Auctions with Contract Design

2026-07-15

AI Summary

This study addresses the moral hazard problem in auctions wherein bidders lack ex ante incentives to improve quality due to sunk costs. To mitigate this issue, the authors propose embedding a post-performance quality-based reward mechanism into auction contracts. By developing a unified framework that integrates linear incentive contracts with various auction formats, they employ Bayesian Nash equilibrium analysis, asymptotic methods, and the revenue equivalence theorem to demonstrate that, as the number of bidders grows large, the optimal reward coefficient converges to the auctioneer’s marginal benefit from quality. Under symmetric equilibrium, this mechanism substantially enhances the auctioneer’s expected revenue and achieves full pass-through of quality value, exhibiting both theoretical robustness and practical efficacy in incentivizing quality improvement.

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.

Problem

• auctions
• contract design
• moral hazard
• quality investment
• revenue maximization

Innovation

• auction with contracts
• moral hazard
• quality incentives
• revenue equivalence
• linear contracts