Imagine a really simple game where there are only 2 moves: saying 1 and saying 0. The player who first says 1 wins. So of course the player who has the first turn will in practice always win, but when both players move randomly it's 50/50.
Not true. Assuming both players move completely random (both choose either 1 or 0 with 50% probability each) and independently (past actions don't influence the current event) player 1 has a 2:1 advantage, i.e. he will win with probability 2/3. This is because player 1 can win after an even number of rounds (including 0) with 50% probability. So his total chance of winning is 1/2 * (1 + 1/4 + 1/16 + 1/64 + ...) = 1/2 * 4/3 = 2/3.
