Prophet Inequalities: Competing with the Top ℓ Items is Easy

2024-08-14

arXiv.org

AI Summary

This paper studies the online selection problem under i.i.d. sequences, aiming to approximate the average reward achieved by a “prophet” who selects the top ℓ largest values (generalizing beyond the classical single-item benchmark) and extending to the setting of selecting k items. Using integral equation analysis, probabilistic inequalities, and optimal stopping theory, we establish the first exact characterization of the competitive ratio CRₗ under the top-ℓ benchmark, proving it converges to 1 exponentially fast (CRₗ > 1 − e⁻ℓ) and obtaining CR₂ ≈ 0.966—substantially surpassing the classic single-selection bound of 0.745. For k-item selection, we derive the optimal asymptotic lower bound on the competitive ratio and identify the tight limiting ratio achievable by static threshold policies. Our results provide the strongest known theoretical guarantees and principled design guidance for multi-item online decision-making under i.i.d. inputs.

Technology Category

• Machine Learning: Online Learning & Bandits
• Reasoning under Uncertainty: Sequential Decision Making
• Search and Optimization: Learning to Search

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Uses of LLMs and GenAI for marketplace design, bidding, and strategic interactions
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

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.

Problem

• Online Decision Making
• Prophet Inequality
• Optimal Selection

Innovation

• CR_ell Methodology
• Exponential Convergence
• Online Decision-making