# Dynamic Pricing with Demand Covariates

arXiv: https://arxiv.org/abs/1604.07463 · HTML: https://mlsanity.com/paper/1604.07463/

## Key facts

- Cited by 31 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Apr 2016.

## Built on by (most cited first, 30 of 31)

- [Post-Contextual-Bandit Inference](https://mlsanity.com/paper/2106.00418/) (2021)
- [Policy Optimization Using Semi-parametric Models for Dynamic Pricing](https://mlsanity.com/paper/2109.06368/) (2021)
- [Logarithmic Regret in Feature-based Dynamic Pricing](https://mlsanity.com/paper/2102.10221/) (2021)
- [Distribution-free Contextual Dynamic Pricing](https://mlsanity.com/paper/2109.07340/) (2021)
- [Dynamic pricing and assortment under a contextual MNL demand](https://mlsanity.com/paper/2110.10018/) (2021)
- [Differential Privacy in Personalized Pricing with Nonparametric Demand Models](https://mlsanity.com/paper/2109.04615/) (2021)
- [Risk Minimization from Adaptively Collected Data: Guarantees for Supervised and Policy Learning](https://mlsanity.com/paper/2106.01723/) (2021)
- [On Dynamic Pricing with Covariates](https://mlsanity.com/paper/2112.13254/) (2021)
- [Online Statistical Inference for Contextual Bandits via Stochastic Gradient Descent](https://mlsanity.com/paper/2212.14883/) (2022)
- [Doubly High-Dimensional Contextual Bandits: An Interpretable Model for Joint Assortment-Pricing](https://mlsanity.com/paper/2309.08634/) (2023)
- [Stateful Offline Contextual Policy Evaluation and Learning](https://mlsanity.com/paper/2110.10081/) (2021)
- [Loss Functions for Discrete Contextual Pricing with Observational Data](https://mlsanity.com/paper/2111.09933/) (2021)
- [Local Metric Learning for Off-Policy Evaluation in Contextual Bandits with Continuous Actions](https://mlsanity.com/paper/2210.13373/) (2022)
- [Pricing with Contextual Elasticity and Heteroscedastic Valuation](https://mlsanity.com/paper/2312.15999/) (2023)
- [Understanding the Training and Generalization of Pretrained Transformer for Sequential Decision Making](https://mlsanity.com/paper/2405.14219/) (2024)
- [Regulatory Instruments for Fair Personalized Pricing](https://mlsanity.com/paper/2202.04245/) (2022)
- [Online Pricing and Allocation with Demand Learning and Fulfillment Cost](https://mlsanity.com/paper/2501.18049/) (2025)
- [Contextual Online Pricing with (Biased) Offline Data](https://mlsanity.com/paper/2507.02762/) (2025)
- [Multi-Task Dynamic Pricing in Credit Market with Contextual Information](https://mlsanity.com/paper/2410.14839/) (2024)
- [Convex Surrogate Loss Functions for Contextual Pricing with Transaction Data](https://mlsanity.com/paper/2202.10944/) (2022)
- [A Reinforcement Learning Approach in Multi-Phase Second-Price Auction Design](https://mlsanity.com/paper/2210.10278/) (2022)
- [Smoothness-Adaptive Dynamic Pricing with Nonparametric Demand Learning](https://mlsanity.com/paper/2310.07558/) (2023)
- [Demistifying Inference after Adaptive Experiments](https://mlsanity.com/paper/2405.01281/) (2024)
- [Localized exploration in contextual dynamic pricing achieves dimension-free regret](https://mlsanity.com/paper/2412.19252/) (2024)
- [Active Learning for Contextual Search with Binary Feedbacks](https://mlsanity.com/paper/2110.01072/) (2021)
- [Machine Learning based Framework for Robust Price-Sensitivity Estimation with Application to Airline Pricing](https://mlsanity.com/paper/2205.01875/) (2022)
- [Learning to Sell a Focal-ancillary Combination](https://mlsanity.com/paper/2207.11545/) (2022)
- [Phase Transitions in Learning and Earning under Price Protection Guarantee](https://mlsanity.com/paper/2211.01798/) (2022)
- [Bayesian Optimization for Dynamic Pricing and Learning](https://mlsanity.com/paper/2510.12447/) (2025)
- [Learning Fair Demand Models](https://mlsanity.com/paper/2606.06830/) (2026)

## Abstract (arXiv)

We consider a firm that sells products over T periods without knowing the demand function. The firm sequentially sets prices to earn revenue and to learn the underlying demand function simultaneously. A natural heuristic for this problem, commonly used in practice, is greedy iterative least squares (GILS). At each time period, GILS estimates the demand as a linear function of the price by applying least squares to the set of prior prices and realized demands. Then a price that maximizes the revenue, given the estimated demand function, is used for the next time period. The performance is measured by the regret, which is the expected revenue loss from the optimal (oracle) pricing policy when the demand function is known. Recently, den Boer and Zwart (2014) and Keskin and Zeevi (2014) demonstrated that GILS is sub-optimal. They introduced algorithms which integrate forced price dispersion with GILS and achieve asymptotically optimal performance. In this paper, we consider this dynamic pricing problem in a data-rich environment. In particular, we assume that the firm knows the expected demand under a particular price from historical data, and in each period, before setting the price, the firm has access to extra information (demand covariates) which may be predictive of the demand. We prove that in this setting GILS achieves asymptotically optimal regret of order log(T). We also show the following surprising result: in the original dynamic pricing problem of den Boer and Zwart (2014) and Keskin and Zeevi (2014), inclusion of any set of covariates in GILS as potential demand covariates (even though they could carry no information) would make GILS asymptotically optimal. We validate our results via extensive numerical simulations on synthetic and real data sets.