Over many repetitions (to balance out bad luck streaks), it’s rational to play the game for prices of $20 or $19. If you’re saying that SB should reject those prices when presented it while waking, that’s irrational, because an SB that accepts those prices will make money.
This is assuming that if asked twice, she’ll always make the same decision because her decision-making is deterministic. (We could assume there’s a chance she makes a different decision on Tuesday than on Monday but I think that’s an unnecessary and irrelevant complication?)
Just to be sure: in your original formulation, the bet counted only once for the two wake-up events in the Tails case, and it was canceled if she didn’t accept it on Monday and Tuesday.
That is not the scheme I described in previous comment. Each wake-up event is independent. She is awoken then asked to pay the fee now and gets the rewards now. If she pays twice she gets the rewards twice. It’s a bet that can be done, it is no less valid than yours. In this bet (when asked during a wake-up event), her gain expectation is $18. Accepting for $19 will lose money on average. Her decision-making process on the other day is irrelevant. Test it if you don’t believe it.
Over many repetitions (to balance out bad luck streaks), it’s rational to play the game for prices of $20 or $19. If you’re saying that SB should reject those prices when presented it while waking, that’s irrational, because an SB that accepts those prices will make money.
This is assuming that if asked twice, she’ll always make the same decision because her decision-making is deterministic. (We could assume there’s a chance she makes a different decision on Tuesday than on Monday but I think that’s an unnecessary and irrelevant complication?)
Just to be sure: in your original formulation, the bet counted only once for the two wake-up events in the Tails case, and it was canceled if she didn’t accept it on Monday and Tuesday.
That is not the scheme I described in previous comment. Each wake-up event is independent. She is awoken then asked to pay the fee now and gets the rewards now. If she pays twice she gets the rewards twice. It’s a bet that can be done, it is no less valid than yours.
In this bet (when asked during a wake-up event), her gain expectation is $18. Accepting for $19 will lose money on average. Her decision-making process on the other day is irrelevant. Test it if you don’t believe it.