Prophets Inequalities with Uncertain Acceptance

A team of four computer scientists—Martinez, Garrido-Lucero, Grandi, and Pérez-Salazar—has published a new theoretical framework on arXiv titled 'Prophets Inequalities with Uncertain Acceptance.' The paper tackles a core problem in online decision-making: how to sequentially choose between options when each has a known value and a probability of being accepted, but you only discover if you're successful after attempting selection. This models real-world scenarios like job candidates who may reject an offer or limited admissions slots.

The researchers analyze the performance of a decision-maker who only knows the distributions of values and acceptance probabilities in advance. They compare this agent against two powerful benchmarks: a 'value-aware' agent who knows all future values beforehand, and a 'full-knowledge prophet' who knows everything. Their key finding is that, in the worst case, the online decision-maker can achieve a competitive ratio of 1/2 against the prophet, meaning her expected reward is at least half of the prophet's. They also show conditions where knowing values better is more crucial than knowing acceptance outcomes, providing guidance on what information to prioritize for better decisions.

This work extends the classic 'prophet inequality' problem, a cornerstone of optimal stopping theory, by adding a layer of uncertainty in the acceptance phase. The 19-page paper includes a mathematical reduction of the problem to a simpler form involving scaled Bernoulli distributions, making it more tractable for analysis. The result establishes a fundamental limit (the 1/2 barrier) for algorithm design in stochastic environments with two-sided uncertainty.