Game Theory Q: Proving Player Win w/Axiom of Choice

[dhong]

Hey PF!

Can you help me with something:

Players alternately choose 0's or 1's. A play of this infinite game is thus a sequence of 0's and 1's. Such a sequence can be considered as the binary expansion of a real number between 0 and 1. Given a set $E$ of real numbers satisfying $0 < x < 1 \forall x \in E$, say that player 1 wins if the play corresponds to a number in $E$ and player two wins if the way corresponds to a number in $[0,1] \backslash E$.

Evidently the Axiom of Choice implies there exists a set $E$ for which the game has no value. Can you help me out with showing this?

 

[Erland]

1. What about ambiguities similar to 0.99999...=1, for example the two sequences 011111... and 100000, which both correspond to the real number 0.1?

2. If this ambiguity is resolved, it is certain that either player 1 or player 2 wins, since every real number in [0,1] lies in either E or its complement. But you meant perhaps something else with the "value" of the game?