Prophets Inequalities with Uncertain Acceptance

2026-03-23

AI Summary

This work addresses a novel variant of the prophet inequality problem in sequential decision-making, where each option is associated with a random value and an acceptance probability drawn from a known joint distribution. The decision-maker must irrevocably decide in real time whether to attempt selection; the value is realized only if the option is accepted. The authors introduce the “prophet inequality with uncertain acceptance” model and transform the problem via a distributional mapping into a classical setting with scaled Bernoulli rewards. By comparing a value-aware decision-maker against an omniscient prophet and employing probabilistic modeling alongside competitive ratio analysis, they establish a worst-case competitive ratio of 1/2. Moreover, they identify sufficient conditions under which this bound can be surpassed, revealing that information about values holds higher decision priority than information about acceptance outcomes.

Technology Category

• Reasoning under Uncertainty: Sequential Decision Making
• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Machine Learning: Online Learning & Bandits

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

We introduce the \textit{prophet inequality with uncertain acceptance} model, in which a decision maker sequentially observes a sequence of independent options, each characterized by a value $x_i$ and an acceptance probability $p_i$, both sampled from a known joint distribution. At time $i$, the decision maker observes the value $x_i$ and must irrevocably and immediately decide whether to attempt to select it or to continue to the next time step. If the option is selected, the process terminates with probability $p_i$ and the decision maker obtains $x_i$; otherwise, she continues searching. In this setting, two natural benchmarks arise: the \textit{value-aware decision-maker}, who knows all value realizations in advance but not the acceptance outcomes, and the \textit{full-knowledge prophet}, who knows all realizations beforehand and can choose the best option among those that will be accepted. We characterize the worst-case competitive ratios between our defined agents and show that all these values equal $1/2$. In addition, we provide sufficient conditions under which the value-aware decision-maker surpasses the $1/2$-barrier against the more informed prophet. This demonstrates the (crucial) interest for the decision maker to improve her knowledge over the values rather than over the acceptances, and is obtained via a more general result that reduces the value-aware decision-maker's problem to a classical prophet inequality with scaled Bernoulli distributions, followed by a sequence of transformations that further reduce the problem to a unique optimization problem.

Problem

• prophet inequality
• uncertain acceptance
• competitive ratio
• sequential decision making
• online selection

Innovation

• prophet inequality
• uncertain acceptance
• competitive ratio
• value-aware decision-maker
• online selection