Prophet Inequalities via the Expected Competitive Ratio

Tomer Ezra, Stefano Leonardi, Rebecca Reiffenhäuser, Matteo Russo, Alexandros Tsigonias-Dimitriadis

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.