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

TL;DR

This work analyzes prophet inequalities under an iid valuation model, measuring online performance against the average of the top-\ell items. It introduces a quantile-based algorithm that achieves the optimal worst-case competitive ratio CR_\ell(n) and shows CR_\ell is the unique solution to an integral equation, with CR_\ell > 1- e^{-\ell} and CR_2 ≈ 0.966, illustrating exponential convergence to 1 as \ell grows. The authors further extend the framework to the multi-unit setting (k,\ell), establish limiting ODEs in the large-n limit, prove asymptotic guarantees and tightness results, and analyze static-threshold policies with near-optimal bounds. These results yield precise asymptotic guarantees and practical, learnable policies for selecting multiple high-value items online, along with extensions to non-iid and nonuniform benchmarks. Collectively, the paper deepens understanding of how quickly online selection can approach the top order statistics under iid distributions and provides actionable algorithmic and analytical tools for multi-item and threshold-based variants.

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.

Figures (4)

• Figure 1: $\operatorname{CR}_{\ell}$ as a function of $\ell$: the competitive ratio converges exponentially fast to $1$.
• Figure 2: Solution to the continuous boundary value problem for $k=\ell=5$.
• Figure 3: For $k,\ell \in [15]$: on the left $\operatorname{CR}_{k,\ell}^S$, on the right $\operatorname{CR}_{k,\ell}^S \cdot \min(\ell/k,1)$
• Figure 4: Competitive ratio for $\mathbf{p}=(1-\alpha,\alpha)$, for $\alpha \in [0,1/2]$.

Theorems & Definitions (48)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Proposition 2.1
• proof : Sketch of proof
• Lemma 2.2
• proof : Sketch of proof
• Remark
• Proposition 2.3