An adaptive $O(\log n)$-optimal policy for the online selection of a monotone subsequence from a random sample

Given a sequence of $n$ independent random variables with common continuous distribution, we propose a simple adaptive online policy that selects a monotone increasing subsequence. We show that the expected number of monotone increasing selections made by such a policy is within $O(\log n)$ of optimal. Our construction provides a direct and natural way for proving the $O(\log n)$-optimality gap. An earlier proof of the same result made crucial use of a key inequality of Bruss and Delbaen (2001) and of de-Poissonization.