"Who Is Next in Line?'' On the Significance of Knowing the Arrival Order in Bayesian Online Settings

TL;DR

This paper introduces the order-competitive ratio, a new measure for online algorithms in Bayesian settings, quantifying the performance loss due to unknown arrival order, with tight bounds established for prophet inequality problems.

Contribution

It defines and analyzes the order-competitive ratio, providing tight bounds for various algorithm classes and objectives, highlighting the importance of knowing the arrival order in Bayesian online algorithms.

Findings

Order-competitive ratio captures the impact of arrival order on algorithm performance.

Adaptive algorithms outperform single-threshold algorithms in the order-competitive setting.

Tight bounds are established for prophet inequalities under different algorithm classes.

Abstract

We introduce a new measure for the performance of online algorithms in Bayesian settings, where the input is drawn from a known prior, but the realizations are revealed one-by-one in an online fashion. Our new measure is called order-competitive ratio. It is defined as the worst case (over all distribution sequences) ratio between the performance of the best order-unaware and order-aware algorithms, and quantifies the loss that is incurred due to lack of knowledge of the arrival order. Despite the growing interest in the role of the arrival order on the performance of online algorithms, this loss has been overlooked thus far. We study the order-competitive ratio in the paradigmatic prophet inequality problem, for the two common objective functions of (i) maximizing the expected value, and (ii) maximizing the probability of obtaining the largest value; and with respect to two families of algorithms, namely (i) adaptive algorithms, and (ii) single-threshold algorithms. We provide tight bounds for all four combinations, with respect to deterministic algorithms. Our analysis requires new ideas and departs from standard techniques. In particular, our adaptive algorithms inevitably go beyond single-threshold algorithms. The results with respect to the order-competitive ratio measure capture the intuition that adaptive algorithms are stronger than single-threshold ones, and may lead to a better algorithmic advice than the classical competitive ratio measure.