[Paper Review] Nonparametric Pricing Analytics with Customer Covariates

Ningyuan Chen, Guillermo Gallego|arXiv (Cornell University)|May 3, 2018

Advanced Bandit Algorithms Research37 references19 citations

TL;DR

This paper proposes a nonparametric pricing policy that adaptively partitions the customer covariate space into hyper-rectangular bins, clusters customers with similar covariates, and sets personalized prices to maximize revenue without assuming parametric demand models. It achieves a regret of $ O(\log(T)^2 T^{(2+d)/(4+d)}) $, which is near-optimal, as no policy can achieve lower regret than $ O(T^{(2+d)/(4+d)}) $ under mild smoothness and local concavity conditions.

ABSTRACT

Personalized pricing analytics is becoming an essential tool in retailing. Upon observing the personalized information of each arriving customer, the firm needs to set a price accordingly based on the covariates such as income, education background, past purchasing history to extract more revenue. For new entrants of the business, the lack of historical data may severely limit the power and profitability of personalized pricing. We propose a nonparametric pricing policy to simultaneously learn the preference of customers based on the covariates and maximize the expected revenue over a finite horizon. The policy does not depend on any prior assumptions on how the personalized information affects consumers' preferences (such as linear models). It is adaptively splits the covariate space into smaller bins (hyper-rectangles) and clusters customers based on their covariates and preferences, offering similar prices for customers who belong to the same cluster trading off granularity and accuracy. We show that the algorithm achieves a regret of order $O(\log(T)^2 T^{(2+d)/(4+d)})$, where $T$ is the length of the horizon and $d$ is the dimension of the covariate. It improves the current regret in the literature \citep{slivkins2014contextual}, under mild technical conditions in the pricing context (smoothness and local concavity). We also prove that no policy can achieve a regret less than $O(T^{(2+d)/(4+d)})$ for a particular instance and thus demonstrate the near optimality of the proposed policy.

Motivation & Objective

• To address the learning/earning dilemma in personalized dynamic pricing when the firm lacks historical data on customer demand.
• To design a pricing policy that does not rely on parametric assumptions (e.g., linearity) about how covariates affect demand.
• To simultaneously learn customer preferences based on covariates and maximize expected revenue over a finite time horizon.
• To achieve near-optimal regret in the presence of high-dimensional covariates while maintaining adaptivity and accuracy.
• To establish a theoretical lower bound on regret to demonstrate the near-optimality of the proposed policy.

Proposed method

• The policy adaptively splits the d-dimensional covariate space into hyper-rectangular bins using a recursive partitioning strategy based on observed customer responses.
• Customers are clustered by their covariates and assigned similar prices within each bin to balance granularity and estimation accuracy.
• The algorithm uses a UCB-type exploration strategy to balance learning and exploitation, with confidence bounds derived from the observed Bernoulli purchase outcomes.
• The method assumes smoothness and local concavity of the underlying demand function to ensure convergence and regret control.
• A binning scheme is employed where each bin is split into 2^d sub-bins in the worst case, but the algorithm adapts based on observed data to avoid overfitting.
• The regret analysis is conducted using a minimax lower bound construction on a family of functions $ \mathcal{C} $, where functions differ only in a single bin’s behavior.

Experimental results

Research questions

• RQ1 Can a nonparametric pricing policy achieve sublinear regret without assuming a parametric form for the demand function?
• RQ2 How does the dimensionality of the covariate space affect the regret rate in personalized pricing with unknown demand functions?
• RQ3 Is it possible to design a policy that adapts to local structure in the covariate space while maintaining near-optimal regret?
• RQ4 What is the fundamental limit (lower bound) on regret for any policy in this personalized pricing setting?
• RQ5 Can the regret be significantly reduced when the true demand depends only on a sparse subset of the covariates?

Key findings

• The proposed policy achieves a regret of $ O(\log(T)^2 T^{(2+d)/(4+d)}) $, which is sublinear and thus ensures diminishing average regret per period as T increases.
• The regret rate is near-optimal, as the paper establishes a matching lower bound of $ \Omega(T^{(2+d)/(4+d)}) $ for any non-anticipating policy under mild technical conditions.
• The lower bound is derived using a carefully constructed family of functions $ \mathcal{C} $ that differ only in a single bin, making it difficult to distinguish without extensive exploration.
• The construction shows that the cost of distinguishing between similar demand functions in small bins with sparse observations leads to unavoidable regret.
• The result implies that the proposed policy is asymptotically optimal up to logarithmic factors in the regret rate.
• The framework suggests that exploiting sparsity in the covariates—by identifying the d' relevant dimensions—can reduce the regret from $ T^{(2+d)/(4+d)} $ to $ T^{(2+d')/(4+d')} $, highlighting the importance of feature selection.