The Ski Rental Problem
lesves.github.ioThis 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.
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.
In the secretary problem, you're trying to maximize the probability of selecting the absolutely best candidate. In other words you assume that you “win” if you select the best candidate and “lose” otherwise (even if you end up picking the second best who is almost as good!), and you're trying to maximize the probability of winning. (The optimal solution says you can win with probability 1/e ≈ 37%, meaning that ≈63% of the time you lose!)
This has always seemed the most unsatisfying assumption in the problem to me, with application to no real-life case that I can think of. The Wikipedia article has some stuff on relaxing this assumption, in its section titled “Cardinal payoff variant” (it seems that the optimal at least under one set of assumptions is √n rather than n/e, though those assumptions also seem unrealistic): https://en.wikipedia.org/w/index.php?title=Secretary_problem...
I would add that (having simulated this problem in code myself), the reason you have bad outcomes is that you run out of candidates and take a bad one because you have no choice. In real life, at some point you would grab a decent candidate even if s/he were not as good as a prior passed candidate. It is also true that even under the original assumptions, there is a wide range of thresholds around 1/e that yield a similar outcome.
I think most people learning CS101 will at some point attempt to merge-sort a stack of physical papers alphabetically, and give up half way through. Everyone should have this experience: it teaches a lot about the importance of assumptions about the problem. Not to say that the math isn't important, but you have to think critically, because spherical cows are pretty rare.
It also assumes that the sequence is random and that there is no a priori information about the quality of the candidates.
A similar problem occurs in dating today, when people tend to dismiss the current option in the hope or expectation of finding someone better later, intuitively treating it as a secretary problem. But too many people fail to take full advantage of the knowledge they have of themselves and the type of partner they can attract, of the dating market and of the world in general, and end up bitter and disappointed and victims of their own poor choices.
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.
I highly disagree with this, whats the point of dating others if you are perfectly happy with the one you are dating right now? Do you really lose out in life if there might be someone "better", even though you are happy?
I've seen stats that correlated number of previous partners with divorce risk. But I always wondered if that was a correlation rather than causation (i.e. both of those factors sharing underlying causes).
An obvious example is that a person from a culture where sex before marriage is unacceptable is also from a culture where divorce is frowned upon.
Someone from Uzbekistan told me that a girl can only date one man, and must marry him or her reputation is ruined forever. So when her now-husband wanted to date her, he had to take care to avoid making it official before they were sure to marry.
The US has a lot of subcultures, so I too doubt the usefulness of such studies.
Then again, for some leaving the subculture they know carries a very high social cost. Hopefully new generations will break some of these barbarous traditions.
it depends on how many people you expect to date in your lifetime though
these days with dating apps can prob date way more than 18..
I think it was 1/e but close enough (37%)
https://en.wikipedia.org/wiki/Secretary_problem#1.2Fe-law_of...
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. :)
Marry the next one that tangents the best experienced after 37 (or other optimal number).
> 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.
I just scrapped an ICE mower for a battery powered one. No more winterizing, changing oil, or worrying about filling with gas. I still don’t like mowing, but it sucks a little less now
Sure, but you're just deferring moderate yearly maintenance cost for a rig that will need to be totally overhauled in 5 years due to battery degradation with current battery tech offered in mowers
At least, that's the conclusion I came to this year when researching ride-on battery mowers vs ICE. Electric push mowers seem like a no brainer though
LiFePO4 will be even cheaper in 5 years, so go for it and have a project for then?
That's actually the situation I was in though. When your electric mower breaks you're probably on your own. I would not want to go back, but going forward definitely has it's own issues.
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.
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.
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.
I have relatives that have spent $60/mo for a storage unit for 12 years for some furniture they "might want to use some day if they buy a new house". Even pointing out that the furniture will likely look extremely dated by now and they've spent almost $10k so far doesn't seem to click at all.
I’ve just moved house. I have a few high quality pieces of furniture that I’d like to keep but don’t have space for, so considered storage. They are things that I’ve bought second hand and at auctions, and part of the pleasure is the hunt. So I felt I’d invested both money and time.
But then I realised that I can sell them for pretty much what I paid for them, put that money into a savings account or investments and buy them same things back in a few years. So that’s what I’m doing.
And, of course, I probably won’t buy the same things back in a few years. If I do then I’ll have all the pleasure of the hunt again. Or maybe I buy something else. Or maybe I just have the cash sitting and I can count my gold, Smaug style.
But either way I won’t have spent thousands on storage.
That realisation was extremely liberating.
Basically, don't rent a storage locker unless you have a fairly specific exit plan. There may be exceptions like I have no intention of moving out of a condo/apartment but I do sports that require fairly large equipment. But it's a good rule-of-thumb.
But now it's a sunk cost, might as well keep it going, right?
Exactly! Why cut your losses now when it’ll surely pay off in 10 more years
Yeah, that's what you think the storage locker is for.
> the S.K.I. model — Storage Kills Investment
What is this? Google doesn't return anything for me...
/s
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.
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.
Nothing wrong with the rental skis but time ain’t free
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.
also ski prices probably vary throughout the season
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
> 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.
This doesn't include the "Checked Bag fee" variable. Which is a significant component if you live somewhere you have to fly to ski.
Rent until you know what you want to buy. Done
Buy second-hand and worst case resell. Best of both worlds.
This kind of blew my mind when I realized that if you buy second hand and resell, you can own a product (often for years) for net zero dollars.
I was also delighted when I realized that instead of going through the hassle and cost of moving everything across the country, I could just sell it and buy roughly the same thing in the new place... again for net zero dollars! Teleportation!
This assumes your time is free. The used market has tons of friction.
