The Ski Rental Problem

[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.

[eterm]

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.

[svachalek]

Interesting. I once read somewhere that you should date at least... 6? people and leave before it gets too serious, before settling down with anyone. It seemed to imply there was math involved but it didn't explain. I think it must be the same statistics here, with some estimate of how many people you could meet and burn through without getting too old. I think people just don't really work this way but otherwise it makes some sense.

[pmalynin]

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

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

[jedberg]

Some would say this advice applies to finding a spouse as well. Date 37 people and keep track of the best. Then marry the next one that's better. :)

[david422]

> 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,

I bought mine, ran great for 4 years, then ran into a bunch of trouble, which made me recognize the other hidden cost of ownership is simply just maintenance. A very expensive mower just sitting there, nearest potential repair shop far away, no idea how I'd even get it there let alone the cost. And if I decide I don't want it, I've got to pay to get rid of it now too.

Luckily I was able to watch a bunch of youtube videos and order myself some parts to get it up and running again, but definitely sunk quite a bit of time and energy into it.

[mitch_f]

Derek Muller did a good overview of this concept on his Veritasium channel: https://www.youtube.com/watch?v=d6iQrh2TK98. Even more interesting for HN is a link in the description to one of the speaker's (Tom Magliery) websites: http://www.thirty-seven.org.

[matchagaucho]

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.

[Cerium]

At least yours are in the garage. I have a friend who spent years storing a couple of hundred dollars of outdoor gear in a rental storage locker costing $50 per month.

[gyomu]

> the S.K.I. model — Storage Kills Investment

What is this? Google doesn't return anything for me...

[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.

[Noumenon72]

I have rented skis perhaps 70 times in my life and I have nothing bad to say about them. Maybe a broken buckle once or twice.

[phillipcarter]

Yeah it's a fun problem but not really related to reality. Some important factors like: proximity to different ski resorts, travel plans, whether to rent demo skis or not, quality of rental skis, skiing proficiency and desire to grow, and many more all factor into the decision. Suffice to say, it's not much of an optimization problem if you're set on skiing every year.

[geokon]

its been a while since i was serious about skiing. but my impression was that when it comes to comfort, the most important factor was getting good boots

Renting skiis is okay. lets you try out a lot of different kinds. They all ride different

[bmacho]

> Oh man, I had no idea that the decision of whether to rent or buy skis required calculus to solve.

The best decision is literally a bunch of equations that you want to solve / optimize. It is sometimes school level math, but that's rare.

[trillic]

This doesn't include the "Checked Bag fee" variable. Which is a significant component if you live somewhere you have to fly to ski.

[thrawa8387336]

Rent until you know what you want to buy. Done

[xandrius]

Buy second-hand and worst case resell. Best of both worlds.

[newsclues]

This is what I recommend to people who are interested in a new expensive hobby. Try it before you buy it, make sure you love it and get an idea of what you like and plan your investment from there.

[JohnKemeny]

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?

[mikestew]

10 minutes? Always walk. Walking then becomes a known quantity, unlike your bus, and your health will benefit. And, yeesh, it’s only a ten minute walk.

[xandrius]

Always decide to walk, especially for just 10 minutes. Good for health, mental wellbeing and it's just easy.

If the question was 1h+ then maybe the answer would be different.

[porridgeraisin]

Don't you need to know the inter arrival time to solve this? I think the point is that it's a memory less distribution so you're expected to wait for the same time regardless of how long you've already waited.

[poulsbohemian]

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.

[cwmoore]

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?

[pfedak]

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.

[rzzzt]

Why stop there and instead buy when 33% of the equipment cost is spent on renting?

[comrade1234]

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.

[1659447091]

> You should own your own custom-fitted boots though

I would preach this with snowboard boots (+ helmet), made travel easier as well when you live nowhere near snow. Trying to take up skiing now and have no idea why I didn't think to do this with ski boots. Would probably help, a lot.

As for rentals it's easy to avoid beat-up janky gear. Have to go places just outside the ski town areas. Usually have to find a shop outside the resort for snowboards if you bring your own boots anyway, but easier to find better gear options. I remember getting a new (or basically new) K2 board from a general sports store in Reno, same for getting rentals in Queenstown or Denver or Vancouver before hopping on a bus.

Shops outside the resorts tend to have reasonably priced demo rentals, newer high end gear they are hoping you buy afterwards. Far better equipment that is nicely tuned than anything the resorts offer.

Not paying the oversize/ski baggage fee and lugging that gear around the whole trip while having quality rentals available levels the rent vs buy equation -- if buying a season lift pass makes sense so does buying your own gear, otherwise it's more hassle than it needs to be, imo.

[bee_rider]

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.

[amelius]

Obviously, the skiing is not what the article is about, really.

[taminka]

it's cheaper to just buy your own pair at that point, not to mention that those skis are usually beat up, and mostly beginner/intermediate level...

[leifmetcalf]

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

[armanboyaci]

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

[tantalor]

Is this related to the secretary problem?

[polivier]

Kind of, in the sense that you need to make a decision about something mid-way when there is still some unknown information ahead of you.

[rkagerer]

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

Stop paying the SaaS tax.

[Tycho]

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.

[_kyran]

There’s a large spectrum between skiing on an uncrowded slope alone, a crowded beginner trail, hucking cliffs and backcountry in avy prone terrain.

I don’t think the risk profile can be all lumped together.

Much in the same way that cycling on the road has a different risk profile to on a bike path vs downhill or freeride mountain biking.

[mritterhoff]

Cycling on roads could be safer, but in the US at least, we're numb to car-caused deaths.

[bix6]

It seems many fun things are dangerous :)