Prophet Inequalities: Competing with the Top $\ell$ Items is Easy

We explore a prophet inequality problem, where the values of a sequence of items are drawn i.i.d. from some distribution, and an online decision maker must select one item irrevocably. We establish that $\mathrm{CR}_{\ell}$ the worst-case competitive ratio between the expected optimal performance of an online decision maker compared to that of a prophet who uses the average of the top $\ell$ items is exactly the solution to an integral equation. This quantity $\mathrm{CR}_{\ell}$ is larger than $1-e^{-\ell}$. This implies that the bound converges exponentially fast to $1$ as $\ell$ grows. In particular for $\ell=2$, $\mathrm{CR}_{2} \approx 0.966$ which is much closer to $1$ than the classical bound of $0.745$ for $\ell=1$. Additionally, we prove asymptotic lower bounds for the competitive ratio of a more general scenario, where the decision maker is permitted to select $k$ items. This subsumes the $k$ multi-unit i.i.d. prophet problem and provides the current best asymptotic guarantees, as well as enables broader understanding in the more general framework. Finally, we prove a tight asymptotic competitive ratio when only static threshold policies are allowed.