Pricing Query Complexity of Multiplicative Revenue Approximation
Wei Tang, Yifan Wang, Mengxiao Zhang
TL;DR
The paper investigates pricing query complexity for single-buyer revenue maximization with unknown value distributions under binary feedback, focusing on multiplicative guarantees. It introduces two scale-hint models—the value-range hint and the one-sample hint—and develops a unified algorithm for regular and MHR distributions that exploits half-concavity to achieve near-optimal revenue with a polylogarithmic factor overhead in queries. The work provides nearly tight upper and lower bounds across Regular, MHR, and General distribution classes under these hints: Regular value-range $\widetilde{\mathcal{O}}(H/\varepsilon^{2})$, MHR value-range $\widetilde{\mathcal{O}}(1/\varepsilon^{2})$, Regular one-sample $\widetilde{\mathcal{O}}(\varepsilon^{-3})$, MHR one-sample $\widetilde{\mathcal{O}}(\varepsilon^{-2})$, and General value-range $\widetilde{\mathcal{O}}(H/\varepsilon^{3})$ with matching lower bounds. Collectively, these results delineate how scale-information shapes pricing-query complexity and bridge gaps with existing sample-based analyses, while highlighting intrinsic hardness in the absence of scale hints.
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.