One step futher: an explicit solution to Robbins' problem when $n=4$

Fix some $n \in \mathbb{N}$ and let $X_1, X_2,\dots, X_n$ be independent random variables drawn from the uniform distribution on $[0,1]$. A decision maker is shown the variables sequentially and, after each observation, must decide whether or not to keep the current one, with payoff the overall rank of the selected observation. Decisions are final: no recall is allowed, no regret is tolerated. The objective is to act in such a way as to minimise the expected payoff. In this note we give the explicit solution to this problem, known as Robbins' problem of optimal stopping, when $n=4$.