Optimal Stopping Problem with Several Search Areas

The present paper deals with an optimal stopping problem with several possible search areas in which travel costs are assumed among the areas. In terms of the future availability of an offer once obtained and passed up, the following two cases are considered: (1) it becomes instant.ly and forever unavailable and (2) it remains forever available, called a no recall model and a recall model. respect.ively. The main results obtained here are as follows: 1. Bot.h models have a reservation value property, and the reservation vallles are non decreasing in the number of periods, t, remaining up to deadline and converge as t -+ 00; 2. Their limits in both models do not always become the same, which coincide in conventional optimal stopping problems; 3. In the recall model, there may exist double critical points w. and w' (w. < w') in terms of the present offer w in the sense that, if w < w., then the opt.imal next search area is i, if w. :s tu :s w', then j i= i, and if w' < w, then again i; and 4. Suppose the travel cost is independent. of the start.ing search area. Then, in the recall model, t.he reservat.ion value is independent. of both the remaining periods and the current search area. Furthermore, in this case, the reservation values in both models converge to the same value as t -+ 00.