Prophet Inequalities via the Expected Competitive Ratio

Abstract

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 Ratio of Expectations ($\textsf {RoE}$). However, $\textsf {RoE}$ 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 Expected Ratio ($\textsf {EoR}$), namely the expectation of the ratio between algorithm’s and prophet’s value. $\textsf {EoR}$ offers robust guarantees, e.g., a constant $\textsf {EoR}$ implies achieving a constant fraction of the offline optimum with constant probability. For the special case of single-choice problems the $\textsf {EoR}$ coincides with the well-studied notion of probability of selecting the maximum. However, the $\textsf {EoR}$ 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, $\textsf {RoE}$ and the $\textsf {EoR}$ are at most a constant factor apart. Additionally, we show that the $\textsf {EoR}$ is a stronger benchmark than the $\textsf {RoE}$ in that, for every instance (constraint and distribution), the $\textsf {RoE}$ is at least a constant fraction of the $\textsf {EoR}$, but not vice versa. Both these reductions imply a wealth of $\textsf {EoR}$ results in multiple settings where $\textsf {RoE}$ results are known.