We consider prophet inequalities under downward-closed constraints. In this problem, a decision-maker makes immediate and irrevocable choices on arriving elements, subject to constraints. Traditionally, performance is compared to the expected offline optimum, called the \textit{Ratio of Expectations} (RoE). However, RoE has limitations as it only guarantees the average performance compared to the optimum, and might perform poorly against the realized ex-post optimal value. We study an alternative performance measure, the \textit{Expected Ratio} (EoR), namely the expectation of the ratio between algorithm's and prophet's value. EoR offers robust guarantees, e.g., a constant EoR implies achieving a constant fraction of the offline optimum with constant probability. For the special case of single-choice problems the EoR coincides with the well-studied notion of probability of selecting the maximum. However, the EoR naturally generalizes the probability of selecting the maximum for combinatorial constraints, which are the main focus of this paper. Specifically, we establish two reductions: for every constraint, RoE and the EoR are at most a constant factor apart. Additionally, we show that the EoR is a stronger benchmark than the RoE in that, for every instance (constraint and distribution), the RoE is at least a constant fraction of the EoR, but not vice versa. Both these reductions imply a wealth of EoR results in multiple settings where RoE results are known.

翻译：我们考虑了向下封闭约束下的先知不等式问题。在该问题中，决策者对到达的元素做出即时且不可撤销的选择，并受限于特定约束。传统上，算法性能通过与期望离线最优值的比较来评估，即所谓的“期望比值”（Ratio of Expectations, RoE）。然而，RoE存在局限性，它仅能保证与最优值相比的平均性能，在面对实际实现的事后最优值时可能表现不佳。本文研究了一种替代性能度量——“期望比值”（Expected Ratio, EoR），即算法价值与先知价值之比的期望值。EoR提供了鲁棒性保证，例如，常数EoR意味着以恒定概率实现离线最优值的常数比例。对于单选择问题的特例，EoR与广为人知的“选取最大值的概率”概念一致。然而，EoR自然地推广了组合约束下选取最大值的概率，而这正是本文的主要关注点。具体而言，我们建立了两种归约：对于任意约束，RoE与EoR至多相差一个常数因子。此外，我们证明了EoR是比RoE更强的基准，即对于每个实例（约束和分布），RoE至少是EoR的常数比例，反之则不成立。这两种归约意味着在已知RoE结果的多种设置下，可获得丰富的EoR结果。