[OskarS]

But... but... you can't "pick a number uniformly distributed in (-inf, inf)". Like, that's not a thing that is possible [0], so saying anything about the probabilities once you've done so is not reasonable. You can pick a random number in infinite range non-uniformly (e.g. to pick a number in the range [1,inf), pick a number X in (0,1] and use 1/X), but then the probabilities are no longer so neat and tidy.

The game only makes sense if you attach limits to the numbers the other person is allowed to pick, and then the only "split" that works is in the middle of the range. At which point the game is sort-of "obviously true".

[0] https://math.stackexchange.com/a/14169/641751

[MauranKilom]

It is not necessary for the distribution to be uniform, it just has to be nonzero everywhere.

[kgwgk]

If the distribution of A and B is uniform, the probability that C from this proper distribution is between A and B is zero.

If the distribution of A and B is also a proper distribution the probability that C is between them is positive. But then the distribution of A and B has a middle point such that 50% of the numbers are above and 50% below and the problem is trivial.