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.

翻译：本文研究单一买家收益最大化的定价查询复杂度问题，其中买家的私有估值服从未知分布。在此设定下，卖家仅能通过公布价格并观察二元购买决策（而非实际估值）来学习最优垄断价格。已有研究在估值分布支撑集为$[0,1]$时，建立了以加性误差$\varepsilon$学习近似最优价格的紧致查询复杂度边界。然而，对于如何学习能达到最优收益至少$(1-\varepsilon)$比例的近似最优价格，目前理解仍然有限。本文在多种设定下研究此类乘性误差保证下的单一买家收益最大化问题的定价查询复杂度。需要指出的是，当定价查询是获取买家分布信息的唯一来源时，任何算法都无法实现非平凡近似，因为仅通过定价查询无法学习分布的尺度信息。受这一基本不可能性的启发，我们考虑两种提供"尺度提示"的自然且动机明确的模型：(i) 单样本提示模型，算法在进行定价查询前观测到单个已实现估值；(ii) 值域提示模型，已知估值支撑集位于$[1, H]$区间内。针对每种提示类型，我们为包括单调风险率分布、正则分布和一般分布在内的多类分布，建立了精确至多对数因子的定价查询复杂度保证。