Optimal Pricing Schemes for Identical Items with Time-Sensitive Buyers
Zhengyang Liu, Liang Shan, Zihe Wang
TL;DR
The paper addresses optimal dynamic pricing for identical items with time-sensitive buyers in a Bayesian framework, modeling buyer utility as $\min_t\{p(t)+\theta t\}$ and revenue as $p(t^*(\theta))$. It develops a separation-function formulation with $\ell_p$ that yields a concave, non-decreasing boundary, enabling a polynomial-time algorithm for discrete distributions and a discretization-based approach for continuous ones, with provable bounds on revenue loss. It analyzes the trade-off between revenue and wasted time, establishing that $k$-step schemes can outperform fixed pricing by up to a factor of $k$ and that time-waste can be substantial, especially under many types. The work further shows that positive correlation between item value and cost-per-unit-time can enhance revenue and provides closed-form instances illustrating how the optimal scheme adapts to correlation, contributing to a computational understanding of time-factor pricing.
Abstract
Time or money? That is a question! In this paper, we consider this dilemma in the pricing regime, in which we try to find the optimal pricing scheme for identical items with heterogenous time-sensitive buyers. We characterize the revenue-optimal solution and propose an efficient algorithm to find it in a Bayesian setting. Our results also demonstrate the tight ratio between the value of wasted time and the seller's revenue, as well as that of two common-used pricing schemes, the k-step function and the fixed pricing. To explore the nature of the optimal scheme in the general setting, we present the closed forms over the product distribution and show by examples that positive correlation between the valuation of the item and the cost per unit time could help increase revenue. To the best of our knowledge, it is the first step towards understanding the impact of the time factor as a part of the buyer cost in pricing problems, in the computational view.