Thanks! I wasn't thinking too hard when I wrote that, but your example shows exactly what I had in mind. The game just 'fails earlier', than I thought.
To play a similar-ish game, you could perhaps have Alice and Bob alternate to pick open (or closed etc) balls, with the constrained that subsequent balls have to be contained in each other.
1. In an ultrametric space like the p-adics (ℚₚ), two open balls are either totally disjoint or one is a subset of the other
2. In an ultrametric space like the p-adics (ℚₚ), every ball is both open and closed (clopen)
3. The p-adics are spherically complete, which means the intersection of any sequence of nested balls is non-empty (remember balls are clopen, so it doesn't matter if the balls are "open" or "closed")
4. Let ℤₚ denote the p-adic integers (not the integers-mod-p). Then ℚₚ has a countable basis consisting of sets that look like: {q + pⁿℤₚ : q ∈ ℚ, n ∈ ℤ}
This doesn't prove anything, but, to me, the game has the "smell" of requiring an ordered field to even make sense. Then for the trick to work, the field has to be Dedekind-complete. But the real numbers are the only such field (up to isomorphism).
Overall the game feels very similar to one of Cantor's early, more analytic proofs of the fact that the reals are uncountable. Those proofs were all very tightly coupled to the analytics structure of the reals.
But in the process of writing that proof he "saw" that it didn't depend on any of the analytic stuff. He wrote up a rough version of the diagonal argument which he sent to Dedekind in a letter, who then refined it into the proof that is typically taught today.
There's something really special to me about Cantor's diagonal argument because diagonalization gets at the heart of the set-ness of (un-)countability. It might not be the most comfortable or natural, but it distills the essence of the concept.
Remember, Cantor came to the concept of (un-)countability via harmonic analysis. His early proofs were all very analytic, so it wasn't comfortable or natural to him, either — at least at first!
To play a similar-ish game, you could perhaps have Alice and Bob alternate to pick open (or closed etc) balls, with the constrained that subsequent balls have to be contained in each other.