  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.
