To put it another way: the reason that Beauty should guess tails in my first scenario above, is that we’re rewarding her twice as much for correctly guessing tails than for correctly guessing heads! It’s just exactly the same thing as if I offered you a reward for guessing the outcome of a simple coin flip, and paid you $2 if the coin landed tails (and you guessed right), but only $1 if it landed heads (and you guessed right). Of course you should guess tails!
To put it another way: the reason that Beauty should guess tails in my first scenario above, is that we’re rewarding her twice as much for correctly guessing tails than for correctly guessing heads! It’s just exactly the same thing as if I offered you a reward for guessing the outcome of a simple coin flip, and paid you $2 if the coin landed tails (and you guessed right), but only $1 if it landed heads (and you guessed right). Of course you should guess tails!