A Primal-Dual Online Learning Approach for Dynamic Pricing of Sequentially Displayed Complementary Items under Sale Constraints
Francesco Emanuele Stradi, Filippo Cipriani, Lorenzo Ciampiconi, Marco Leonardi, Alessandro Rozza, Nicola Gatti
TL;DR
This paper tackles dynamic pricing of sequentially displayed complementary items under a sale constraint by formulating the problem as a constrained Markov decision process (CMDP) and introducing a primal-dual online learning algorithm. The PD-DP method uses occupancy-measure representations, optimistic estimations, and epoch-based transition confidence sets to learn in environments with non-stationary demand and adversarial or stochastic rewards/constraints, without assuming known reward or constraint functions. Empirical evaluation on synthetic data generated from real-world LastMinute datasets demonstrates sublinear regret and effective constraint satisfaction, with a convex-combination policy addressing partial observability in intermediate states. The work advances practical dynamic pricing by enabling coordinated pricing of interdependent items under realistic online learning constraints, and suggests avenues for real-world deployment and function-approximation enhancements.
Abstract
We address the challenging problem of dynamically pricing complementary items that are sequentially displayed to customers. An illustrative example is the online sale of flight tickets, where customers navigate through multiple web pages. Initially, they view the ticket cost, followed by ancillary expenses such as insurance and additional luggage fees. Coherent pricing policies for complementary items are essential because optimizing the pricing of each item individually is ineffective. Our scenario also involves a sales constraint, which specifies a minimum number of items to sell, and uncertainty regarding customer demand curves. To tackle this problem, we originally formulate it as a Markov Decision Process with constraints. Leveraging online learning tools, we design a primal-dual online optimization algorithm. We empirically evaluate our approach using synthetic settings randomly generated from real-world data, covering various configurations from stationary to non-stationary, and compare its performance in terms of constraints violation and regret against well-known baselines optimizing each state singularly.