The Ski Rental Problem (lesves.github.io)

Do like many people in Switzerland and just rent skis for the full season. That way you get a new pair every year. You should own your own custom-fitted boots though.

Is this related to the secretary problem?

Maybe the relatable concept is just a stepladder to the general ongoing scenario, eg. modeling all consumers from a retailer’s perspective. Otherwise, the continuous to discrete assumption reads as a hand-wavy fiat.

Could someone who groks this math tell me why not buy the skis once you’ve paid half their price on rentals?

Oh man, I had no idea that the decision of whether to rent or buy skis required calculus to solve. I just figured that if you ski more than say, 3 times a season, it's probably better to own your own gear for reasons unrelated to the entry cost, but more to do with comfort, tuning, quality, and so on. Anyone who has rented skis knows that the rental fleets are trashed.

What happens if you have a prior knowledge for $k$ as a probability distribution?

This feels very similar to the “radio” or “restaurant” problem:

You’re driving down the street trying to decide which restaurant to stop at (or scanning through the radio trying to decide which song to stop on).

If you stop at the first, there’s a good chance something better is ahead. But if you wait too long then you risk getting stuck with something you don’t really like (the problem assumes you can’t go back).

If I remember correctly, mathematically you skip the first 1/3, but keep track of your “best”. Then stop at the next option that’s >= than your current best or maybe the next thing you like.

With respect to skis, I have the same issue every year with a ride on lawn mower. Do I just pay someone weekly or buy one outright and do it myself? In this case I loathe mowing, so I don’t mind paying. But with skis it’s a question of just how much I’ll ski after this stretch, regardless of whether or not this stretch is 1 or 20 days. Because there are additional costs (and benefits) to ownership beyond the initial purchase.

Rent until you know what you want to buy. Done

Every time my wife brings up the skis collecting dust in our garage, I’m reminded of the S.K.I. model — Storage Kills Investment.

Here's a different version of the problem.

It takes 10 minutes to walk home from the bus central. The bus is late but should be here any minute now. The bus takes one minute. Do you wait or walk?

Skiing is incredibly fun but I wonder if it should be put in the same category as cycling (on roads): too dangerous to be sane.

Or just buy the skis and sell them on the used market when you no longer need them.

Stop paying the SaaS tax.

Why do we have that E[max_k alg(k)/opt(k)] is equal to max_k E[alg(k)]/opt(k) ?

This is kind of an interesting problem, but it overlooks another variable, at least in the case of skis - it's not just how many days I'm going to use them this year, but also for the next few years. Yes, there are people who buy new skis regularly, but more commonly the person that makes the buy vs rent decision decides that over the next multiple seasons they intend to ski enough to justify the buy decision. This is especially true if you are buying new skis rather than say, rental skis at the end of the season (think kinda like buying a used car that has been depreciated, you can buy used skis that still have a lot of miles...). So my point is simply that the real world problem is actually even more interesting than this hypothetical.