  For $\tau$ a stopping rule adapted to a sequence of $n$ iid observations, we
define the loss to be $\ex [ q(R_\tau)]$, where $R_j$ is the rank of the $j$th
observation, and $q$ is a nondecreasing function of the rank. This setting
covers both the best choice problem with $q(r)={\bf 1}(r>1)$, and Robbins'
problem with $q(r)=r$. As $n\to\infty$ the stopping problem acquires a limiting
form which is associated with the planar Poisson process. Inspecting the limit
we establish bounds on the stopping value and reveal qualitative features of
the optimal rule. In particular, we show that the complete history dependence
persists in the limit, thus answering a question asked by Bruss in the context
of Robbins' problem.
