[Paper Review] Pricing Query Complexity of Multiplicative Revenue Approximation

Wei Tang, Yifan Wang+1 more|arXiv (Cornell University)|Feb 11, 2026

Auction Theory and Applications0 citations

TL;DR

The paper studies the pricing query complexity to achieve multiplicative revenue approximations for a single buyer with unknown valuation distribution, under scale hints (one-sample or value-range). It provides near-tight bounds for regular, MHR, and general distributions.

ABSTRACT

We study the pricing query complexity of revenue maximization for a single buyer whose private valuation is drawn from an unknown distribution. In this setting, the seller must learn the optimal monopoly price by posting prices and observing only binary purchase decisions, rather than the realized valuations. Prior work has established tight query complexity bounds for learning a near-optimal price with additive error $\varepsilon$ when the valuation distribution is supported on $[0,1]$. However, our understanding of how to learn a near-optimal price that achieves at least a $(1-\varepsilon)$ fraction of the optimal revenue remains limited. In this paper, we study the pricing query complexity of the single-buyer revenue maximization problem under such multiplicative error guarantees in several settings. Observe that when pricing queries are the only source of information about the buyer's distribution, no algorithm can achieve a non-trivial approximation, since the scale of the distribution cannot be learned from pricing queries alone. Motivated by this fundamental impossibility, we consider two natural and well-motivated models that provide "scale hints": (i) a one-sample hint, in which the algorithm observes a single realized valuation before making pricing queries; and (ii) a value-range hint, in which the valuation support is known to lie within $[1, H]$. For each type of hint, we establish pricing query complexity guarantees that are tight up to polylogarithmic factors for several classes of distributions, including monotone hazard rate (MHR) distributions, regular distributions, and general distributions.

Motivation & Objective

• Motivate and formalize the pricing query complexity problem under multiplicative revenue guarantees.
• Introduce scale-hint models (one-sample and value-range) to overcome impossibility results without scale information.
• Derive (nearly) tight upper and lower bounds for regular, MHR, and general distributions in both hint settings.
• Develop a unified learning algorithm applicable to regular and MHR distributions.
• Analyze how a single sample or known value range affects query complexity and achievable revenue guarantees.

Proposed method

• Model the seller-buyer interaction as posting prices and observing binary purchase decisions from i.i.d. valuations drawn from an unknown F.
• Define revenue RevF(p)=p(1−F(p)) and optimize via multiplicative approximation with respect to Rev(p^opt).
• Propose Algorithm 1 a unified learning procedure leveraging half-concavity/unimodality of the revenue curve for regular (and MHR) distributions.
• Use a ternary-search-like approach on a geometric price grid with a winning-probability lower bound γ to prune low-sale regions.
• Provide a general query-complexity bound O(γ^{−1} ε^{−2} log(1/δ) log^4(r/(εℓ))) and refine per distribution class and hint model.
• Introduce a reduction that locates a search interval from a single sample for regular distributions to guarantee (1−ε)-approximation (Lemma 4.3).
• Show that for general distributions the one-sample hint is insufficient, hence focus on the value-range hint with tight bounds.

Experimental results

Research questions

• RQ1 What is the pricing query complexity to achieve a (1−ε) multiplicative revenue approximation under unknown valuation distributions?
• RQ2 How do scale hints (one-sample vs value-range) affect the query complexity for regular, monotone hazard rate (MHR), and general distributions?
• RQ3 Can a unified algorithm handle both regular and MHR distributions efficiently under these hints?
• RQ4 How does the one-sample hint compare to the value-range hint in terms of achievable approximation and query complexity, especially for regular distributions?
• RQ5 What are the fundamental impossibility results without scale information in the multiplicative revenue setting?

Key findings

• Under a value-range hint, regular distributions admit a near-tight bound of Õ(ε^{−2} H) queries and Ω(ε^{−2} H) lower bound.
• Under a one-sample hint, regular distributions require Õ(ε^{−3}) queries and Ω(ε^{−3}) lower bound, matching the sample-access baseline up to polylog factors.
• For MHR distributions, both value-range and one-sample hints yield Õ(ε^{−2}) queries with corresponding Ω(ε^{−2}) lower bounds.
• For general distributions, the value-range hint gives Õ(ε^{−3} H) queries with Ω(ε^{−3} H) lower bound, while the one-sample hint is insufficient for resolving scale.
• The paper provides a unified algorithm (Algorithm 1) that achieves near-optimal revenue with a polylogarithmic factor overhead and a γ-winnability pruning step, applicable to both regular and MHR classes.
• A key technical contribution is a reduction turning one-sample hints into a search-interval localization that preserves multiplicative revenue guarantees (Lemma 4.3).