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arxiv: 2604.17923 · v1 · submitted 2026-04-20 · 💰 econ.TH

Recognition: unknown

Optimal linear-payment auction design with aftermarket collaboration

Dazhong Wang , Ruqu Wang , Xinyi Xu
Authors on Pith no claims yet

Pith reviewed 2026-05-10 03:50 UTC · model grok-4.3

classification 💰 econ.TH
keywords optimal auction designaftermarket collaborationdouble moral hazardlinear payment schemesvalue sharingvirtual surplusseller revenueascending auctions
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The pith

The optimal auction allocates the asset to the bidder with highest virtual surplus and uses linear payments with proportional value sharing, delivering strictly higher seller revenue when the seller takes the pivotal post-auction role.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper shows that when the winning bidder and seller must exert non-contractible efforts after the auction to create value, the designer faces both adverse selection from private types and double moral hazard in the aftermarket. A reader would care because ignoring the collaboration stage leads to misallocation and lost revenue, while the mechanisms here restore full type revelation and align effort incentives through simple linear contracts. The core result is that seller-pivotal collaboration always outperforms winner-pivotal collaboration in seller revenue for any type distribution, with the seller claiming a larger value share and the pivotal party working harder.

Core claim

The optimal direct mechanism allocates the asset to the bidder with the highest virtual surplus, employs a deterministic value-sharing rule, and achieves full type revelation through the signal realization rule. Under seller-pivotal collaboration the seller extracts the entire value from low-type winners and always secures a strictly higher share than under winner-pivotal rules; the pivotal party exerts more effort than the supporting party in either structure; and seller-pivotal collaboration yields strictly higher seller revenue than winner-pivotal collaboration for any type distribution. These mechanisms can be implemented through ascending auctions with endogenously determined linear pro

What carries the argument

Linear payment schemes that combine cash transfers with proportional value sharing between seller and winner, which simultaneously screen types and induce efficient post-auction efforts under either winner-pivotal or seller-pivotal role assignments.

If this is right

  • The seller secures a strictly higher value share under seller-pivotal collaboration, extracting the full value from sufficiently low-type winners.
  • The pivotal party always exerts higher post-auction effort than the supporting party, and each party works harder when pivotal than when supporting.
  • Seller-pivotal collaboration yields strictly higher seller revenue than winner-pivotal collaboration for any bidder type distribution.
  • The optimal mechanisms can be implemented by ascending auctions with endogenously chosen linear contracts.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Designers facing ongoing partnerships after sale could assign the lead collaboration role to the seller to raise revenue without changing the allocation rule.
  • The same linear-contract approach may extend to other aftermarket settings such as technology licensing where effort remains non-contractible.
  • If value creation were nonlinear in efforts, the deterministic sharing rule might no longer be optimal and revenue rankings could reverse.

Load-bearing premise

Post-auction efforts are non-contractible and the value generated is linear in the two parties' efforts.

What would settle it

A type distribution in which seller-pivotal collaboration produces weakly lower seller revenue than winner-pivotal collaboration would falsify the revenue comparison.

Figures

Figures reproduced from arXiv: 2604.17923 by Dazhong Wang, Ruqu Wang, Xinyi Xu.

Figure 1
Figure 1. Figure 1: Linear-payment auction with aftermarket collaboration [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Optimal value-sharing comparison across different distributions [PITH_FULL_IMAGE:figures/full_fig_p024_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Effort combinations under different collaboration structures [PITH_FULL_IMAGE:figures/full_fig_p025_3.png] view at source ↗
read the original abstract

