The Problem of Renting versus Buying

The problem studied here is a member of a large class of on-line decision problems. The basic question is to choose from two possible alternatives at a given instance, for the use of an item. This choice is based solely on previous experience. One alternative di ers the permanent acquisition of the item (rents it) and has a small recurring xed cost (rent) associated with its use. The other choice secures the item (buys it), but has a larger, though one time, cost (purchase price) associated with it. The item is used regularly. The choices have to be made to minimize the total cost of the item. As an example, we describe the car rental problem. We present deterministic and randomized algorithms against an unrestricted adversary and analyze them. We, also, de ne a -statistical adversary and present a deterministic algorithm against this adversary. All the algorithms presented are also shown to be optimal.