Dynamic pricing policies with time-based price limits guarantee half optimal revenue
Inter-Temporal Price Constraints in Dynamic Pricing: Performance Guarantees Under Price Monotonicity and Promotion Fatigue
Discrete Mathematics
Summary
This paper looks at how to set prices over time for products when certain rules must be followed, such as prices only going up or down steadily or limits on how often discounts can happen. Since finding the best exact pricing strategy is too hard, the authors use simpler approximate methods to create pricing plans that still do well. Their plans guarantee to make at least half the revenue compared to the best possible strategy when resources are large enough. This work also sets limits on how well any strategy can do under these constraints.
What this means in practice
- •For retail pricing teams: Plan time-based discount strategies that obey limits on price changes and discount frequency, ensuring predictable revenue outcomes with large inventories.
- •For inventory managers: Manage the selling schedule of limited-stock products while respecting pricing rules to maximize returns under capacity constraints.
Authors
Weiyuan Li, Paat Rusmevichientong, Huseyin Topaloglu
Abstract
We study dynamic pricing problems under inter-temporal price constraints. We have resources with limited capacities. At each time period, we decide which products to make available and what prices to charge for the available products. The sale probability for a product depends on its price. If we make a sale for a product, then we collect a revenue reflecting the price and consume the capacities of a combination of resources. We work with two types of inter-temporal constraints. In price monotonicity, the prices charged for a product at different time periods have to be monotone. In promotion fatigue, we can discount a product at most once over each time interval of a fixed length. Computing the optimal policy is intractable. We use fluid approximations to construct policies. Traditionally, policies from fluid approximations make randomized decisions at each time period by following an optimal solution to the fluid approximation, but such randomized decisions easily violate price monotonicity or promotion fatigue constraints. We develop policies that sample price paths according to an optimal solution to the fluid approximation, while satisfying the inter-temporal constraints. Letting $c_{\min}$ be the smallest initial capacity of a resource and $L$ be the maximum number of resources used by a product, our policies have a performance guarantee of $\max\Big\{ \frac{1}{8L}, \, \frac{1}{2} - \sqrt{\frac{\log c_{\min}}{2 \,c_{\min}}} - \frac{L}{c_{\min}}\Big\}$. Thus, under large resource capacities, our policies are guaranteed to obtain at least half of the optimal total expected revenue. The latter performance guarantee is tight in the sense that no policy can, in general, obtain more than half of the optimal objective value of the fluid approximation even under large resource capacities. We unify our approach to open the path for extensions to other inter-temporal price constraints.