Sophie - Musings on a math problem before sleep
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Date: 2012-01-02 07:18
Security: Public
xposthttp://soph.livejournal.com/228251.html
Tags:big posts, math, steam
Subject: Musings on a math problem before sleep

I had problems getting to sleep today, because I was thinking about an interesting mathematical problem. My brain isn't going to let me sleep unless I post about it here.

For the past two weeks, Steam (the game distribution platform run by Valve) has been having a sale, as it has been doing every year at this time. This time around, there are also giveaways to encourage people to play the games it has on sale. These take the form of extra achievements for certain games. When you manage to get one of these achievements, you'll get something for your trouble. Prizes come from what Valve calls "The Great Gift Pile", and it includes games and coupons.

However, there's only a 25% chance that you'll actually get something from the GGP (as I'll call it from now on). 75% of the time, you'll receive coal instead. Coal isn't completely useless, though; if you collect 7 coal, you can 'craft' them into a random item from the GGP. In addition, each piece of coal that remains in your inventory at the end of the promotion (which is tomorrow) will count as one entry into the "Epic Holiday Giveaway", which is a big raffle which takes place at the end. The prizes for this raffle can be seen at the bottom of http://store.steampowered.com/holidaysale/details . Since the page is no longer online, the prizes were as follows:

  • Grand Prize (1 winner): Every single game on Steam
  • First Prize (50 winners): Top 10 items on wishlist
  • Second Prize (100 winners): Top 5 items on wishlist
  • Third Prize (1,000 winners): Valve Complete Pack
The Steam forums have had an interesting discussion going on about this last raffle. One person (whose name was "X01" on the forums) casually mentioned how having 100 coal would hardly increase your chances to win, and that you wouldn't be 100 times more likely to win. Another person replied to say that 100/<number of entries> is 100 times bigger than 1/<number of entries>, so yes, you would be 100 times more likely to win. From there, there was a big argument between X01 insisting they were correct, and most of the other people on the forum saying that, c'mon, this is basic math. It eventually ended up with X01 deleting their comments in anger and disappearing from the thread.

(If you're interested, this is the thread in question. The fun begins at post #9, but remember while reading that X01 deleted their posts, so the quotes are all there is to go on.)

At first, the answer seems really obvious. After all, with 100 coal you're 100 times more likely to win than someone who only has 1 coal... right?


The answer to the question above as stated is that yes, if you have 100 coal then you are 100 times more likely to win than someone who has 1 coal. Thus, the people arguing against X01 are right.

However, you are not 100 times more likely to win than if you only had 1 coal. Thus, X01 is also right. (There is a caveat to this, but I'll cover it later in this post.)

The thing is, the problem was never clearly stated, which is why the discussion ballooned. In reality, both groups were absolutely correct because they each viewed the problem differently:
  • X01 viewed the problem as "If you have 100 coal, are you 100 times more likely to win than if you only had 1 coal?".
  • Most other people viewed it as "If you have 100 coal, are you 100 times more likely to win than someone else who only has 1 coal?"
The reason the distinction is important is that nothing you do can affect the amount of coal anybody else gets. If someone else has 100 coal at the end of the promotion, then that can be held as always true in any calculations you make. In contrast, the amount of coal *you* get is dependent on your actions, and is thus variable. Both figures, however, contribute to the total number of entries into the raffle, which is determined by how much coal has been given out across the entire userbase.

Now, let's assume that it's the night before the raffle, and there have been 10,000,000 pieces of coal given out. Of those, one person has 100 coal and everybody else has 1 coal. From there, it's easy to conclude that if nothing changes in the following night, then the person with 100 coal is 100 times more likely to win than one of the people with 1 coal, because 100/10,000,000 is 100 times greater than 1/10,000,000.

Now, though, let's say that one of the people who only had 1 coal manages to somehow get 99 coal overnight just before the raffle ends. Thus, the total amount of coal is now 1,000,099. Is this person 100 times more likely to win than if they only had 1 coal? No, because now you're comparing 100/10,000,099 to 1/10,000,000. That's not precisely 100 times bigger; it's actually only approximately 99.99901 times bigger.

