In the ski rental problem, the decision-maker (let’s call him Joe) is going to spend multiple days skiing. He can purchase a pair of skis for a price of $K and use them one or more days (including the day on which he purchases the skis). He can also rent a pair of skis for one day for $1. (He won't rent skis if he has purchased some.) He can rent skis as many times as he wants. The critical assumption is that Joe has no idea how many days he will go skiing; he will ski until something stops him – an injury, bad weather, a loss of interest, or something urgent at home or work, for example – but he does not put a probability distribution on the number of days that he will ski.
What is interesting is that there are two ways to model the problem. The first model assumes that, before day 1, Joe will select a day (D) on which he will purchase skis if he is still skiing that day, but he will rent skis every day until then. Thus, if he skis a total of N days, then his total cost for skis equals N if N < D, and D – 1 + K if N >= D. This version seems to capture the risky nature of the decision. Selecting a small value of D is bad if N is only slightly larger than D; selecting a large value of D is bad if N is close to but not greater than D. (Small and large are relative to K.)
The second model considers the sequential nature of the problem: if Joe rented skis on the one day, then he has a decision to make the next day (rent again or purchase). Interestingly, there seems to be a memoryless property here: because Joe has no idea how many days he will ski, then he has no idea how many more days he will ski, so the decision on every day is the same. If he decides to rent a pair of skis on Day 1, he should rent a pair of skis every day, for there is no reason to change his choice. If, on any day, he would decide to purchase a pair of skis, then he should do that on Day 1. This would imply that Joe’s only reasonable choices are to purchase a pair of skis on Day 1 or never.
I suspect that I’ve run into a situation that occurs in other infinite-horizon sequential decisions. If so, I’d love to know which model is more reasonable and why.