The Optimal Stopping Problem in Which the Sum of the Accepted Offer's Value and the Remaining Search Budget Is an Objective Function

The paper deals with an optimal stopping problem with a finite planning horizon where an available search budget, the total amount of money that can be invested in search activities throughout the planning horizon, is limited and where both the probability of an offer being obtained at each point in time and the probability distribution function of an obtained offer's value may depend on the search cost invested at that time. The objective is to maximize the expected present discounted value of the sum of the accepted offer's value and the remaining search budget at that point. The optimal decision strategy for the problem consists of the two decision rules: an optimal investment rule, prescribing how much of the search budget to invest in search activities of each point in time and an optimal stopping rule, prescribing how to stop the search with accepting an offer. The present paper examines the properties of the optimal decision strategy analytically and numerically. The most interesting and important properties revealed here are the two: (1) In a recall model, the optimal stopping rule has a reservation value properly, and if the given search budget tends to infinity, the reservation value becomes time-independent, implying that the optimal stopping rule has a myopic properly, and (2) in both of a recall model and a no recall model, the optimal investment does not always become mon010ne in the amount of search budget remaining then, possibly increasing or decreasing drastically with its very slight change.