Best is not Always First: Optimizing Offer Deadlines and Offer Sequence with Delayed Feedback

Abstract

We study a setting where a proposer (e.g., an employer) with a single offer to make sequentially approaches a set of responders (e.g., job candidates) each of whom is also searching for alternative offers. We are interested in the proposer's decision of which responder to offer to first, and the optimal deadline to set for the responder's reply. We provide a full characterization of the optimal deadlines. When the proposer's choice set is composed of only two responders, we also fully characterize the proposer's sequence decision: She should offer to the higher-utility responder first if and only if she approaches the responders later than a time threshold. Moreover, when the choice set is only composed of exploding responders who are able to provide immediate feedback on the proposer's offer, the proposer should always prioritize the highestutility responder. This policy among exploding responders continues to hold even when the proposer's choice set also includes non-exploding responders whose decisions on the proposer's offer are only revealed with a delay-as long as the responders are homogeneous in the probabilities of their availability at any given time. In addition, when all but one of the responders are exploding responders, we show that as the proposer approaches this choice set later in the time horizon, the sole non-exploding responder should be prioritized over fewer exploding responders. Finally, our analysis of a static greedy policy that always prioritizes the highest-utility responder indicates that a greedy policy can be rather efficient when the market is not very competitive; otherwise, our numerical examples show that the proposer can incur substantial loss. Overall, our findings shed light on, for example, hiring decisions: We recommend that employers entering a market late or operating in less competitive enviornments adopt a simple greedy approach, which is likely to be close to optimal. In contrast, early entrants in highly competitive markets should carefully consider the popularity of candidates, as it may be optimal to prioritize those who are less sought after.

Keywords: search theory, ultimatum, offer sequence, offer deadline