Persistence of the asymptotic improvement over memoryless strategies

Determine whether the improvement achieved by the optimal memoryless threshold strategy over a baseline memoryless strategy remains nonvanishing as the number of observations tends to infinity.

Background

The paper discusses memoryless threshold strategies for Robbins’ problem, which base the stopping decision on the current observation and the time index while ignoring the preceding observed values. Such strategies yield useful asymptotic upper bounds, but prior work has shown that the optimal memoryless strategy can always be improved for each finite number of observations.

It remains unresolved whether this finite-horizon improvement persists asymptotically. Specifically, the paper notes that no proof establishes that the improvement does not vanish as the number of observations tends to infinity; a related Poisson-embedded version of Robbins’ problem is cited as evidence that non-memoryless limiting behavior may occur in a related setting.

References

It is however important to note that it was shown that the memoryless optimal strategy can always be improved, see e.g.,. However, so far nobody has proved that the improvement will not disappear as $n\to \infty$.

— Algorithms for Robbins' Problem using Markov Decision Processes  (2608.27419 - Brice et al., 27 Aug 2026) in Section 1, paragraph “Known results for Robbins’ problem,” item (iii)