Almost every number chosen randomly between negative infinity and infinity would have so many digits that it would not fit on any computer currently in existence. It would not fit on a piece of paper. In the original problem, a number is written down on one piece of paper. That limits the possible numbers involved considerably.

If instead, the game was played between two supernatural beings with access to an inifinite piece of paper, and infinite time to write the numbers down, that would be a very different game.

This demonstrates that a sequence of coin flips which could be carried out in the lifetime of a human, or by a computer while the universe exists, converted to a number between 0 and 1, and then mapped to the whole number line is not random between negative infinity and positive infinity. Because of the small number of coin flips, you will never end up with any of the really big numbers. And the really big numbers is where it's at.