Near-Optimal Bisection Search for Nonparametric Dynamic Pricing with Inventory Constraint

Abstract

We consider a single-product revenue management problem with an inventory constraint and unknown, noisy, demand function. The objective of the firm is to dynamically adjust the prices to maximize total expected revenue. We restrict our scope to the nonparametric approach where we only assume some common regularity conditions on the demand function instead of a specific functional form. We propose a family of pricing heuristics that successfully balance the tradeoff between exploration and exploitation. The idea is to generalize the classic bisection search method to a problem that is affected both by stochastic noise and an inventory constraint. Our algorithm extends the bisection method to produce a sequence of pricing intervals that converge to the optimal static price with high probability. Using regret (the revenue loss compared to the deterministic pricing problem for a clairvoyant) as the performance metric, we show that one of our heuristics exactly matches the theoretical asymptotic lower bound that has been previously shown to hold for any feasible pricing heuristic. Although the results are presented in the context of revenue management problems, our analysis of the bisection technique for stochastic optimization with learning can be potentially applied to other application areas.3 revenue loss? It turns out that it is possible: If we use Stochastic Approximation algorithms (i.e., Kiefer-Wolfowitz and Robbins-Monro, see Broadie et al. [10]) during the exploitation phase instead of another bisection search, then the resulting revenue loss is exactly Θ( √ θ). Thus, we have provided an "optimal" nonparametric pricing heuristic for the setting of a single-product problem with inventory constraint. (In the case where the firms know the functional form of the demand function, i.e., parametric model, the Ω( √ θ) lower bound has been repeatedly shown to be tight. For example, in the setting without inventory constraints, Keskin and Zeevi [27], den Boer and Zwart [19], and Broder and Rusmevichientong [11], each proposes a parametric pricing heuristic that guarantees a revenue loss of the order of O( √ θ) . As for the setting with inventory constraints, recently Chen et al. [13] propose a heuristic that exactly matches this lower bound. Their result holds for a general parametric model with an arbitrary set of inventory constraints. Thus, they have resolved the parametric dynamic pricing problem with inventory constraints.)

A Primal-dual Learning Algorithm for Personalized Dynamic Pricing with an Inventory Constraint

“…Before proceeding, we introduce a modified stochastic system. Technically, if the inventory is depleted at t, then P m (t) must be switched to p ∞ , a choke price at which the demand of type-m customers is turned off, for all m. We use a similar simplification to Lei et al (2017) and consider a slightly different problem. When the inventory is depleted, instead of forced to set p ∞ for all types of customers, the firm can still use prices between [p, p].…”

“…Therefore, the firm has to balance the exploration/exploitation trade-off, which is usually referred to as the learning-andearning problem in this line of literature. Among them, our paper is related to those with nonparametric formulations and inventory constraints (Besbes and Zeevi, 2009;Wang et al, 2014;Lei et al, 2017). In addition, we consider personalized dynamic pricing for multiple types of customers, while most of the above papers consider a single type.…”

Dynamic Online Pricing with Incomplete Information Using Multiarmed Bandit Experiments

“…The interpretation here is that if the product is sold for v L (can be zero) all consumers will purchase for sure, and if the product is sold for v H , no consumers will buy. Consistent with the robust pricing literature [Bergemann and Schlag, 2008, Besbes and Zeevi, 2009, Handel et al, 2013, Wang and Hu, 2014, Handel and Misra, 2015, Lei et al, 2014 we assume within this range the firm does not know the distribution of consumer preferences across or within segments. Our motivation for this assumption is that it is infeasible for the manager to have credible priors for millions of products.…”

“…The firm has to ex-ante set the length of experimentation stage. More recent additions to this literature Wang and Hu [2014] and Lei et al [2014] improve the convergence results, yet the algorithms proposed in these paper also consider distinct phases for exploration then exploitation, or as we refer to it, "learning then earning." Instead, in our paper, we consider the learning and earning phases simultaneously, accounting for the potential value from learning at each point in time.…”

Uncertainty Quantification for Demand Prediction in Contextual Dynamic Pricing

“…In recent years, data-driven sequential decision-making has received a lot of attentions and finds a wide range of applications in operations management, such as dynamic inventory control (see, e.g., Huh et al (2011), Chen and Plambeck (2008), Chen et al (2019b,a), Lei et al (2019)), dynamic pricing (see, e.g., Zeevi (2009, 2015), Wang et al (2014), Chen et al (2019c), Broder and Rusmevichientong (2012)), dynamic assortment optimization (see, e.g., Rusmevichientong and Topaloglu (2012), Saure and Zeevi (2013), Agrawal et al (2019), Wang et al (2018), Chen et al (2018)). Take the personalized/contextual dynamic pricing as an example; it is usually assumed that the underlying demand, which is a function of the price and customer's contextual information, follows a certain probabilistic model with unknown parameters.…”