I take back one piece, when I said $18 doesn’t correspond to anything. I believe that corresponds to how much Beauty should expect to be making per day on average: (1/2)(30) + (1/2)(6).
If we’re talking about expectation per iteration, though, then it should be (1/2)(30) + (1/2)(12) = $21. If you’re using the $18 value there, then you can get money pumped.
Let’s be sure we agree on the parameters of the bet. If SB agrees to bet, she pays a fee, then:
she gains $30 if the coin came up heads.
she gains $12 if the coin came up tails.
And the question is: How much should she be willing to pay to bet? (assuming she’s ok to break even)
Answer: depends on when you ask SB.
If you propose the bet to SB on Sunday or Wednesday, she should be willing to pay $21.
If you propose the bet during a wake-up event, she should be willing to pay only $18. That’s the meaning of the $18. The difference comes from the fact that when she wakes up, the coin has only 1⁄3 chance of having landed heads. That’s where the thirder position comes from. Test it if you don’t believe it checks out.
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.
I take back one piece, when I said $18 doesn’t correspond to anything. I believe that corresponds to how much Beauty should expect to be making per day on average: (1/2)(30) + (1/2)(6).
If we’re talking about expectation per iteration, though, then it should be (1/2)(30) + (1/2)(12) = $21. If you’re using the $18 value there, then you can get money pumped.
Let’s be sure we agree on the parameters of the bet. If SB agrees to bet, she pays a fee, then:
she gains $30 if the coin came up heads.
she gains $12 if the coin came up tails.
And the question is: How much should she be willing to pay to bet? (assuming she’s ok to break even)
Answer: depends on when you ask SB.
If you propose the bet to SB on Sunday or Wednesday, she should be willing to pay $21.
If you propose the bet during a wake-up event, she should be willing to pay only $18. That’s the meaning of the $18. The difference comes from the fact that when she wakes up, the coin has only 1⁄3 chance of having landed heads. That’s where the thirder position comes from. Test it if you don’t believe it checks out.
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.