Constant-sample ex-ante prophet inequalities for alpha at most one

Determine whether a constant number of samples per distribution suffices for a constant-competitive ex-ante alpha-fair prophet inequality when alpha is in (0,1], or whether the sample complexity must grow with the number of agents.

Background

For ex-ante alpha-fairness with alpha in (0,1], the paper establishes a constant competitive ratio using O(n log n) samples per distribution. This leaves unresolved whether the dependence on n is an artifact of the learning and implementation procedure or is information-theoretically necessary.

The question is motivated by the classical utilitarian prophet inequality, where a single sample per distribution suffices to achieve the optimal one-half competitive ratio. Resolving it would clarify whether nonlinear alpha-fair objectives require fundamentally more statistical information than utilitarian welfare.

References

For example, in the ex-ante sample-access model, our algorithm uses $O(n\log n)$ samples per distribution for $\alpha\in(0,1]$. It would be interesting to determine whether a constant number of samples per distribution suffices, as in the classical utilitarian setting, or whether a growing number of samples is necessary.

— Fair Prophets  (2609.21826 - Duetting et al., 18 Sep 2026) in Section Conclusion and Future Directions