A Nonhomogeneous Poisson Process Approach to the Optimal Selection from a Sequence of Relatively Best Objects

A fixed known number n of rankable objects appear one at a time with all n! permutations equally likely. An object is called candidate if it is relatively best. As each candidate appears, we must decide either to choose it, or reject it and continue observations until the next candidate appears. Denote by $$C_{k}$$ C k the kth to last candidate. For the one-choice problem, a reward $$\alpha _{k}$$ α k is earned if $$C_{k}$$ C k is chosen and the objective is to find a stopping rule that maximizes the expected reward of the chosen candidate. Some cases with particular reward sequences $$\left\{ \alpha _{k}\right\} $$ α k are examined in the limiting form. The two-choice problem is also considered, where the reward is $$\alpha _{i, j}$$ α i , j if $$C_{i}$$ C i and $$C_{j}$$ C j are both chosen for $$i