Personalized Dynamic Pricing with Machine Learning: High Dimensional Covariates and Heterogeneous Elasticity

We consider a seller who can dynamically adjust the price of a product at the individual customer level, by utilizing information about customers’ characteristics encoded as a $d$-dimensional feature vector. We assume a personalized demand model, parameters of which depend on $s$ out of the $d$ features. The seller initially does not know the relationship between the customer features and the product demand, but learns this through sales observations over a selling horizon of $T$ periods. We prove that the seller’s expected regret, i.e., the revenue loss against a clairvoyant who knows the underlying demand relationship, is at least of order $s\sqrt{T}$ under any admissible policy. We then design a near-optimal pricing policy for a “semi-clairvoyant” seller (who knows which s of the d features are in the demand model) that achieves an expected regret of order $s\sqrt{T}log(T)$. We extend this policy to a more realistic setting where the seller does not know the true demand predictors, and show this policy has an expected regret of order $s\sqrt{T}(log(d)＋log(T))$, which is also near-optimal. Finally, we test our theory on simulated data and on a data set from an online auto loan company in the United States. On both data sets, our experimentation-based pricing policy is superior to intuitive and/or widely-practiced customized pricing methods such as myopic pricing and segment-then-optimize policies. Furthermore, our policy significantly improves upon the loan company’s historical pricing decisions in terms of annual expected revenue.