Intertemporal Price Discrimination via Randomized Promotions

Problem definition: The undesirable but inevitable consequence of running promotions is that consumers can be trained to time their purchases strategically. In this paper, we study randomized promotions, where the firm randomly offers discounts over time, as an alternative strategy of intertemporal price discrimination. Methodology/results: We consider a base model where a monopolist sells a single product to a market with a constant stream of two market segments. The segments are heterogeneous in both their product valuations and patience levels. The firm precommits to a price distribution, and in each period, a price is randomly drawn from the committed distribution. We characterize the optimal price distribution as a randomized promotion policy and show that it serves as an intertemporal price discrimination mechanism such that high-valuation customers would make a purchase immediately at a regular price upon arrival, and low-valuation customers would wait for a random promotion. Compared against the optimal cyclic pricing policy, which is optimal within the strategy space of all deterministic pricing policies, the optimal randomized pricing policy beats it if low-valuation customers are sufficiently patient and the absolute discrepancy between high and low customer valuations is large enough. We extend the model in three directions. First, we consider the case where a portion of customers are myopic and would never wait. We show that the existence of myopic customers is detrimental to the firm’s profitability, and the expected profit from an optimal randomized pricing policy decreases as the proportion of myopic customers in the population increases. Second, we consider Markovian pricing policies where prices are allowed to be intertemporally correlated in a Markovian fashion. This additional maneuver allows the firm to reap an even higher profit when low-valuation customers are sufficiently patient by avoiding consecutive promotions but, on average, running the promotion more frequently with a smaller discount size. Lastly, we consider a model with multiple customer segments and show that a two-point price distribution remains optimal, and our conclusion from the two-segment base model still holds under certain conditions that are adopted in the literature. Managerial implications: Our results imply that the firm may want to deliberately randomize promotions in the presence of forward-looking customers. Funding: This work was supported by the National Natural Science Foundation of China [Grant 72201124], the Natural Sciences and Engineering Research Council of Canada [Grants RGPIN-2015-06757 and RGPIN-2021-04295], and the Youth Project of the Humanities and Social Science Foundation of the Ministry of Education of China [Grant 22YJC630006]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2023.1194 .