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The Ski Rental Problem

lesves.github.io

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Re: The Ski Rental Problem

#11

Post by rkomorn »

Loic wrote: post #10
tempay wrote: post #9

Earlier quoted context omitted.

In Switzerland it varies. Many places will also offer expert hire where you get brand-new skis and at the end of the season you can choose to buy them for cost.

FYI, the rental skis, even if they look the same as the same skis you could buy retail are not the same. They have bigger edges and a dicker base. The bindings are not the same. This is because they are built to go through the machine after each rental. Good retails skis have less "robust" but faster, thinner base, they would be dead after 3 months of rental. Source: I spend way too many hours each season in a ski sh…

I always suspected my rental skis had dicker bases! (Sorry for typo-sniping for cheap laughs.)

Re: The Ski Rental Problem

#12

Post by pfedak »

cwmoore wrote: post #7

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?

Another aspect of the solution that makes it rather abstract is it effectively assumes we know nothing about the distribution of the number of days.

Paying at 1/2 will be optimal if it ends before you buy, very bad (3x optimal) if it ends right after you buy, and slightly better than the solution in the post if it lasts at least twice that long (1.5x optimal vs e/(e-1)).

The metric in the post is just the worst of those ratios. Assuming the unproven statement in the post (that the solution which is a constant factor worse than optimal is best), any solution of the form you suggest is going to have similar tradeoffs. If we had a distribution, we could choose.

Re: The Ski Rental Problem

#13

Post by lisbbb »

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.

Re: The Ski Rental Problem

#15

Post by massung »

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.

Re: The Ski Rental Problem

#16

Post by pmalynin »

massung wrote: post #15

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 think it was 1/e but close enough (37%)

https://en.wikipedia.org/wiki/Secretary_problem#1.2Fe-law_of...

Re: The Ski Rental Problem

#18

Post by bee_rider »

Loic wrote: post #10
tempay wrote: post #9

Earlier quoted context omitted.

In Switzerland it varies. Many places will also offer expert hire where you get brand-new skis and at the end of the season you can choose to buy them for cost.

FYI, the rental skis, even if they look the same as the same skis you could buy retail are not the same. They have bigger edges and a dicker base. The bindings are not the same. This is because they are built to go through the machine after each rental. Good retails skis have less "robust" but faster, thinner base, they would be dead after 3 months of rental. Source: I spend way too many hours each season in a ski sh…

I’ve rented skis in the US for a weekend out skiing, and of course they are usually pretty beat up (to be expected), and I don’t expect any amazing performance (wasted on me anyway). But, they are taking about something that seems a bit different, renting skis for a whole season. Almost seems more like leasing a car vs renting one?

Re: The Ski Rental Problem

#19

Post by eterm »

massung wrote: post #15

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…

It's known to wikipedia as the "Secretary Problem":

https://en.wikipedia.org/wiki/Secretary_problem

The optimum is actually based on 1/e rather than 1/3 but 1/3 is a good enough practical approximation.

Re: The Ski Rental Problem

#20

Post by bee_rider »

comrade1234 wrote: post #2

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.

When people in the US think of rental skis, we think of weekend rentals that are usually not very specialized and pretty beat up.

What you describe here sounds more like leasing a car vs renting one—technically a lease is a rental, but practically it is a bit closer to owning the thing.

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