Your Game 2 is poorly defined. What does it mean to "guess correctly" when you're asked to guess twice? If you systematically take the first (or the last) answe

Your Game 2 is poorly defined. What does it mean to "guess correctly" when you're asked to guess twice? If you systematically take the first (or the last) answer, then it's equivalent to being awaken only once on Tails. If you take the logical OR of the correctness of two guesses, then the sleeping beauty can win 5/8 of the time by answering randomly, which is better than 1/2 by guessing a fixed answer (the only strategy you analyzed).