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

TL;DR

This paper analyzes prophet inequalities for selecting top items from i.i.d. sequences, establishing exact competitive ratios and asymptotic bounds, showing improved performance over classical bounds especially for larger top sets.

Contribution

The paper derives exact formulas for the competitive ratio R_ll, proves exponential convergence to 1, and extends results to multi-unit scenarios with asymptotic guarantees.

Findings

R_ll exceeds classical bounds and converges exponentially to 1

For ll=2, R_ll pprox 0.966, closer to optimal than 0.745

Provides asymptotic lower bounds for multi-unit prophet problems

Abstract

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.