Matroid Prophet Inequality

A matroid prophet inequality is a prophet inequality in which the accepted elements must form an independent set of a known finite matroid . For a vector of nonnegative values , the offline benchmark is

The online rule decides irrevocably whether to accept each arriving element.

OpenAI (2026) reports a measurable rule using one independent sample from each element's value statistical distribution before arrivals. The random variables and are mutually independent across all elements and draws, with each having the law of its own . The rule uses the samples and the arrival history without knowing the value laws. If the offline optimum has finite expectation value, the claimed guarantee for the accepted set is

The bound is claimed even when the order of arrival is chosen after seeing the samples, values, value laws, and the rule's random seed. The AI-generated proof gives a measurable rule without a polynomial time bound.

Abdi et al. (2026) consider an order unknown to the rule and fixed independently of the samples, values, and the rule's randomness. With independent nonnegative values and one independent sample from each value's statistical distribution, their algorithm is reported to attain the optimal guarantee of half the expectation value of the offline optimum, using independence queries for a known matroid with elements. They credit GPT Astra and GPT Sol 5.6 with assistance in deriving the algorithm and its guarantee.