Question:
Subset Game involving Intervals and Subintervals
In the following infinite game, Alice and John take turns moving. First, Alice picks a closed interval I1 of length <1. Then, Bob picks a closed subinterval I2 which is a subset of or equal to I1, of length 1/2.
Next, Alice picks a closed subinterval I3 which is a subset of or equal to I2. The game continues in this way for infinitely many turns. At the endof the game, Alice and Bob have thus picked a sequence of intervals ...(subset of or equal to)I3(subset of or equal to)I2(subset of or equal to)I1, whose intersection consists of a single point x&infin
A referree examines x&infin. If it is rational, Alice wins. If it is irrational, Bob wins.
Does either Alice or bob have a winning strategy for the game; a rule by which Alice picks her intervals I1, I2, I3, etc, possibly depending on all the intervals previously chosen by both players, such that Alice wins the game no matter how Bob plays?
If there is a winning strategy for one player, give the strategy and prove how it works. If there is no winning strategy, explain why.
Note: Alice and Bob are not allowed to play intervals on length 0.