On Dynamic Pricing with Covariates

TL;DR

This paper develops algorithms for dynamic pricing with covariates that achieve near-optimal regret bounds without assuming i.i.d. covariates, extending the theory to more general and realistic settings.

Contribution

It introduces UCB and Thompson sampling algorithms for dynamic pricing with covariates that do not require i.i.d. assumptions and match lower bounds, improving upon prior work.

Findings

Achieve $O(d ext{ } \sqrt{T} ext{ }\log T)$ regret bounds without i.i.d. assumptions.

Regret bounds are independent of the covariance matrix's eigenvalues.

Demonstrated effectiveness through numerical experiments.

Abstract

We consider dynamic pricing with covariates under a generalized linear demand model: a seller can dynamically adjust the price of a product over a horizon of $T$ time periods, and at each time period $t$, the demand of the product is jointly determined by the price and an observable covariate vector $x_t\in\mathbb{R}^d$ through a generalized linear model with unknown co-efficients. Most of the existing literature assumes the covariate vectors $x_t$'s are independently and identically distributed (i.i.d.); the few papers that relax this assumption either sacrifice model generality or yield sub-optimal regret bounds. In this paper, we show that UCB and Thompson sampling-based pricing algorithms can achieve an $O(d\sqrt{T}\log T)$ regret upper bound without assuming any statistical structure on the covariates $x_t$. Our upper bound on the regret matches the lower bound up to logarithmic factors. We thus show that (i) the i.i.d. assumption is not necessary for obtaining low regret, and (ii) the regret bound can be independent of the (inverse) minimum eigenvalue of the covariance matrix of the $x_t$'s, a quantity present in previous bounds. Moreover, we consider a constrained setting of the dynamic pricing problem where there is a limited and unreplenishable inventory and we develop theoretical results that relate the best achievable algorithm performance to a variation measure with respect to the temporal distribution shift of the covariates. We also discuss conditions under which a better regret is achievable and demonstrate the proposed algorithms' performance with numerical experiments.