The Query Complexity of Uniform Pricing
Houshuang Chen, Yaonan Jin, Pinyan Lu, Chihao Zhang
TL;DR
The paper investigates the query complexity and regret of Uniform Pricing in a multi-buyer setting, showing that distributional structure (regularity or MHR) does not yield learning advantages when at least two distributions are present. It develops a unified information-theoretic framework, leveraging base-plus-hard-instance constructions and a pricing-to-identification reduction to prove near-tight lower bounds. Specifically, for two regular or three MHR distributions, both pricing query complexity and minimax regret match the general-case bounds, at $\widetilde{\Theta}(\varepsilon^{-3})$ and $\widetilde{\Theta}(T^{2/3})$ respectively; for two MHR distributions, the bounds are $\Omega(\varepsilon^{-5/2})$ and $\Omega(T^{3/5})$, with small gaps to the general bounds. These results reveal a dichotomy between single-agent and multi-agent environments in mechanism design, indicating that competition among buyers erases the benefits of distributional structure on learning efficiency and suggesting the need for robust pricing strategies in multi-distribution markets.
Abstract
Real-world pricing mechanisms are typically optimized using training data, a setting corresponding to the $\textit{pricing query complexity}$ problem in Mechanism Design. The previous work (LSTW23, SODA) studies the $\textit{single-distribution}$ case, with tight bounds of $\widetildeΘ(\varepsilon^{-3})$ for a $\textit{general}$ distribution and $\widetildeΘ(\varepsilon^{-2})$ for either a $\textit{regular}$ or $\textit{monotone-hazard-rate (MHR)}$ distribution. This can be directly interpreted as ''the query complexity of the $\textsf{Uniform Pricing}$ mechanism, in the $\textit{single-distribution}$ case''. Yet in the $\textit{multi-distribution}$ case, can the regularity and MHR conditions still lead to improvements over the tight bound $\widetildeΘ(\varepsilon^{-3})$ for general distributions? We answer this question in the negative, by establishing a (near-)matching lower bound $Ω(\varepsilon^{-3})$ for either $\textit{two regular distributions}$ or $\textit{three MHR distributions}$. We also address the $\textit{regret minimization}$ problem and, in comparison with the folklore upper bound $\widetilde{O}(T^{2 / 3})$ for general distributions (see, e.g., SW24, EC), establish a (near-)matching lower bound $Ω(T^{2 / 3})$ for either $\textit{two regular distributions}$ or $\textit{three MHR distributions}$, via a black-box reduction. Again, this is in stark contrast to the tight bound $\widetildeΘ(T^{1 / 2})$ for a single regular or MHR distribution.