Sample-Driven Optimal Stopping: From the Secretary Problem to the i.i.d. Prophet Inequality

Abstract

We take a unifying approach to single selection optimal stopping problems with random arrival order and independent sampling of items. In the problem we consider, a decision maker (DM) initially gets to sample each of N items independently with probability p, and can observe the relative rankings of these sampled items. Then, the DM faces the remaining items in an online fashion, observing the relative rankings of all revealed items. While scanning the sequence the DM makes irrevocable stop/continue decisions and her reward for stopping the sequence facing the item with rank i is Yi. The goal of the DM is to maximize her reward. We start by studying the case in which the values Yi are known to the DM, and then move to the case in which these values are adversarial. For the former case we are able to recover several classic results in the area, thus giving a unifying framework for single selection optimal stopping. For the latter, we pin down the optimal algorithm, obtaining the optimal competitive ratios for all values of p. Funding: This work was partially supported by The Center for Mathematical Modeling at the University of Chile (ANID FB210005), Grant Anillo Information and Computation in Market Design (ANID ACT210005), FONDECYT 1220054 and 1181180, and a Meta Research PhD Fellowship.