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- W3186366996 abstract "We consider the Unimodal Multi-Armed Bandit problem where the goal is to find the optimal price under an unknown unimodal reward function, with an additional constraint that requires that the price exploration is non-increasing. This markdown optimization problem faithfully models a single-product revenue management problem where the objective is to adaptively reduce the price over a finite sales horizon to maximize expected revenues.We measure the performance of an adaptive exploration-exploitation policy in terms of the regret: the revenue loss relative to the maximum revenue that could have been attained when the demand curve is known in advance. For the case of $L$-Lipschitz-bounded unimodal revenue functions with infinite inventory, we present a natural policy that explores the price space at a uniform optimal speed in $T$ steps and has regret $O(T^{3/4} (Llog T)^{1/4})$. On the other side, we provide an almost-matching lower bound of $Omg(L^{1/4}T^{3/4})$ on the regret of any policy. Further, under mild assumptions, we show that the above tight bounds also hold when the inv is finite but is on the order of $Omg(T)$. Our tight regret bounds highlight the additional complexity of the markdown constraint, and are asymptotically higher than the corresponding bounds without this markdown requirement of $tilde{Theta}(T^{1/2})$ for unimodal bandits and $tilde{Theta}(L^{1/3} T^{2/3})$ for $L$-Lipschitz bandits. We finally consider a generalization called Dynamic Pricing with Markup Penalty where the seller is allowed to increase the price by paying a markup penalty of magnitude $O(T^c)$ per markup where $cin [0,1]$ is a given constant. We extend our results to a tight $tilde O(T^{mathrm{med}{frac{2}{3}, frac{3}{4}, c}})$ regret bound for this variantfootnote{$mathrm{med}{a,b,c}$ denotes the median of the numbers $a,b,c$." @default.
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- W3186366996 date "2021-01-01" @default.
- W3186366996 modified "2023-10-18" @default.
- W3186366996 title "Markdown Pricing Under Unknown Demand" @default.
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- W3186366996 doi "https://doi.org/10.2139/ssrn.3861379" @default.
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