Improved pricing method adapts to unknown demand and inventory changes
Optimal Nonparametric Dynamic Pricing with Censored Demand and Adversarial Inventory
Machine Learning
Summary
This work tackles the challenge of setting product prices when the sellers don’t know exactly how customers will respond and have changing inventory limits. The authors extend previous methods by handling more complex and realistic demand patterns without assuming simple linear relationships. They propose two new algorithms to learn good pricing strategies over time, with the second one making smarter use of past sales data to improve results. Their approach achieves the best possible theoretical guarantees on pricing performance and works well across many scenarios in experiments.
What this means in practice
- •For ecommerce pricing teams: Set prices dynamically when inventory limits change unpredictably and customer demand patterns are complex and unknown.
- •For supply chain planners: Adjust pricing strategies online using demand and inventory data to improve revenue without assuming simple demand models.
Authors
Mengxiao Zhang, Yingfei Wang, Haipeng Luo
Abstract
We study online dynamic pricing with censored demand, where an arbitrary inventory level is revealed before pricing and may adapt to past observations, while demand follows an unknown, price-dependent distribution that is stationary over time. For a horizon of $T$ rounds, Xu et al. [2026] achieved $\widetilde{\mathcal{O}}(\sqrt{T})$ regret under restrictive structural assumptions including linear demand, price-independent additive noise, and conditions relating inventory levels to the noise support. Our first contribution is to extend this framework to a substantially more general and statistically harder nonparametric setting, requiring only the natural assumption that expected sales are nonincreasing in price and allowing nonlinear demand curves and price-dependent noise. For this model, we first propose a simple baseline, Double-Grid-UCB, which discretizes both price and inventory and achieves $\widetilde{\mathcal{O}}(T^{3/4})$ expected regret using separate revenue estimates for each price-inventory grid pair. Then, we develop Threshold-UCB, which improves the expected regret to $\widetilde{\mathcal{O}}(T^{2/3})$. Unlike Double-Grid-UCB, Threshold-UCB reuses sales observations across inventory levels through shared estimates of demand-tail probabilities, allowing the same data to support revenue upper bounds for multiple inventories rather than a single inventory bin. We also complement this upper bound with an $Ω(T^{2/3})$ lower bound via a reduction from stochastic posted pricing, establishing its minimax optimality. Finally, extensive experiments across inventory processes, demand functions, and noise models demonstrate consistently superior performance of Threshold-UCB over benchmark algorithms.