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

Let X 1 ,X 2 , . . . ,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 being the overall rank of the selected observation. Decisions are final: no recall is allowed. The objective is to minimize the expected payoff. In this note we give the explicit solution to this problem, known as Robbins' problem of optimal stopping, when n = 4.