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!