The postdoc variant of the secretary problem

The classical secretary problem involves sequentially interviewing a pool of n applicants with the aim of hiring exactly the best one in the pool-nothing less is good enough. The optimal decision strategy is easy to describe and the probability of success is 1/e. In this paper, we consider a minor variant of this classical problem. We wish to pick not the best, but the second best (the best is going to Harvard). In this case, an explicit solution can be given both for the optimal strategy and the associated optimal success probability. The probability of success is k^∗₀(n-k^∗₀)/(n(n-1)) where k^∗₀= [n/2]. Clearly, as n goes to infinity, the probability of success tends to 1/4. Apparently, it is easier to pick the best than the second best.