This paper studies optimal auction design when valuations depend endogenously on post-auction collaboration between the seller and the winning bidder. Both parties exert non-contractible efforts after the auction, generating a double moral hazard problem alongside adverse selection. We analyze two role structures -- winner-pivotal and seller-pivotal collaboration -- and characterize optimal direct mechanisms using linear payment schemes that combine cash transfers with proportional value sharing. The optimal mechanism allocates the asset to the bidder with the highest virtual surplus, employs a deterministic value-sharing rule, and achieves full type revelation through the signal realization rule. Comparing the two scenarios yields three main findings. First, regarding value sharing, the seller secures a strictly higher share under seller-pivotal collaboration: for sufficiently low-type winners, the seller extracts the entire value, whereas under winner-pivotal collaboration every winner must retain a positive share to sustain his critical effort. Second, regarding effort exertion, the pivotal party always exerts higher post-auction effort than the supporting party, and each party exerts greater effort when pivotal than when providing support. Third, seller-pivotal collaboration yields strictly higher seller revenue than winner-pivotal collaboration for any type distribution. Finally, these optimal mechanisms can be implemented through ascending auctions with endogenously determined linear contracts.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper studies optimal auction design when the winner's valuation is endogenous to post-auction collaboration with the seller, creating a double moral-hazard problem in addition to adverse selection. It examines two role structures (winner-pivotal and seller-pivotal collaboration) and characterizes optimal direct mechanisms restricted to linear payment schemes that combine cash transfers with proportional value sharing. The optimal mechanism allocates the asset to the bidder with the highest virtual surplus, employs a deterministic value-sharing rule, and induces full type revelation via the signal realization rule. Comparative analysis yields three findings: the seller obtains a strictly higher share (extracting the entire value for low-type winners) under seller-pivotal collaboration; the pivotal party always exerts higher effort than the supporting party, with each exerting more when pivotal; and seller-pivotal collaboration generates strictly higher seller revenue than winner-pivotal collaboration for any type distribution. The mechanisms are implementable via ascending auctions with endogenous linear contracts.

Significance. If the derivations hold, the paper makes a meaningful contribution to mechanism design by extending the virtual-surplus approach to settings with endogenous values and double moral hazard. The revenue ranking between collaboration structures is robust (holding for arbitrary type distributions) and the implementation result via ascending auctions with linear contracts adds practical value. The restriction to linear payments enables a tractable characterization while still delivering clear comparative statics on shares and efforts.

minor comments (3)
  1. The abstract states that the analysis 'characterizes optimal direct mechanisms using linear payment schemes,' but it would be useful to clarify early whether the restriction to linear schemes is without loss of generality or a maintained modeling choice that could be relaxed in future work.
  2. In the model section, the effort cost functions and the precise form of the linear value-production technology should be stated explicitly with notation (e.g., c_w(e_w), c_s(e_s), v(e_w, e_s)) before the mechanism-design analysis begins, to aid readability.
  3. The three comparative findings are clearly listed, yet the implementation result via ascending auctions is presented separately; consider whether it can be folded into the main findings or given its own numbered result for structural clarity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the paper, recognition of its contribution to mechanism design under endogenous values and double moral hazard, and recommendation for minor revision. The referee correctly identifies the key results on virtual-surplus allocation, deterministic value sharing, full type revelation, the revenue ranking between collaboration structures, and implementability via ascending auctions with linear contracts. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper constructs the model from explicit first-principles assumptions on non-contractible post-auction efforts, linear value production in the two efforts, and double moral hazard. It then applies standard virtual-surplus maximization to characterize the optimal linear-payment direct mechanism, derives the deterministic value-sharing rule and signal realization rule by solving the resulting optimization program, and obtains the revenue ranking between seller-pivotal and winner-pivotal regimes by direct comparison of the induced effort levels and seller shares under those same assumptions. No equation reduces to a self-definition, no fitted parameter is relabeled as a prediction, and no load-bearing step relies on a self-citation chain or imported uniqueness theorem. The central claims therefore remain independent of the paper's own outputs.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard mechanism-design primitives plus two modeling choices that are not derived inside the paper: (1) non-contractible efforts whose costs are private and (2) a linear value-production technology that permits proportional sharing rules. No new entities are postulated.

free parameters (2)
  • type distribution
    The virtual-surplus rule and revenue comparison are stated to hold for any type distribution, but the distribution itself is an exogenous input.
  • effort cost functions
    The double moral-hazard problem is solved under unspecified but assumed convex cost functions for each party’s effort.
axioms (2)
  • standard math Quasi-linear utilities and independent private values
    Implicit in the use of virtual surplus and direct mechanisms.
  • domain assumption Efforts are non-contractible and chosen after the auction
    Stated in the abstract as the source of the double moral-hazard problem.

pith-pipeline@v0.9.0 · 5520 in / 1509 out tokens · 54127 ms · 2026-05-10T03:50:39.633332+00:00 · methodology

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