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

Abstract

A firm is selling a product to different types (based on the features such as education backgrounds, ages, etc.) of customers over a finite season with non-replenishable initial inventory. The type label of an arriving customer can be observed but the demand function associated with each type is initially unknown. The firm sets personalized prices dynamically for each type and attempts to maximize the revenue over the season. We provide a learning algorithm that is near-optimal when the demand and capacity scale in proportion. The algorithm utilizes the primal-dual formulation of the problem and learns the dual optimal solution explicitly. It allows the algorithm to overcome the curse of dimensionality (the rate of regret is independent of the number of types) and sheds light on novel algorithmic designs for learning problems with resource constraints.

Network revenue management with online inverse batch gradient descent method

“…This assumption is widely adopted by many revenue management literature (see, e.g., Besbes & Zeevi, 2012; N. Chen & Gallego, 2018; Gallego & Van Ryzin, 1997; Song et al., 2021). Many demand models satisfy this concavity property, such as the multinomial logit model and the nested logit model (see, e.g., Dong et al., 2009; Li & Huh, 2011; Song et al., 2021).…”

“…There are some other approaches used for the online learning problems in revenue management, such as the bisection search method (see Lei et al., 2014; Wang et al., 2014) and the least square method (see Besbes & Zeevi, 2015) for the settings with a single product and the primal‐dual method (see N. Chen & Gallego, 2018) for the settings with single‐resource shared by all products. These approaches heavily exploit the special features of the single‐product or single‐resource system.…”

Technical Note—Joint Learning and Optimization of Multi-Product Pricing with Finite Resource Capacity and Unknown Demand Parameters

“…This observation also raises an open question: Can O( √ k) regret be attained in the nonparametric setting with multiple products, multiple resources, and a continuum of prices? The most general case in the literature where regret bounds that are only up to logarithmic multiplicative terms larger than O( √ k) is achievable is the case of single resource and multiple independent products (Chen and Gallego 2019). Two features of this setting greatly simplify the analysis: First, it has the nice structure that the optimal solution is either binding or non-binding at the resource constraint; second, due to separable demand, the nonparametric estimation problem is effectively single-dimensional.…”

“…Therefore, there is also another stream of literature that investigates the case where the seller can choose from a continuum of prices. Most of the work in this stream has focused on the so-called nonparametric approach: The seller has no idea about the functional form of the demand function (Besbes and Zeevi 2009, Wang et al 2014, Besbes and Zeevi 2012, Lei et al 2014, Chen and Gallego 2019.…”

“…Unfortunately, strong analytical results under the nonparametric approach are limited to special cases. When there is a single product and a single resource, Wang et al (2014) developed an algorithm that attains O( √ k log 9 2 (k)); Chen and Gallego (2019) improved/generalized the result to the case of multiple products with no demand substitution (in the sense that the demand of product i is independent of the prices of other products) and a single resource, and developed a primaldual learning method that attains a regret of O( √ k log 2 (k)). The problem becomes very difficult with substitutable demand and multiple resource constraints because the seller needs to estimate a multi-variate vector-valued demand function in order to optimally allocate multiple resources over time.…”