Sleeping Beauty is right not to cancel the bet if she is woken up on not-Wednesday (so Monday or Tuesday, but she does not know which). But this is not the optimal strategy. instead she should she roll a 4-sided die and cancel the bet if she rolls a 1. In other words she should keep the bet with 75% chance.
If she always keeps the bet, her expected payout on Wednesday is 0.5 1.5 − 0.5 1 = 0.25. That is she makes 25 cents per dollar bet, on average. But if she cancels the bet with 25% chance each time she wakes up, her expected win is 0.5 0.75 1.5 − 0.5 0.75 0.75 * 1 = 0.28125. So now she makes 28.125 cents for every dollar bet.
I think this illuminates the apparent paradox too. The fact that she is woken up on a not-Wednesday is extra information. She should lower her confidence in winning the bet, knowing the fact that she has been woken up. And she can use this information to her benefit.
If she always cancels the bet though, she destroys this extra information, since the bet will end up cancelled regardless of whether the coin came up heads or tails.
Sleeping Beauty is right not to cancel the bet if she is woken up on not-Wednesday (so Monday or Tuesday, but she does not know which). But this is not the optimal strategy. instead she should she roll a 4-sided die and cancel the bet if she rolls a 1. In other words she should keep the bet with 75% chance.
If she always keeps the bet, her expected payout on Wednesday is 0.5 1.5 − 0.5 1 = 0.25. That is she makes 25 cents per dollar bet, on average. But if she cancels the bet with 25% chance each time she wakes up, her expected win is 0.5 0.75 1.5 − 0.5 0.75 0.75 * 1 = 0.28125. So now she makes 28.125 cents for every dollar bet.
I think this illuminates the apparent paradox too. The fact that she is woken up on a not-Wednesday is extra information. She should lower her confidence in winning the bet, knowing the fact that she has been woken up. And she can use this information to her benefit.
If she always cancels the bet though, she destroys this extra information, since the bet will end up cancelled regardless of whether the coin came up heads or tails.