Pricing with a Hidden Sample

TL;DR

This work studies robust, prior-independent pricing for a single-item sale when the seller observes one hidden sample and the buyer knows the valuation distribution. It introduces hidden pricing mechanisms that map a single sample (and buyer-reported statistics) to payments, enabling the seller to implement concave pricing policies previously requiring distributional statistics. A tractable reduction shows worst-case distributions for monotone policies lie in simple two-parameter families, enabling precise characterization of optimal hidden pricing and approximate ratios, with a tight ≈0.79 ratio for MHR distributions and near-optimality relative to broader prior-independent methods. The framework further extends to statistic-based robust pricing, demonstrating equivalences that unify sample-based and statistic-based approaches and computing concrete prices under mean, L^η-norm, and CVaR information, thereby informing practical pricing under extreme data scarcity.

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.

Figures (4)

• Figure 1: Illustration of Lemma \ref{['lem:small']}, where $R(q)=q\cdot \mathbb{F}^{-1}(1-q)$ is the revenue as a function of the purchase probability $q\in [0,1]$. We construct a distribution $\widecheck{\mathbb{F}}_{\lambda,a}\preceq_{1} \mathbb{F}$ that coincides with $\mathbb{F}$ at the optimal price $p^*(\mathbb{F})$. By the monotonicity of the pricing policy $p$, the resulting revenue under $\widecheck{\mathbb{F}}_{\lambda,a}$ is weakly smaller than that under $\mathbb{F}$, yielding a strictly smaller approximation ratio.
• Figure 2: Illustration of Lemma \ref{['lem:large']}. We first construct a distribution $\widehat{\mathbb{F}}_{\lambda,b}\succeq_{1} \mathbb{F}$ that agrees with $\mathbb{F}$ at the price $p(\mathbb{F})$. By the monotonicity of the pricing policy $p$, the resulting revenue under $\widehat{\mathbb{F}}_{\lambda,b}$ is weakly smaller than that under $\mathbb{F}$, yielding a strictly smaller approximation ratio. However, $\widehat{\mathbb{F}}_{\lambda,b}$ may not lie in $\widehat{\mathcal{F}}$, since its optimal sale probability is not necessarily equal to $1$. This issue arises only when $\alpha > 0$. To address it, we further construct a distribution $\widehat{\mathbb{F}}_{\lambda^\dagger, b^\dagger}$ by truncating values below the optimal monopoly price and rescaling the remaining mass.
• Figure 3: Approximation ratio upper bound $\Gamma_\alpha$
• Figure 4: Performance Ratio and Optimal Discount with $L^\eta$-norm and Superquantile Information

Theorems & Definitions (66)

• Theorem 2.1
• proof
• Definition 2.1: Proper Pricing Rules
• Definition 2.2: Hidden Pricing Mechanisms
• Theorem 2.2: savage1971elicitationgneiting2007strictly
• Corollary 2.1
• Definition 2.3: Monotone Pricing Policies
• Definition 2.4
• Definition 2.5
• Lemma 2.1: schweizer2015quantitative