Pricing Query Complexity of Multiplicative Revenue Approximation

TL;DR

This paper investigates the number of pricing queries needed to approximate the maximum revenue within a multiplicative factor for a single buyer, under different informational models and distribution classes.

Contribution

It introduces and analyzes two models with scale hints, providing tight bounds on query complexity for revenue approximation across various distribution classes.

Findings

Tight bounds established for MHR, regular, and general distributions.

Scale hints enable non-trivial revenue approximation with limited queries.

Analysis covers multiple distribution classes and informational settings.

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.