Markdown Pricing Under Unknown Demand

67 Pages Posted: 8 Jun 2021 Last revised: 15 Aug 2023

See all articles by Su Jia

Su Jia

Cornell University

Andrew Li

affiliation not provided to SSRN

R. Ravi

Carnegie Mellon University - David A. Tepper School of Business

Date Written: June 7, 2021


We consider a variant of the Continuum-Armed Bandit problem where the arm sequence is required to be non-increasing. This problem models a single product revenue management problem where the objective is to {\em reduce} the price over a finite sales horizon to maximize expected revenue. A policy that satisfies this monotonicity constraint is often called a {\em markdown} policy.

We focus on the scenario where the demand model is unknown. A policy's performance is measured by the {\it regret}, that is, the revenue loss due to not knowing the true demand function.
We first observe that to achieve sublinear regret, it is necessary to assume that the revenue function (defined as the product of the price and the mean demand) is unimodal and Lipschitz.
We then present a markdown policy that explores prices monotonically before a certain stopping condition is satisfied, and show that it has $\tilde O (L^{1/4} \min\{n,m\}^{3/4})$ regret for any inventory level $m$, number of time periods $n$, and the Lipschitz constant $L\ge 1$. This bound is nearly optimal: We prove that no markdown policy achieves $o(L^{1/4}\min\{m,n\}^{3/4})$ regret for {\em all} $m, n$. In contrast, under the same assumptions, the optimal regret is $\tilde{\Theta} (L^{1/3} n^{2/3})$ {\em without} monotonicity for $m=\infty$, which is \asym ally lower; see \cite{kleinberg2005nearly}.

We also consider a variant where price increases are allowed, but subject to a limited budget.
We present a policy with $O(\log n)$ price increases and $\tilde O(L^{1/3} \min\{m,n\}^{2/3})$ regret, which matches the optimal regret without monotonicity.

Keywords: markdown pricing, dynamic pricing, multi-armed bandits, revenue management, online learning

Suggested Citation

Jia, Su and Li, Andrew and Ravi, R., Markdown Pricing Under Unknown Demand (June 7, 2021). Available at SSRN: or

Su Jia (Contact Author)

Cornell University ( email )

New York
United States

Andrew Li

affiliation not provided to SSRN ( email )

R. Ravi

Carnegie Mellon University - David A. Tepper School of Business ( email )

5000 Forbes Avenue
Pittsburgh, PA 15213-3890
United States

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