I try to buy higher quality things, used if possible, and sell when I’m done with them. But this is primarily for ecological reasons. The complete cycle can be pretty annoying/time-consuming to the point that it’s probably economically a small loss.
Buy-side can take a lot of trawling FB/Craigslist to find an item that hasn’t been thrashed. Plus coordinating pickup time/location. This usually requires driving somewhere I wasn’t planning on going.
Sell-side you have to deal with all the flakes and ghosts, not to mention the people who will show up hours late. Giving away the thing instead seems to counterintuitively make the flake problem even worse.
If you buy and sell online the platform and shipping fees are 15-20% of item value.
Indeed, this is how it's supposed to work in an ideal world anyway. You only need to have some knowledge of the items and their tech to evaluate if they are useful second-hand or not for you. What parts wear and break and what are you able and willing to repair?
I have bought pretty much all my tools used. I usually buy low-tech solutions so my biggest concern most of the time is that the carbon brushes need to be replaced (and some models make this simple operation practically impossible, while in some models it takes under a minute). And I always look for dirty things that need a lot of cleaning since they usually go for cheap, and I'm cheap.
I've seen this described as using Craigslist/similar for long term storage.
I read an article which described the free market as storage. People sell X when there is an oversupply and buy when there is an undersupply. The guy on the other end of those transactions acts as a storage provider.
The second-hand market is a bit of a mess unless you're comparing among the most popular skis. The depreciation is also brutally steep.
You can avoid the depreciation issue with old skis, but that comes with other problems.
Ski swaps are a great place for second hand or almost second hand. I’ve been to many where the local ski shops dump their old inventory. You get pretty much pristine gear for a fraction of the price.
In many stores you can ask for "test skis". These would have been used for a few weeks as rentals and you usually get a significant discount from the normal retail price
You also get 'test bindings' that both toe and heel can be shifted forward/back to adjust to fit most boots. Disadvantage: weight. Advantage: adjustable.
On normal bindings the toe is often screwed in at one location to fit your boots and cannot be shifted later.
I never treat rented with the same love I treat my owned ones
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.
For tools, I buy the cheapest one. If (when) it fails, I replace it with one that is better along the metric that the original failed in.
For hobbies, they say "buy once, cry once", but there are so many ways to be unhappy! I won't limit myself! I say buy all, cry all, and learn all the different ways to cry. I don't ski, but for the analogy, I'd try short ones, long ones, cross country ones, racing ones, red ones, blue ones, etc and then only buy really nice ones once I understand exactly what the nice ones do that the others don't. There's a good chance that I'll learn I like skis more than I like skiing, and that's okay.
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?
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.
It's a problem from computer science, not Dr Oz.
You want to optimize for when you get home, not for your health or environment.
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.
It's a computational problem, I thought (mistakenly) that the HN crowd would understand.
Your explanation was too good, people go straight for the answer to your example. There might be a bit of cheekiness too!
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.
Suppose you repeat this every day, and every day the bus arrives a random time between "now" and in (let's say) 30 minutes.
There is a strategy that allows you to never be worse than 2x if you knew exactly when the bus arrived: Wait for 10 minutes, and if the bus didn't arrive, walk home.
In all cases when the bus arrives between now and in 10 minutes, you do the optimal thing, and whenever the bus arrives after 10 minutes have passed, you will be home after 20 minutes, which is not worse than 2x worse than optimal.
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.
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.
Why stop there and instead buy when 33% of the equipment cost is spent on renting?
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.
> 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.
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.
Are you sure that there aren't multiple tiers of rental skies in resorts in the USA? I have friends in the rental business in Norway, and they rent out skies from the kind of beginner friendly skies that would fit your description. To top of the line Stockli and Van Deer kit. I think they keep all of their gear in at least decent shape, with the top gear kept in the best shape, as it makes the most difference for those customers. Btw. haven't rented much in the Alps. But my impression is that you can rent gear for all ability levels there aswell, and that that gear is kept in good shape.
I’m not sure there aren’t, in fact I bet there are! I’m just talking about the typical case that I think most people think of.
When I was downhill skiing, there were typically regular rentals and performance rentals. I'd usually bring my own boots but, if traveling by air, usually rent the skis.
Here in NY region seasonal rentals are pretty common too. Esp for kids who will upsize every year or two.
Obviously, the skiing is not what the article is about, really.
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...
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 shop taking care of a mix of rental and competitive hardware.
>This is because they are built to go through the machine after each rental
What machine are you referring to here? It sounds like you are referring to some kind of waxing / edge sharpening / cleaning device, which would be extreme luxury compared to the rental shop at my local hill where they intake the rentals for the day and dump them back into a bucket for the next skier.
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?
I always suspected my rental skis had dicker bases! (Sorry for typo-sniping for cheap laughs.)
French living in Germany, sometimes I am mixing up things. What is interesting is that the Swiss person (probably German speaking) did not notice. Thank you for allowing to laugh after the fact :-D
This is a double bonus thread for me. I did not know (or remember from high school German 30 years ago...) that dicker meant thicker. :D
Renting for a whole season seems pretty different from renting for a weekend or whatever. I wouldn’t be surprised if it was the more expensive option than buying—but you get brand new skis every year.
Maybe for us in the US, this is more equivalent to buying every year and then reselling? Haha.
Why do we have that E[max_k alg(k)/opt(k)] is equal to max_k E[alg(k)]/opt(k) ?
What happens if you have a prior knowledge for $k$ as a probability distribution?
Is this related to the secretary problem?
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.
Or just buy the skis and sell them on the used market when you no longer need them.
Stop paying the SaaS tax.
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.
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.
Most people get hurt because they try to learn in an uncontrolled fashion and overestimate their skill.
Cycling on roads could be safer, but in the US at least, we're numb to car-caused deaths.
It seems many fun things are dangerous :)