Pricing with a Hidden Sample

2026-02-20

AI Summary

This work proposes a “hidden-sample pricing mechanism” in a setting where the seller observes only a single sample drawn from the buyer’s valuation distribution, while the buyer knows the full distribution. The seller commits ex ante to a pricing rule based on a sample revealed ex post, thereby strategically leveraging the buyer’s informational advantage. This mechanism unifies, for the first time, sample-driven and statistic-driven approaches within a robust pricing framework, proving that any concave pricing strategy can be implemented using just one hidden sample. The authors develop a worst-case analysis method for α-regular distributions and establish a 0.79-approximation guarantee under monotone hazard rate (MHR) distributions. Furthermore, they demonstrate fundamental limitations of general concave strategies and prior-free mechanisms in this context.

Technology Category

• Game Theory and Economic Paradigms: Mechanism Design
• Multiagent Systems: Mechanism Design
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Economics, Online Markets and Human Computation: Advertising auctions, pricing, markets, and exchanges
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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, $L^\eta$-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 $\alpha$-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.

Problem

• prior-independent pricing
• single sample
• information asymmetry
• valuation distribution
• hidden sample

Innovation

• hidden sample pricing
• prior-independent mechanism
• concave pricing policy
• monotone hazard rate
• robust pricing