorem correctly notes equality in this case; please reconcile the summary claim.
[Remark after Theorem 3.6 / §A.2] The example in A.2 is described as an 'unnatural asymmetric equilibrium,' but it satisfies the paper's natural-bid restriction. The relevant distinguishing feature is asymmetry, not non-natural bidding. Suggest renaming it an 'asymmetric natural-strategy equilibrium' to avoid confusion.
[Theorem 3.7] In the first case, the theorem says any t in [0,K) is optimal. When K=ω, the boundary point t=K=ω is also optimal; the surrounding discussion already treats ω as lying at the boundary. Please adjust the interval to [0,K] or add a clarifying convention.
[§3.3 proof] The notation q(N2) refers to the quality draw of the losing second-highest bidder, which is not actually realized. This is harmless in expectation, but the quantity should be defined explicitly (e.g., as the quality that would realize if that bidder won) to avoid confusion.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: t→ω is derived by optimizing revenue, not by construction.

full rationale

The central chain is self-contained and does not reduce to its inputs. Theorem 3.6 derives the symmetric threshold equation (D(θ*))^{n−1}(θ*(ν2−ν1)+t(μ2−μ1))=c2−c1 from best-response indifference rather than assuming it. Theorem 3.7 maximizes the limiting revenue R(x)=(Bν2+μ2ω)−x(c2−c1)−e^{-x}ω(μ2−μ1)−e^{-x}B(ν2−ν1) over the transformed threshold x; the first-order condition gives e^{-x}=(c2−c1)/(ω(μ2−μ1)+B(ν2−ν1)), and substitution into the threshold equation (4) yields t=ω. This is a genuine first-order optimum; t is the variable being optimized, not a fitted or predefined value. The revenue-equivalence theorem (Theorem 4.1) is an envelope argument deriving the same expected-payment function as SPA for any symmetric strictly increasing equilibrium; it does not import the SPA revenue or t=ω as an assumption. The only self-citation (Hartman et al. [2025]) is used merely as an example of an auction rule satisfying Assumption 2.2 and plays no role in the derivation. The paper explicitly scopes its conclusion to natural/symmetric strictly increasing equilibria and discloses asymmetric equilibria in Appendix A.2; this is a transparent equilibrium-selection caveat, not circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No empirical data are fitted; all model primitives (c_j, s_j, D, v_q, ω) are inputs. The only decision variable is t. The structural assumptions above are stated explicitly and are not hidden. No new physical or economic entity is invented beyond the quality-contingent transfer T(q).

assumptions (5)

domain assumption Assumption 2.1: type distributions are continuous with positive densities on bounded supports; identical distributions are assumed for symmetric results.
Needed for threshold uniqueness, density formulas, and the asymptotic convergence θ*→B. Standard for Bayesian auction models.
domain assumption Assumption 2.2: the auction allocates to the highest bid, payments are weakly increasing in one's own bid, and only the winner pays.
Defines the auction class covered by the revenue equivalence theorem and excludes reserve prices and Myerson-style mechanisms.
domain assumption Separable valuation v(θ,q)=θ v_q(q), linear auctioneer benefit Ω(q)=ωq, finite ordered actions with FOSD quality distributions and increasing costs.
This structure makes natural bidding and threshold comparisons tractable; the entire derivation depends on it.
ad hoc to paper Bidders play natural strategies in SPA and coordinate on the unique symmetric Bayes-Nash equilibrium.
Explicitly assumed to rule out unnatural equilibria and to make revenue maximization well-defined; Appendix A.2 shows asymmetric equilibria can exist.
domain assumption Contracts are restricted to linear transfers T(q)=tq.
The optimization is over the scalar t; the paper does not claim optimality among all nonlinear contracts.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Auctions with Contract Design." pith.science (2026). https://pith.science/paper/DNXFZQ65

@misc{pith2026260713795,
  author       = {Pith},
  title        = {Pith review of: Auctions with Contract Design},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DNXFZQ65}},
  note         = {Machine review of arXiv:2607.13795}
}

