Designing Auctions When Sellers Know Prices but Not Bidders

Designing for what the data leave unknown

In Robust Mechanism Design with Anonymous Information, Zhihao Gavin Tang and Shixin Wang study a seller who knows the distribution of one ranked bidder value, but not each bidder’s individual value distribution. The article concerns version three of their preprint, revised on November 5, 2026.

An “order statistic” is simply a value’s position after sorting: the highest, second-highest, or lowest, for example. Knowing its distribution means knowing how often that ranked value falls below different thresholds across auctions—not merely observing one transaction.

The baseline model sells one item to bidders with independent, nonnegative private values. Their distributions may differ. The seller considers every independent combination consistent with the known statistic, then chooses an auction to maximize its lowest expected revenue across those possibilities. Eligible mechanisms make truthful reporting a dominant strategy and satisfy individual rationality after values are realized.

This is a worst-case revenue objective. It does not promise to match what a seller could earn with complete knowledge of every bidder’s distribution.

Different statistics favor different mechanisms

When the highest-value distribution is known, the supremum of revenues from take-it-or-leave-it prices equals optimal worst-case revenue; a price attaining it need not exist. Its expected revenue is the price multiplied by the probability that at least one bidder will pay it. Both quantities can be determined without identifying who has the highest value.

The reason no mechanism can guarantee more is revealing: the available information is consistent with just one active bidder and everyone else always valuing the item at zero. That possibility limits what competition can reliably contribute. Unlike most results in the paper, this posted-pricing conclusion also extends to correlated values.

Knowing the lowest-value distribution produces a different answer. For a fixed number of bidders, it identifies a unique hypothetical model in which bidders’ values are independent and identically distributed, or i.i.d. The authors prove that the Myerson revenue-optimal auction designed for that model is robustly optimal, using their specified implementation with uniformly random priority for ties.

The hypothetical i.i.d. model is not a discovery that actual bidders are identical. It is a benchmark that delivers the auction’s lowest revenue among all consistent independent models.

A reserve can protect revenue

The second-highest value is especially relevant because it determines the payment in a truthful second-price auction without a reserve. With a reserve, the highest bidder wins only if their value reaches that minimum price, and pays the larger of the reserve and the second-highest bid.

The paper proves that, for every fixed reserve, this auction’s worst-case revenue is attained by the consistent i.i.d. model. Sellers can therefore optimize its guaranteed revenue by calculating performance in that benchmark rather than searching through every possible heterogeneous market.

A stronger conclusion needs an additional condition. If the implied common distribution is “regular above the reserve,” the auction with the corresponding monopoly reserve is optimal among all eligible mechanisms for the worst-case objective. Roughly, this condition prevents the revenue-relevant ranking of values from reversing above that threshold; it is weaker than requiring regularity everywhere. The result also applies when other order statistics below the highest are known.

Without regularity, knowing the second-highest-value distribution still gives a universal guarantee. The best reserve-based second-price revenue is at least 2 − √2, approximately 58.6%, of optimal worst-case revenue. This is a lower bound, not a measured improvement or a claim that the bound is tight. When optimal worst-case revenue is finite, a reserve can approach the guarantee with arbitrarily small additive loss; bounded support ensures attainment.

Why the benchmark is not always enough

The authors also construct counterexamples to simply running the full Myerson auction optimized for the consistent i.i.d. model. When its procedure pools different values into equal revenue-relevant scores, allocation and payments can depend on lower-ranked bidders through ties. Its benchmark revenue can then overstate its robust guarantee. Particular implementations matter: the paper does not show that every Myerson implementation fails.

The broader positive result likewise has a precise boundary. The i.i.d. benchmark is worst-case for mechanisms whose total payments are sums of nondecreasing functions of the relevant leading order statistics. Applications include multi-unit and laddered advertising-slot auctions. This permits robust reserve optimization within those formats, not unconditional optimality among all mechanisms.

What remains unresolved

These are mathematical guarantees, not evidence from a live-market deployment. The model assumes the statistic’s full distribution is known; it does not establish how estimation error from finite transaction records changes the guarantees. Independence remains central outside the highest-value pricing result.

The authors connect anonymous information with nondiscrimination because random relabeling cannot reduce worst-case revenue. That symmetry argument is not a general privacy guarantee or proof of broader social fairness. Characterizing optimal auctions for unrestricted intermediate-statistic settings, and finding the tight approximation ratio, remain open.