Modeling the search for the least costly opportunity

With the continuing growth in the number of opportunities available at virtual stores over the Internet there is also a growing demand for the services of computer programs capable of scanning a large number of stores in a very short time. We assume that the cost associated with each scan is linear in the number of stores scanned, and that the resulting list of price quotes is not always satisfactory to the customer, in which case an additional scan is performed, and so on. In such a reality the customer, wishing to minimize her expected cost, must specify the requested sample size and a rule (control limit) to stop the search. In the context of search theory, the above search model can be categorized as "fixed-sample-size, sequential, with infinite horizon". According to this model the expected search cost is a function of two decision variables: the sample size and the control limit. We prove that for arbitrary sample size the expected search cost is either quasi-convex or strictly decreasing in the control limit, and that the optimal expected search cost is quasi-convex in the sample size. These properties allow an efficient calculation of the optimal policy. We also develop analytic formulas to calculate the cost's variance, allowing customers to choose a slightly higher expected cost if there is a considerable decrease in the variance. Finally, we present detailed examples for price quotes that are distributed uniformly or exponentially.