Improve the general sample-based finite-horizon stopping guarantee

Improve the prophet constant \(\beta_C(K)\) guaranteed by the finite-horizon stopping theorem for nonnegative reward sequences satisfying \(E[\max_{i\le j\le n}(Y_j-Y_i)\mid\mathcal F_i]\le C E[M]\), when the decision-maker has access to \(K\) independent samples of the maximum \(M\).

Background

The appendix establishes a finite-horizon algorithmic result for a general class of nonnegative reward sequences satisfying a conditional bound on expected future increments. Given KK independent samples from the distribution of the maximum, the theorem guarantees a prophet constant βC(K)=sup⁡{x−Cx2:x∈AK}\beta_C(K)=\sup\{x-Cx^2:x\in A_K\}.

The authors explicitly identify improving this theorem as unresolved. The question is broader than the random-walk specialization because it concerns the theorem's general structural assumptions and its sample-based guarantee.

References

We leave improving the above theorem open, but remark that, as noted by Wittmann, that for C=1 the best possible prophet constant with full information is at most 1/2.881 and that to achieve 1/e in the many-samples limit requires more exploitation of the random walk structure.

— Sample-Based Prophet Inequalities for Random Walks  (2609.26017 - Kleer et al., 22 Sep 2026) in Appendix, Section 7, immediately after the finite-horizon algorithmic theorem

This prophet constant can be improved with more refined arguments, but we leave this as an open problem.

— Sample-Based Prophet Inequalities for Random Walks  (2609.26017 - Kleer et al., 22 Sep 2026) in Appendix, Section 7.1, concluding paragraph