So, to conclude - both groups are absolutely right, but in reality the difference is so tiny that you may as well just call it 100 times and be done with it.

Now that I've typed all that out, I'm going to see if my brain will let me sleep now. Good night. :)

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cxcvi
User: [personal profile] cxcvi
Date: 2012-01-02 09:24 (UTC)
Subject: (no subject)

Replying with trying to not look at your answer:

For small amounts of total coal (let's go extreme and say that you have 1, they have 1), you go from 1/2 to 2/3 to 3/4... all the way up to 100/101. This becomes rapidly dimishing returns.

For large amounts of total coal (let's say that there's a million other pieces out there; I suspect that this is on the low side of how many there actually are), your chances go from 1/1000001 to 100/1000100, which is essentially 100 times more, as by this point, the denominator is essentially static.

Given the number of people on Steam is high, the amount of total coal will also be high, and so therefore, X01 was wrong.

However, given that the non-coal prizes doesn't seem like they're worth anything (I'm not reading the forums here, only going by the 4 discount coupons sitting in my inventory that I'm probably never going to use), I'm not going to be crafting my coal...

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cxcvi
User: [personal profile] cxcvi
Date: 2012-01-02 10:12 (UTC)
Subject: (no subject)

I'm not convinced that that's the end of the story, however.

The reason the distinction is important is that nothing you do can affect the amount of coal anybody else gets.

However, groupthink dictates that it will be somewhere around (amount of coal that someone can reasonably get, without having to buy lots of games) * (number of people registered on Steam who actually care about the raffle).

Say, for instance, that I'm holding a raffle for something, and that there's a single top prize that doesn't grow in value with the amount of tickets bought. Let's also say that there are 10 other people, and that as the organiser, I'm not going to buy any tickets (to make it appear fair, you understand)...

The first 8 people all come to my table, and buy a single ticket. Person 9 sees this, and then decides that they're going to buy 10 tickets (it's a nice round number, and at that point, it gives them about a 56% chance of winning). Person 10 see this then, and then realises that, to be competitive, in terms of chances to win the raffle, that they are going to need to buy more than 1 ticket. So they also buy 10. The other eight people all realise this, and then soon they have 10 tickets each, as well. Or more, if they're being greedy. But they all also realise

This escalating number of tickets only stops when people decide that they're not going to buy any more tickets. Either because "I can't afford it" (including variations of "my partner won't let me spend that much"), or because the cost to buy that many tickets has exceeded the cost or value of the prize in the first place (much less likely), with a remote chance of someone realising that everyone still pretty much has the same chance as everyone else of winning at this point, and it's senseless that they've had to buy more than one ticket to reach this point, and then convincing everyone else about this; or someone realising that I haven't stated where the money from the tickets is going, and so therefore I'm only legally obliged to give 20% to charity, and that the other 80% is probably going straight into my pocket (this is why the UK has laws about these things, I suspect...)

At least with Steam, the first few tickets have very little cost. Although there's no way in hell I'm linking my account to failbook for another ticket...

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who needs to think when your feet just go
User: [personal profile] jd
Date: 2012-01-03 07:56 (UTC)
Subject: (no subject)

Some of the non-coal prizes were actual games, so I crafted my (single set, I only managed to earn 8) coal in hopes of getting another.

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User: [personal profile] rho
Date: 2012-01-02 11:04 (UTC)
Subject: (no subject)

In the limit as the number of coal owned by other people tends to infinity, the ratio of winning probabilities for owning 1 coal and 100 coal does, indeed, tend to exactly 100. Of course, in the same limit, the probability of actually winning tends towards zero.

Of course, the actual number of entries isn't infinite, but they are large enough that you can happily state that you improve your odds by almost exactly 100 times if you have 100 coal, but that your chance of winning is still almost exactly 0.

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metawidget
User: [personal profile] metawidget
Date: 2012-01-02 13:02 (UTC)
Subject: (no subject)

If you like this problem, then you may like design-based finite-population sampling and estimation. That question of what process you're taking the limit of comes up a lot.

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