Pricing Query Complexity of Multiplicative Revenue Approximation

2026-02-11

AI Summary

This work addresses the problem of efficiently learning an approximately optimal monopoly price—achieving at least a $(1-\varepsilon)$-fraction of the optimal revenue—when the seller interacts with a single buyer whose valuation distribution is unknown and can only observe binary purchase feedback through pricing queries. The paper establishes the first systematic theory of pricing query complexity in this setting and introduces a novel “scale hint” mechanism to overcome the fundamental limitation that pure pricing queries cannot identify the scale of the underlying distribution. Tight (up to polylogarithmic factors) upper and lower bounds on query complexity are provided for MHR, regular, and general distributions under two models: one with a single-sample scale hint and another where the valuation is known to lie in $[1, H]$.

Technology Category

• Search and Optimization: Learning to Search
• Multiagent Systems: Mechanism Design
• Game Theory and Economic Paradigms: Mechanism Design

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 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.

Problem

• pricing query complexity
• revenue maximization
• multiplicative approximation
• monopoly pricing
• unknown valuation distribution

Innovation

• pricing query complexity
• multiplicative revenue approximation
• scale hint
• monotone hazard rate
• revenue maximization