[Paper Review] Pricing with a Hidden Sample

Zhihao Gavin Tang, Yixin Tao+1 more|arXiv (Cornell University)|Feb 20, 2026

Auction Theory and Applications0 citations

TL;DR

The paper introduces hidden pricing mechanisms that use a single hidden sample to implement concave pricing policies, bridging statistic-based and sample-based robust pricing.

ABSTRACT

We study prior-independent pricing for selling a single item to a single buyer when the seller observes only a single sample from the valuation distribution, while the buyer knows the distribution. Classical robust pricing approaches either rely on distributional statistics, which typically require many samples to estimate, or directly use revealed samples to determine prices and allocations. We show that these two regimes can be bridged by leveraging the buyer's informational advantage: pricing policies that conventionally require the seller to know statistics such as the mean, $L^η$-norm, or superquantile can, in our framework, be implemented using only a single hidden sample. We introduce hidden pricing mechanisms, in which the seller commits ex ante to a pricing rule based on a single sample that is revealed only after the buyer's participation decision. We show that every concave pricing policy can be implemented in this way. To evaluate performance guarantees, we develop a general reduction for analyzing monotone pricing policies over $α$-regular distributions, enabling a tractable characterization of worst-case instances. Using this reduction, we characterize the optimal monotone hidden pricing mechanisms and compute their approximation ratios; in particular, we obtain an approximation ratio of approximately $0.79$ for monotone hazard rate (MHR) distributions. We further establish impossibility results for general concave pricing policies and for all prior-independent mechanisms. Finally, we show that our framework also applies to statistic-based robust pricing, thereby unifying sample-based and statistic-based approaches.

Motivation & Objective

• Motivate prior-independent pricing with limited seller information by leveraging the buyer’s distribution knowledge.
• Introduce hidden pricing mechanisms where a single sample is used to implement target distribution statistics.
• Develop a tractable reduction to analyze monotone pricing policies under alpha-regular distributions.
• Characterize the optimal monotone hidden pricing mechanism and quantify approximation guarantees.
• Show connections between hidden pricing and statistic-based robust pricing, unifying the two paradigms.

Proposed method

• Define hidden pricing mechanisms with a pricing rule h(s, F') and a buyer who reports a distribution F' after observing a sample s from F.
• Show that a pricing rule is proper if it corresponds to a concave functional p on distributions, enabling implementation via hidden pricing Rules.
• Prove a reduction showing that for any deterministic monotone pricing policy, nature’s worst-case distribution lies in a simple two-parameter (or one-parameter for certain statistics) family.
• Compute the optimal monotone hidden pricing mechanism (theorem-based) and obtain quantitative approximation ratios for alpha-regular distributions, especially MHR.
• Provide impossibility results showing limits for concave pricing policies and all prior-independent mechanisms, and discuss equivalence with statistic-based robust pricing.

Experimental results

Research questions

• RQ1 Can a seller with only one observed sample achieve the same performance as classical statistic-based mechanisms that know distribution statistics?
• RQ2 How can hidden pricing rules implement common statistics (mean, L^eta norm, CVaR) using a single sample?
• RQ3 What is the best possible approximation ratio achievable by monotone hidden pricing mechanisms under alpha-regular (notably MHR) distributions?
• RQ4 How do hidden pricing mechanisms relate to and unify statistic-based robust pricing approaches?

Key findings

• A single hidden sample suffices to match classical guarantees for mean pricing, L^eta-norm pricing, and superquantile pricing.
• For MHR distributions, the optimal monotone hidden pricing mechanism achieves an approximation ratio of about 0.79.
• There exist lower and upper bounds showing limits: no concave pricing rule can exceed 0.801 under MHR, and general prior-independent mechanisms have an upper bound of 0.838.
• A toy uniform-distribution example yields an approximation ratio of 0.875, optimal among prior-independent mechanisms in that setting.
• The framework also extends to statistic-based robust pricing with monotone statistics beyond concave functionals, unifying sample- and statistic-based approaches.