Oscillation criteria for stopping near the top of a random walk

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 [1] proved that the optimal strategy is determined by an interesting sequence of constants \(p_{n}\) $p_n$ and he conjectured \(p_{n} \rightarrow \ 1/2\) $p_n\to 1/2$ as \(n \rightarrow \infty \) $n\to \infty$ . A major difficulty to prove this is the absence of close form expressions when optimizing in some cases. In this note, the best lower bound for this sequence is found and more of its properties are proven toward solving the conjecture.