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 »

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.
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Re: Higher or Lower

Post by David Williams »

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 »

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 »

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 »

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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Re: Higher or Lower

Post by Charlie Reams »

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!
Gavin Chipper
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Re: Higher or Lower

Post by Gavin Chipper »

I thought I might give this a revisit, for a laugh.

Basically we have two competing things that both seem correct, but are also seemingly in conflict with each other.

1. We just have a single random number from a distribution we don't really know anything about and no other relevant information about it, so can have no way of knowing if another number taken from that distribution is likely to be higher or lower, so it must be 50/50.

2. It is true that for any distribution (one that's relevant for this anyway), you can pick any number (c) and if you pick a number (a) from the distribution, then more often than not, another number (b) from the distribution will be c-side of a.

What those setting the puzzle do is double down on (2). They say it's correct, so (1) must be incorrect. What those being set the puzzle do is double down on (1). It's correct so (2) must be incorrect. But there's little interaction between (1) and (2) in the discussions, so both sides go away thinking that they must still be right. (It reminds me of the aeroplane on a treadmill in that respect, though I can't remember any of the details about that.)

So what's the resolution? Well, I would suggest it might be a bit more philosophical than mathematical. And it's largely what I said in this post, but I'll make it more consise here.

Basically, in the abstract, you can pick any distribution and show that the reasoning in (2) follows. Say you pick a normal distribution with mean 4, standard deviation 5, and arbitrarily choose 6 for your "starting number". If the first number out is 4.3, that's lower than 6, so you go higher. If you played this lots of times, you'd win overall. It works for any distribution and any "starting number". So it will work for the distribution used in the actual game. Great.

But then the first number out is 4. Someone has picked out a distribution from the set of distributions that spewed out a 4. What was the thinking behind this distribution? Is this a likely thing to have happened a priori?

Analogously, if someone has 10 balls and they put them randomly into 10 boxes labelled 1 to 10, then you can say confidently that the ball in box 10 has at most a 10% chance of being the "least white". (There might be ties so it could be less than 10%.) And you can make money by betting according to these odds. But when you open the first 9 boxes and see they're all white, you no longer have this confidence. Someone chose the balls. What are the chances that they chose just one different ball versus all the same? You end up with unanswerable Bayesian stuff, and you can no longer give a clear answer. That happens as soon as you know anything about the distribution. It breaks your reasoning power.

And that happens when the 4 comes out. That solves the conflict. (1) and (2) are both right, but as soon as you see the first number, it breaks (2). (1) remains correct and you cannot win more than 50% of the time.

Fight me.
Gavin Chipper
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Re: Higher or Lower

Post by Gavin Chipper »

Yeah, but what if the guy doesn't look at the number that comes out and sets up a bot to deal with it?

Also given that it works for all distributions and all starting numbers, it will work for the one used in the game. So are we saying it would work on over 50% of occasions but just not the first one?
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