Strongly Optimal Policies in Sequential Search with Random Overlook Probabilities

Abstract

Assume a searcher is hunting for an object which has been hidden in one of $N$ regions or cells, with initial prior probability $p_i^1$ that it is in cell $i$. Suppose that to each $i$ there corresponds a sequence $\{\alpha_{ij}\}_{j \geqq 1}$ of random variables, where $\alpha_{ij}$ describes the chances that the searcher will fail to find the object on the $j$th search of $i$, given that the object is in $i$. The joint distribution of $\{\alpha_{ij}: 1 \leqq i \leqq N, j \geqq 1\}$ is known to the searcher. Under a certain monotonicity condition on the $\alpha_{ij}$'s, it is shown that to maximize the probability of finding the object in at most $n_0$ stages of search, the one-stage look ahead rule is optimal. In an earlier paper concerning a related problem, Hall assumed $\{\alpha_{1j}\}_{j \geqq 1}, \cdots, \{\alpha_{N j}\}_{j \geqq 1}$ were independent processes, whereas we allow them to be dependent. Our result is new for independent processes as well.