Optimal stopping problem with double reservation value property

This paper considers an optimal stopping problem with uncertain recall probability where some recall cost must be paid to accept a past offer. Recall cost is an important factor leading to value deterioration of an offer. However, it is assumed in all the conventional models so far presented that the value of a past offer does not change with time, and its original value is guaranteed even if the past offer is recalled at any time in the future. In the model of this paper, the value of a past offer decreases due to the recall cost. The purpose of this paper is to propose a model with such a recall cost factor and uncertain recall probability, and to examine the properties of its optimal decision rule. One of the most distinctive results in this study is that the optimal decision rule may have a seemingly counter-intuitive property, called the DRV (Double Reservation Value) property: the critical point of whether or not to stop the search by accepting an offer is not a single entity. An interpretation of why the property may appear in this model will be given. Furthermore, this paper examines the necessary and sufficient conditions required for the property to appear, and demonstrates that the recall cost is an essential factor to the appearance of the property. To illustrate the importance of considering the recall cost factor and uncertain recall probability, this paper also provides some practical implications to a decision maker confronted with a problem of M&A (Merger & Acquisition), using an actual case of the Ford–Daewoo deal.