i haven't worked it through but i would approach the problem like this:

what is the probability of alice winning on turn 1?

what is the probability of the game reaching turn 3 and alice winning on that turn 3?

what is the probability of the game reaching turn 5 and alice winning on that turn 5?

what is the probability of the game reaching turn 7 and alice winning on that turn 7?

....

hence what is the probability of alice winning?

find the value of pi that makes this 0.5.