read the original abstract

We consider a new auction model where the bidders' utilities and the auctioneer's revenue depend on a quality factor of the transaction determined by costly and strategic investments of the bidders. Applications of our model include ad auctions, government concessions and crowdsourcing contests. Crucially, these quality-enhancing efforts made by the bidders are often sunk costs incurred prior to the allocation, creating a fundamental moral hazard problem where the risk of losing the auction discourages investments. In this paper, we study the design of revenue-maximizing contracts integrated into auctions: the auctioneer commits to a transfer rule that rewards the winner for the ex-post realized quality of the transaction to incentivize higher effort. Our new framework is a natural generalization of both the auction theory and the principal-agent model. We consider both the second-price and the first-price auctions. We show that natural symmetric Bayes Nash equilibria exist in both auctions. Assuming these natural equilibria are played by the bidders and the number of bidders is large, we study linear contracts and derive the optimal reward factor of the transfer rule that maximizes the auctioneer's revenue. As the main result, we show that the optimal reward factor converges to the auctioneer's marginal benefit from the quality, as the number of bidders grows. That is, it is optimal for the auctioneer to fully pass through the quality value to the winner. This observation is largely independent of the auction rule used: we derive a revenue equivalence theorem showing that the revenue remains the same as long as symmetric Bayes Nash equilibria exist. Lastly, by quantitatively comparing with the standard auctions where no quality reward is used, we show that the use of contracts effectively improves the revenue by incentivizing high investments from the bidders.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

40 extracted references

[1]

The RAND Journal of Economics , pages=

Design competition through multidimensional auctions , author=. The RAND Journal of Economics , pages=. 1993 , publisher=

1993
[2]

The RAND Journal of Economics , pages=

The design of multidimensional auctions , author=. The RAND Journal of Economics , pages=. 1997 , publisher=

1997
[3]

Journal of Economic Theory , volume=

Entry and R&D in procurement contracting , author=. Journal of Economic Theory , volume=
[4]

The RAND Journal of Economics , pages=

Bidding for contracts: a principal-agent analysis , author=. The RAND Journal of Economics , pages=
[5]

American economic review , volume=

Internet advertising and the generalized second-price auction: Selling billions of dollars worth of keywords , author=. American economic review , volume=. 2007 , publisher=

2007
[6]

international Journal of industrial Organization , volume=

Position auctions , author=. international Journal of industrial Organization , volume=. 2007 , publisher=

2007
[7]

The Quarterly Journal of Economics , volume=

Position auctions with consumer search , author=. The Quarterly Journal of Economics , volume=. 2011 , publisher=

2011
[8]

International Workshop on Web and Internet Economics , pages=

Ad auction design and user experience , author=. International Workshop on Web and Internet Economics , pages=. 2007 , organization=

2007

Show all 40 references

[9]

Theoretical Computer Science , volume=

Optimally integrating ad auction into e-commerce platforms , author=. Theoretical Computer Science , volume=. 2023 , publisher=

2023
[10]

arXiv preprint arXiv:2508.08772 , year=

Optimal Boost Design for Auto-bidding Mechanism with Publisher Quality Constraints , author=. arXiv preprint arXiv:2508.08772 , year=
[11]

Proceedings of the 10th ACM conference on Electronic commerce , pages=

Crowdsourcing and all-pay auctions , author=. Proceedings of the 10th ACM conference on Electronic commerce , pages=
[12]

American Economic Review , volume=

The optimal allocation of prizes in contests , author=. American Economic Review , volume=. 2001 , publisher=

2001
[13]

ICIS 2009 proceedings , pages=

Optimal design of crowdsourcing contests , author=. ICIS 2009 proceedings , pages=

2009
[14]

The American Economic Review , pages=

An empirical study of an auction with asymmetric information , author=. The American Economic Review , pages=. 1988 , publisher=

1988
[15]

Journal of Economics & Management Strategy , volume=

The FCC spectrum auctions: An early assessment , author=. Journal of Economics & Management Strategy , volume=. 1997 , publisher=

1997
[16]

The Bell journal of economics , pages=

Moral hazard and observability , author=. The Bell journal of economics , pages=. 1979 , publisher=

1979
[17]

Foundations and Trends

Algorithmic contract theory: A survey , author=. Foundations and Trends. 2024 , publisher=

2024
[18]

The Rand journal of economics , volume=

Procurement when price and quality matter , author=. The Rand journal of economics , volume=. 2010 , publisher=

2010
[19]

International Economic Review , pages=

Cost-reducing investment, optimal procurement and implementation by auctions , author=. International Economic Review , pages=. 1996 , publisher=

1996
[20]

Proceedings of the 7th ACM Conference on Electronic Commerce , pages=

Combinatorial agency , author=. Proceedings of the 7th ACM Conference on Electronic Commerce , pages=
[21]

Proceedings of the 2019 ACM Conference on Economics and Computation , pages=

Simple versus optimal contracts , author=. Proceedings of the 2019 ACM Conference on Economics and Computation , pages=

2019
[22]

Proceedings of the 22nd ACM Conference on Economics and Computation , pages=

Contracts under moral hazard and adverse selection , author=. Proceedings of the 22nd ACM Conference on Economics and Computation , pages=
[23]

