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Alex Bilzerian
@alexbilz
'Knowing When to Stop' - Theodore P. Hill on the mathematics of optimal stopping (PDF):
gwern.net/doc/statistics
Nice! But there's no advantage to drawing R from a Gaussian versus fixing it at zero, right (absent information about how the numbers on the cards are generated)?
A fixed number won’t work because it doesn’t have some non-zero probability of always falling between any two numbers, which is what guarantees the edge.
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Super neat. But make sense once explained. I still find the intuition of Monty hall harder.
I think the best intuition of Monty Hall comes from the 100 door variant. You choose a door, 1/100 you’re right. Monty opens 98 doors leaving one left and yours. With 100 doors you can sense that Monty is concentrating the remaining probability mass onto that one door, so switch.
“sample a random number R”
The axiom of choice once again leads to totally deranged results.
The guaranteed improvement hinges on being able to draw from distribution with unbounded support. I guess it’s questionable whether that is literally possible to do in finite time? (from my understanding random number generators are likely to technically not be unbounded)
The arithmetic checks out, but how do you shake the brute intuition that a random process totally disconnected from the game shouldn’t be able to give you any useful information about it?
I believe any monotonically decreasing function p(n), that spits out the probability you switch as a function of the number you see, will do better than 50/50. (The higher the number you see, the less likely you are to switch.) A simple one is switch with 1/number probability. So
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The worst thing about this is that if you tried it in reality it would work, but that's because humans have the same shortcoming as this proof: being unable to actually construct infinite uniform distributions.
(Skip the complex math and just roll again if it's intuitively low)
𝙏𝙝𝙚 𝘾𝙖𝙪𝙩𝙞𝙤𝙪𝙨 𝘾𝙖𝙨𝙚 𝙁𝙤𝙧 𝙃𝙖𝙧𝙧𝙞𝙨
𝘈 𝘒𝘢𝘮𝘢𝘭𝘢 𝘏𝘢𝘳𝘳𝘪𝘴 𝘱𝘳𝘦𝘴𝘪𝘥𝘦𝘯𝘤𝘺 𝘸𝘪𝘭𝘭 𝘯𝘰𝘵 𝘧𝘶𝘯𝘥𝘢𝘮𝘦𝘯𝘵𝘢𝘭𝘭𝘺 𝘤𝘩𝘢𝘯𝘨𝘦 𝘵𝘩𝘦 𝘯𝘦𝘰-𝘭𝘪𝘣𝘦𝘳𝘢𝘭 𝘴𝘵𝘢𝘵𝘶𝘴 𝘲𝘶𝘰, 𝘣𝘶𝘵 𝘸𝘩𝘢𝘵 𝘸𝘪𝘯𝘴 𝘤𝘢𝘯 𝘸𝘦 𝘨𝘦𝘵?
Important note that there’s nothing special about the Gaussian distribution here. You could use any distribution that assigns a positive probability to every real interval, such as the logistic distribution.
If the numbers can be any numbers, and we don't know anything about their distribution (and cannot deduce anything since we observe only one number), the probability that any random number is between hidden numbers is zero.
I don't get it. Supposing the generating process is to pick a random number N from the reals as the first and pick N - 1 as the second. Isn't P(N > R > N - 1) = 0?
The specific amount of benefit you get might take some thought but “can you add literally even a tiny bit of edge to an already 50/50 baseline?” is intuitive enough for this solution to make sense!
This relies on finite distributions of random numbers.
If you assume uniform random, then as your min and max values approach -infinity and infinity, 1-p-q would approach zero.
Therefore, this is saying more about the choice of distributions than anything else.
If this works it violates your premise of complete unknown-ness, no? Any Gaussian you choose ought to have 0% overlap with any (arguably coherent) uniform distribution over the integers.
My conceptual problem with generating a random no. from the infinite number line is that I expect it to have an infinite no. of digits.
If they can generate any two random numbers (with absolutely no limitations) but I am limited to what number I can express given limited resources, it seems I can't effectively (with probability greater than ε) generate and evaluate a number between two such numbers.
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If you didn't specify "two different numbers" then having unknown x_1 = x_2 would make the method a waste of time.
"Two different numbers" puts enough constraint on the unknowns x_1 and x_2 so I'm not surprised this "works".
Referring to the quanta solution: Since A, B and G are unbounded, the probability of G being between A & B is infinitesimally small. Seems like a flaw in the argument (when expressed in terms of limits).
wait ok maybe i'm just not understanding how this works
"pick any R in a gaussian distribution" is this assuming you already know the distribution of possible numbers you can get? in that case it seems like it's way easier to get it 50% of the time? what am i missing
This is great but what parameters do you pick for the Gaussian distribution where you generate R? If you assume the numbers generated on the two paper slips is bounded between 0 and N, the probability of that strategy winning approaches 50% as N approaches infinity.
That’s not surprising. But I didn’t find Monty Hall unintuitive either.
Being curious, but not good at higher maths I had ChatGPT create code that I could run to test on a range between 0-1000000. Running it 100,000 times produced 69.555% success. Changing the logic to keep the first envelope when it’s above the mean yielded 69.357% success. 
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A long time ago I almost convinced myself extend that this strategy could be extended to the German tank problem with n=1.
It can't though.
I think.
Pretty sure this has the same winrate (75%) as just switching or staying respectively when the first card is less than or greater than the median.
Confused. What is the median number of a standard Gaussian?
Isn't it variable?
The kind of problem that makes you fall in love with probability. Blackwell was a genius
The real hidden info you're guessing in this problem isn't the numbers on the paper, but the range in which they were generated. Given that it's a person asking it and not a computer, and the numbers must be written on slips of paper, one can assume a small-ish range.
I dont quite understand. What's the range of possible numbers? All possible integers? 0 to 100? -1 to infinity?
I had to solve this on an exam once and had never seen it before. Didn't know it was famous, so I just solved it exactly like this.
I think you'll both enjoy this. If you haven't seen it before and want to think about it, you can read the first two images w/o spoilers.
This is contingent on the idea that the set of ways to generate random bounded numbers is a equally large infinity to the set of ways to generate a random unbounded number
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It is not possible to generate unbounded numbers.
So surely this could only work if for generating each number we assume the same limit of computational effort (including duration). Because only then can the numbers have the same bound.
Isn’t this basically how blackjack works? Same idea, slightly different parameters.
There’s an infinite number of generating distributions and an infinitesimally small number give an advantage that is not infinitesimally small. It’s a limit that approaches 0.5 from the right
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That's funny, because a method where you're looking at the revealed number and deciding, you probably have an implicit number as r, even if you choose r after anchoring to the revealed number.
Except if you don’t know anything about the numbers this fails because your probability of picking a Gaussian that encompasses both numbers is 0 given the vast number line
Why Gaussian? And presumably, centered on zero? The parameters of your R generation seem irrelevant
Super interesting how you use outside data points to create an instance for applying stopping theory.