  In Robbins' problem of minimizing the expected rank, a finite sequence of $n$
independent, identically distributed random variables are observed sequentially
and the objective is to stop at such a time that the expected rank of the
selected variable (among the sequence of all $n$ variables) is as small as
possible. In this paper we consider an analogous problem in which the observed
random variables are the steps of a symmetric random walk. Assuming
continuously distributed step sizes, we describe the optimal stopping rules for
the cases $n=2$ and $n=3$ in two versions of the problem: a "full information"
version in which the actual steps of the random walk are disclosed to the
decision maker; and a "partial information" version in which only the relative
ranks of the positions taken by the random walk are observed. When $n=3$, the
optimal rule and expected rank depend on the distribution of the step sizes. We
give sharp bounds for the optimal expected rank in the partial information
version, and fairly sharp bounds in the full information version.

Probability