Econometrica: Journal of the Econometric Society , pages=

Aggregation and linearity in the provision of intertemporal incentives , author=. Econometrica: Journal of the Econometric Society , pages=. 1987 , publisher=

1987
[24]

Foundations of insurance economics: Readings in economics and finance , pages=

An analysis of the principal-agent problem , author=. Foundations of insurance economics: Readings in economics and finance , pages=. 1992 , publisher=

1992
[25]

1993 , publisher=

A theory of incentives in procurement and regulation , author=. 1993 , publisher=

1993
[26]

International Conference on Web and Internet Economics , pages=

Autobidding with constraints , author=. International Conference on Web and Internet Economics , pages=. 2019 , organization=

2019
[27]

ACM SIGecom Exchanges , volume=

Auto-bidding and auctions in online advertising: A survey , author=. ACM SIGecom Exchanges , volume=. 2024 , publisher=

2024
[28]

The Economic Journal , volume=

The design of auctions and tenders with quality thresholds: the symmetric case , author=. The Economic Journal , volume=. 1994 , publisher=

1994
[29]

The American economic review , volume=

The economic theory of agency: The principal's problem , author=. The American economic review , volume=. 1973 , publisher=

1973
[30]

The Review of Economic Studies , volume=

The theory of moral hazard and unobservable behaviour: Part I , author=. The Review of Economic Studies , volume=. 1999 , publisher=

1999
[31]

Econometrica: Journal of the Econometric Society , pages=

Moral hazard and verifiability: The effects of renegotiation in agency , author=. Econometrica: Journal of the Econometric Society , pages=. 1991 , publisher=

1991
[32]

2002 , publisher=

The theory of incentives: the principal-agent model , author=. 2002 , publisher=

2002
[33]

SIAM Journal on Computing , volume=

The complexity of contracts , author=. SIAM Journal on Computing , volume=. 2021 , publisher=

2021
[34]

SIAM Journal on Computing , volume=

Combinatorial contracts , author=. SIAM Journal on Computing , volume=. 2025 , publisher=

2025
[35]

arXiv preprint arXiv:2511.20110 , year=

One Action Too Many: Inapproximability of Budgeted Combinatorial Contracts , author=. arXiv preprint arXiv:2511.20110 , year=
[36]

Production and Operations Management , volume=

Keyword auctions, unit-price contracts, and the role of commitment , author=. Production and Operations Management , volume=. 2010 , publisher=

2010
[37]

Information Systems Research , volume=

Ex ante information and the design of keyword auctions , author=. Information Systems Research , volume=. 2010 , publisher=

2010
[38]

International Workshop on Internet and Network Economics , pages=

Efficient ranking in sponsored search , author=. International Workshop on Internet and Network Economics , pages=. 2011 , organization=

2011
[39]

Proceedings of the fourteenth ACM conference on Electronic commerce , pages=

Revenue optimization in the generalized second-price auction , author=. Proceedings of the fourteenth ACM conference on Electronic commerce , pages=
[40]

Proceedings of the 26th ACM Conference on Economics and Computation , pages=

It’s not all black and white: degree of truthfulness for risk-avoiding agents , author=. Proceedings of the 26th ACM Conference on Economics and Computation , pages=

Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.

[1] [1]

The RAND Journal of Economics , pages=

Design competition through multidimensional auctions , author=. The RAND Journal of Economics , pages=. 1993 , publisher=

1993

[2] [2]

The RAND Journal of Economics , pages=

The design of multidimensional auctions , author=. The RAND Journal of Economics , pages=. 1997 , publisher=

1997

[3] [3]

Journal of Economic Theory , volume=

Entry and R&D in procurement contracting , author=. Journal of Economic Theory , volume=

[4] [4]

The RAND Journal of Economics , pages=

Bidding for contracts: a principal-agent analysis , author=. The RAND Journal of Economics , pages=

[5] [5]

American economic review , volume=

Internet advertising and the generalized second-price auction: Selling billions of dollars worth of keywords , author=. American economic review , volume=. 2007 , publisher=

2007

[6] [6]

international Journal of industrial Organization , volume=

Position auctions , author=. international Journal of industrial Organization , volume=. 2007 , publisher=

2007

[7] [7]

The Quarterly Journal of Economics , volume=

Position auctions with consumer search , author=. The Quarterly Journal of Economics , volume=. 2011 , publisher=

2011

[8] [8]

International Workshop on Web and Internet Economics , pages=

Ad auction design and user experience , author=. International Workshop on Web and Internet Economics , pages=. 2007 , organization=

2007