In many decision situations such as hiring a secretary, selling an ass
et, or seeking a job, the value of each offer, applicant, or choice is
assumed to be an independent, identically distributed random variable
. In this paper, we consider a special case where the observations are
auto-correlated as in the random walk model for stock prices. For a g
iven random walk process of n observations, we explicitly compute the
probability that the j-th observation in the sequence is the maximum o
r minimum among all n observations. Based on the probability distribut
ion of the rank, we derive several distribution-free selection strateg
ies under which the decision maker's expected utility of selecting the
best choice is maximized. We show that, unlike in the classical secre
tary problem, evaluating more choices in the random walk process does
not increase the likelihood of successfully selecting the best.