Pilot-corrected policy achieves minimax-optimal dynamic pricing for multimodal revenue

A new paper from Xueping Gong, Zhuoluo Zhang, Zhaowei Miao, and Jiheng Zhang tackles a notoriously hard problem in revenue management: contextual dynamic pricing with arbitrary covariate sequences and bounded, possibly nonbinary purchase quantities. The authors assume demand follows a semiparametric surplus-index model—an unknown linear valuation parameter plus an unknown Hölder-smooth response function. Critically, they do not assume concavity or strong unimodality of revenue, meaning the revenue function can have multiple peaks and the optimal price need not be unique. Most existing algorithms collapse under such generality, but the team's policy handles it directly.

The proposed solution is a "pilot-corrected layered decision-partitioning policy" that combines four techniques: directional pilot estimation to estimate the valuation parameter, local polynomial learning for the response function, predictable data assignment for stable sampling, and global action elimination to prune suboptimal prices. Pilot correction removes the first-order bias caused by valuation-parameter error, while permanent labels ensure concentration even under adaptive sampling. The policy achieves the minimax smoothness-dependent horizon rate up to logarithmic factors—the best possible worst-case performance—and the authors prove a matching lower bound for a constant-context, binary-demand subclass. In practical terms, this means the algorithm is provably optimal across a wide range of demand behaviors, making it a significant theoretical foundation for next-generation dynamic pricing systems.