Finite Optimal Stopping Problems: The Seller’s Perspective

Abstract

We consider a version of an optimal stopping problem, in which a customer is presented with a finite set of items, one by one. The customer is aware of the number of items in the finite set and the minimum and maximum possible value of each item, and must purchase exactly one item. When an item is presented to the customer, she or he observes its value, and determines whether to purchase the item or to permanently dismiss the item. The customer’s objective is to maximize the value of the purchased item. In this paper, we consider the problem from the perspective of the seller, who wishes to maximize profit associated with the sold item. Hence, the seller seeks an optimal sequence of items to sell, given that the customer acts according to some near-optimal decision-making rules. Our paper takes the perspective that the customer may not act optimally due to imperfect decision-making strategies and/or to the seller’s uncertainty in the items’ values to the customer. We consider max-min and max-expectation objectives when customer behavior is not completely predictable, and discuss the problem tractability in these cases.