Dynamic Pricing Under Self-Exciting Arrival Processes

We study dynamic pricing under a stochastic, nonlinear, self-exciting demand arrival process over a finite sales horizon. We adopt such a correlated demand process to capture the phenomenon that customers who have made a purchase can inform and excite future customers to arrive. Specifically, the stochastic arrival intensity is boosted immediately after any purchase and gradually decays between consecutive purchases. The arrival intensity is also affected by the market phase the seller operates in, a growth or saturation stage, which convexly or concavely shapes the number of new customers that can be stimulated by earlier buyers. We show that the optimal pricing policy depends on the current time and excitement level, which measures the currently accumulated influence that earlier buyers exert and is shown to be the only sufficient statistic for our problem. Over time, the seller tends to initially set a low price to attract early buyers, who can excite later arrivals, and later increase the price to maximize profits. Perhaps surprisingly, at a given time, the optimal price rises with the excitement level if the market is in a growth stage, whereas it drops with the excitement level if the market is in a saturation stage. Moreover, we show that adopting the optimal time-varying (open-loop) pricing policy obtained from the deterministic problem (without re-solving the problem) incurs only a constant performance loss relative to the optimal dynamic (i.e., contingent) pricing policy, which would not hold for the classic revenue management problem with a capacity constraint. Finally, we extend our analysis to account for discrete prices and multiple products in an assortment and make a connection between the self-exciting demand arrival process and the immigration-birth process.

Funding: This research was supported by the National Natural Science Foundation of China [Grants 72431010, 72371225, and 72495134] and the Natural Sciences and Engineering Research Council of Canada [Grant RGPIN-2021-04295].

Supplemental Material: All supplemental materials, including the code, data, and files required to reproduce the results, are available at https://doi.org/10.1287/opre.2024.1172.