On the Power of Delayed Flexibility: Balls, Bins, and a Few Opaque Promotions

Abstract

Effective load balancing lies at the heart of many applications in operations. Frequently tackled via the balls-into-bins paradigm, seminal results established the power of two choices in load balancing: a limited amount of costly flexibility goes a long way in order to maintain an approximately balanced load throughout the decision-making horizon. In many applications, however, balance across time may be too stringent a requirement; rather, the only desideratum is approximate balance at the {\it end} of the horizon. Motivated by this observation, in this work we design "delayed-flexibility" algorithms tailored to such settings. For the canonical balls-into-bins problem, we show that a simple policy that begins exerting flexibility toward the end of the time horizon - namely, when $\Theta \left(\sqrt{T\log T}\right)$ periods remain - suffices to achieve an approximately balanced load, i.e., a maximum load within $O(1)$ of the average load. Moreover, with just a small amount of adaptivity, a threshold policy achieves the same result, while only exerting flexibility in $O\left(\sqrt{T}\right)$ periods, thus matching a natural lower bound. We leverage these results to study the design of opaque selling strategies in retail settings, a topic recently identified as a key application of the power of two choices paradigm. For this problem, we prove that late-stage opaque selling strategies achieve the optimal trade-off between exerting costly flexibility - i.e., offering the opaque product at a discount - and achieving inventory cost savings through load balancing. We demonstrate the robustness of our insights via extensive numerical experiments, for a variety of customer choice models.