Choosing the maximum from a sequence with a discount function

Choosing the maximum value from a sequence ofN independent values is a well known problem often called the candidate problem or secretary problem. This paper treats the above problem with a discount penaltyα (0<α<1) for each additional observation taken. It is shown that asN increases indefinitely, the optimal stopping policy is bounded although the maximum expected payoff goes to zero, and that there exists a sequence 0=α0<α1<α2<⋯<1, such that the asymptotic optimal stopping rule is the same for allαi−1<α⩽αi.