Multiunit I.I.D. Prophet Inequalities via Extreme Value Asymptotics

TL;DR

This paper analyzes the limits of online selection algorithms for i.i.d. rewards, establishing asymptotic performance bounds and examining the effectiveness of heuristic approaches like the certainty-equivalent rule.

Contribution

It characterizes the optimal welfare ratio in the i.i.d. prophet inequality problem using extreme value asymptotics and compares it to static-threshold algorithms and heuristics.

Findings

Optimal welfare ratio approaches 1 as k increases.

The CE heuristic's performance improves with k but can have large regret when n/k is large.

The paper provides bounds that outperform static threshold guarantees.

Abstract

We study the i.i.d. $k$-selection prophet inequality problem, where a decision-maker sequentially observes $n$ independent nonnegative rewards and may accept at most $k$ of them without knowledge of future realizations. The objective is to maximize the expected total reward relative to that of a prophet who observes all rewards in advance. This problem captures the performance limits achievable in online resource allocation and underlies posted-price mechanisms in online marketplaces. We characterize the optimal welfare achievable relative to the prophet in terms of $k$ and the extreme value index of the reward distribution, in the asymptotic regime where the number of offers $n$ grows large. This optimal performance ratio turns out to be at least $1-\frac{\log k}{8k}[1+\epsilon]$ for any $\epsilon > 0$ and sufficiently large $k$, improving upon the respective, tight $1 - \frac{1}{\sqrt{2\pi k}}$ guarantee of static-threshold algorithms. We additionally analyze the certainty-equivalent (CE) heuristic, a widely used online allocation algorithm known to yield optimal regret growth in $n$ when evaluated under the fluid scaling assumption. Even in the absence of the fluid scaling, the CE heuristics's performance improves with $k$ to eventually match the leading order terms of the optimal dynamic program's performance ratio. A finer analysis nevertheless reveals that regret can be divergent and large relative to the optimal dynamic program when $n/k \to \infty$. This highlights the sensitivity in viewing the CE heuristic's performance under the commonly adopted, though subjective, fluid scaling assumption.