Optimal ex-post competitive ratio below Nash social welfare

Determine the optimal competitive ratio for the full-information ex-post alpha-fair prophet inequality when alpha is in (0,1), and establish whether the sharp ratio is one-half throughout this range.

Background

For the ex-post objective with alpha in (0,1), the paper proves a universal full-information competitive-ratio lower bound of e{-pi2/6}, approximately 0.193. The authors explicitly state that the optimal constant is unknown.

The paper conjectures that the sharp ratio is one-half, matching the classical utilitarian prophet inequality, although it notes that a one-half guarantee is currently obtained only for the special case alpha=1/2 through a sharpened KL-divergence argument.

References

Additionally, for the ex-post objective with $\alpha\in(0,1)$, our full-information results give a universal constant lower bound, but we do not know the optimal constant. The natural conjecture is that the sharp competitive ratio is $\nicefrac12$, matching the classical prophet inequality, at least throughout the range $\alpha\in(0,1)$.

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

Understanding whether analogous guarantees hold for ex-post $\alpha$-fairness in such combinatorial settings remains open.

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