Sequential Search with Discounted Income, the Discount a Function of the Cell Searched

Abstract

An object is hidden in one of the cells $1,2,\cdots, R$ with probability distribution $s = (s(1), s(2),\cdots s(R))$ and remains in that cell while search is conducted. A searcher is informed of $s$ and continues search until the object is found. He is informed also of $p = (p(1), p(2),\cdots p(R))$ where $p(i)$ is the probability of finding the object if $i$ is searched and the object is in $i$. If the object is found on trial $n + 1$ its worth is discounted by the factor $\prod^R_{i=1}\beta^{N(i)}_i$, where $N(i)$ is the number of inspections of cell $i$ during the first $n$ trials and $0 \leqq \beta_i < 1, i = 1,2,\cdots, R$ is known by the searcher. For each $n$, if $s(\bar{f}(n)) = (s(1|\bar{f}(n)), s(2\mid\bar{f}(n)),\cdots,s(R|\bar{f}(n)))$ denotes the conditional location distribution, given the history $\bar{f}(n)$ of failures for $n$ trials, then it is shown that an optimal procedure selects an $i$ achieving $\overset{\max}{i}\frac{p(i)s(i\mid\bar{f}(n))}{1 - \beta_i}.$ The limiting behavior of the value is investigated as each element in any collection of components of $(\beta_1, \beta_2,\cdots, \beta_R)$ tends to one.