Rank-based selection strategies for the random walk process

In many decision situations such as hiring a secretary, selling an asset, 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 given random walk process of n observations, we explicitly compute the probability that the j-th observation in the sequence is the maximum or minimum among all n observations. Based on the probability distribution of the rank, we derive several distribution-free selection strategies under which the decision maker's expected utility of selecting the best choice is maximized. We show that, unlike in the classical secretary problem, evaluating more choices in the random walk process does not increase the likelihood of successfully selecting the best.