  Consider the problem of maximizing the probability of stopping with one of
the two highest values in a Bernoulli random walk with arbitrary parameter $p$
and finite time horizon $n$. Allaart \cite{Allaart} proved that the optimal
strategy is determined by an interesting sequence of constants $\{p_{n}\}$. He
conjectured the asymptotic behavior to be $1/2$. In this work the best lower
bound for this sequence is found and more of its properties are proven towards
solving the conjecture.
