Higher or Lower

Cerebral distractions of every kind, mostly but not exclusively Countdown-related.

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Gavin Chipper
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Re: Higher or Lower

Post by Gavin Chipper » Tue May 19, 2020 5:40 pm

Paul Howe wrote:
Tue May 19, 2020 4:21 pm
There is no way to know the size of your advantage p, and it may indeed be incredibly small, but it is definitively positive, there is no limit style argument that makes it vanish to zero.
Are you sure about this? I would say that the more "Godlike" (basically less limited to using "small numbers") your adversary becomes, the further away from your pick the median of the distribution is likely to be, and therefore the less effective your pick becomes.

But then I suppose the standard deviation could also increase with this, which perhaps cancels this out (to some extent? Completely?) I suppose it depends on the subjective nature of increasingly advanced adversaries and how they choose to play the game. But certainly they could play it so that it vanishes to zero in the limit - by increasing the range of the possible median point but also keeping the standard deviation low.

For example, all beings use a normal distribution for the main distribution. But they pick the mean of the distribution randomly first using a separate sub-distribution, and always use a standard deviation of 1 for the main distribution. The sub-distribution is just a uniform distribution between -x and x with x corresponding to the being number. So:

Being 1 uses a sub-distribution that is a uniform distribution between -1 and 1 to pick the mean for their main distribution (which is a normal distribution with standard deviation 1).

Being 2 uses a sub-distribution that is a uniform distribution between -2 and 2 to pick the mean for their main distribution (which is a normal distribution with standard deviation 1, not 2).

Being 3 uses a sub-distribution that is a uniform distribution between -3 and 3 to pick the mean for their main distribution (which is a normal distribution with standard deviation 1).

And so on. The higher up the beings you go, the less effective the picking strategy becomes, with it being useless in the limit.

Obviously you can't have this case where your probability becomes exactly 0.5, but you can get arbitrarily close to it.

David Williams
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Re: Higher or Lower

Post by David Williams » Tue May 19, 2020 8:02 pm

David Williams wrote:
Fri Feb 01, 2008 2:38 pm
Surely the whole essence of the original problem was that the numbers were selected randomly from a boundless selection. If there is some sort of probability distribution it's a different matter altogether.
I knew I should have read at least some of this before I started again.

Gavin Chipper
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Re: Higher or Lower

Post by Gavin Chipper » Tue May 19, 2020 9:24 pm

Gavin Chipper wrote:
Tue May 19, 2020 5:40 pm
I would say that the more "Godlike" (basically less limited to using "small numbers") your adversary becomes
Etc. Well, to add to this, I would say that the more Godlike your adversary becomes, the more the distribution should become like a uniform distribution across the real numbers (God wouldn't arbitrarily favour some numbers over others as in my previous example). Achieving this exactly seems to be impossible, but you can have beings that iteratively get closer to it - using something similar to what I did with the normal distribution in my last post. This way we wouldn't have to worry about keeping the standard deviation small and the mean big as I did with the normal distribution, because we no longer have these variables. If we do this, then as we work our way up to more Godlike beings, your probability of winning should approach 0.5.

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Re: Higher or Lower

Post by Paul Howe » Wed May 20, 2020 4:57 pm

Let's fix some terminology - the "dealer" in the game picks two numbers a,b from distribution X. The "player" picks c from distribution Y and plays according to the strategy described above (i.e guess b>a if c>a, guess b<a if c<a)
Gavin Chipper wrote:
Tue May 19, 2020 5:40 pm
Paul Howe wrote:
Tue May 19, 2020 4:21 pm
There is no way to know the size of your advantage p, and it may indeed be incredibly small, but it is definitively positive, there is no limit style argument that makes it vanish to zero.
Are you sure about this? I would say that the more "Godlike" (basically less limited to using "small numbers") your adversary becomes, the further away from your pick the median of the distribution is likely to be, and therefore the less effective your pick becomes.
Yes. The player's advantage is strictly positive under very trivial conditions - the only thing needed to guarantee the result is a non-zero probability of c falling between a and b, i.e. at least one of X or Y must have non-zero probability density everywhere. The player can guarantee it through their choice of Y alone. Certainly the advantage could be very small indeed, but not infinitesimally so. If the game is played perpetually with any choice of X and Y satisfying these basic conditions, the player will always eventually come out ahead, regardless of the size of his advantage (and even if X is of divine provenance!)

So let's say the dealer fixes some probability, e.g. 50.000001%, and asks "is there a distribution X that is guaranteed to reduce the win probability below this level, regardless of the player's choice of Y". In one sense the answer is no, e.g. if the player happened to choose Y=X, then they would do very nicely indeed, regardless of how localised or otherwise complicated X is. Of course that's obviously a cheat as the player has no knowledge of X and is cosmically unlikely to pick Y=X. But this guards against the idea that you can find a sequence of distributions X_1, X_2, ..., X_n,... for which the win probability is guaranteed to approach 1/2. Such a sequence cannot exist independently of the player's choice of Y. Your question probably lives in a setting where X and Y themselves are in some sense "random", and I have a sense that's not an easy space in which to answer even basic questions (e.g. if you think about how you might actually go about choosing Y "at random" from, say, the set of all continuous pdfs that are non-zero everywhere, you'll see it's not an easy thing to formalise).

Ultimately, most of the complication here just seems less interesting than the very straightforward underlying result, i.e. that the player has a guaranteed advantage in this game merely from making a single observation from X. Doesn't feel like it should be true, but it is.
Gavin Chipper wrote:
Tue May 19, 2020 9:24 pm
Gavin Chipper wrote:
Tue May 19, 2020 5:40 pm
I would say that the more "Godlike" (basically less limited to using "small numbers") your adversary becomes
Etc. Well, to add to this, I would say that the more Godlike your adversary becomes, the more the distribution should become like a uniform distribution across the real numbers (God wouldn't arbitrarily favour some numbers over others as in my previous example). Achieving this exactly seems to be impossible, but you can have beings that iteratively get closer to it - using something similar to what I did with the normal distribution in my last post. This way we wouldn't have to worry about keeping the standard deviation small and the mean big as I did with the normal distribution, because we no longer have these variables. If we do this, then as we work our way up to more Godlike beings, your probability of winning should approach 0.5.


So you define a uniform distribution with pdf 1/2x on [-x,x) and 0 outside of this, and claim the limting distribution is what you get as x goes to infinity. But in the limit this assigns zero probability to everything. You're being told that the set of real numbers is so vast that you can't assign the same probability to everything, you simply have to make some events more likely than others. You might as well say 2+2=5, as god surely wouldn't arbitrarily favour 2+2 to equal 4, and go with whatever nonsense follows from that. You can prove everything and nothing in such an environment!

Gavin Chipper
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Re: Higher or Lower

Post by Gavin Chipper » Wed May 20, 2020 5:36 pm

Fair enough. It definitely works for any specific distribution, but I wondered if that itself might not be enough and whether it would always be down to some sort of bias in the distribution, caused by it being set by a human. But if I'm defining bias as any distribution that isn't a uniform distribution across the reals, and such a distribution is impossible, then that would make an unbiased distribution impossible, so it would negate my argument. But then if God would play the game in an unbiased way, then this proves the non-existence of God.

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Charlie Reams
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Re: Higher or Lower

Post by Charlie Reams » Wed May 20, 2020 10:02 pm

I suggest using the probability distribution of human ages, which of course has zero density above 120 years. Finally something we can all agree on!

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