Robust Scale-Free Auctions

2026-08-03

AI Summary

This study addresses the design of prior-independent, dominant-strategy incentive-compatible auctions in i.i.d. settings where bidders’ value distributions exhibit scale-invariant shapes but have unknown functional forms and scales. By modeling scale-invariant mechanisms and employing a minimax optimization framework, the paper establishes that the second-price auction without reserve is optimal under the monotone hazard rate assumption. For regular distributions, it constructs non-standard mechanisms that may allocate to lower bidders and strictly outperform standard mechanisms. The main contributions include precisely characterizing worst-case revenue guarantees for any number of bidders and their exponential decay rate relative to Bayesian optimal revenue; in the two-bidder regular case, it improves the worst-case approximation ratio of standard mechanisms from approximately 0.524413 to 0.524829, demonstrating a strict performance loss induced by the standardness assumption.

Abstract

We study prior-independent auction design when bidder values are independently and identically distributed and the seller knows only a scale-invariant shape restriction on their distribution, but neither the distribution nor the scale of values. We show that the maximin problem over a broad class of dominant-strategy incentive-compatible mechanisms reduces without loss to scale-free mechanisms. For any $n\ge 2$ monotone-hazard-rate bidders, the second-price auction without a reserve is maximin optimal over this class, including randomized mechanisms that may allocate to a lower bidder. We derive its exact guarantee for every $n$ and the sharp exponential rate at which its loss relative to the Bayesian optimum vanishes. Many familiar auctions are standard: they allocate only to a highest bidder, although incentive compatibility does not require this. For two regular bidders, we solve the standard problem exactly: its optimal mechanism mixes the second-price auction with a relative-markup auction and achieves a worst-case ratio of approximately $0.524413$. We construct a nonstandard mechanism that sometimes allocates to the lower bidder and achieves approximately $0.524829$, proving that standardness is strictly costly. The contrast is driven by tail restrictions: monotone hazard rate makes lower-rank allocation unhelpful, whereas regularity permits it to improve worst-case revenue.

Problem

• prior-independent auctions
• scale-free mechanisms
• maximin optimization
• dominant-strategy incentive compatibility
• worst-case revenue

Innovation

• prior-independent auction
• scale-free mechanism
• maximin optimality
• nonstandard allocation
• monotone hazard rate