Bit-level reasoning suggests you should flip all bits, as each bit impacts the total result in 8⁄16 cases, and in seven of those cases the coin came up tails. 7⁄8$28+1/8$4=$25>$21.
What you call ‘bit level reasoning’ just seems like bad reasoning to me. After the flip (but before I am told) I am given the option of switching, a decision which would give me
Do you agree that is also the case in the previous formulation of the problem, after you have been told you are a decider?
I don’t see a problem with either formulation. Just the solution. But it does seem to be the same mistake made with the proposed ‘yea’ solution in the previous formulation. In this case, however, the mistake appears even more obvious. So I could understand people making a bad decision on the previous formulation but a better decision this time. (If they switch here but stay ‘nay’ on the previous one then I was my hands of them and let their flawed thinking remain opaque.)
In this case, however, the mistake appears even more obvious.
I agree. The challenge is articulating why it’s a bad idea, rather than just recognizing it as such, and having an articulation that survives the transition back to the other formulation.
What you call ‘bit level reasoning’ just seems like bad reasoning to me. After the flip (but before I am told) I am given the option of switching, a decision which would give me
50% chance of gaining $7 (’0000000′ → ‘1111111’ :: $21 → $28).
50% chance of losing $17 (‘0’ → ‘1’ :: $21 → $4)
So no, I’m not swapping. Your ‘bitwise’ transition involves arbitrarily assigning too much weight to the ‘tails’ possibility.
Do you agree that is also the case in the previous formulation of the problem, after you have been told you are a decider?
I don’t see a problem with either formulation. Just the solution. But it does seem to be the same mistake made with the proposed ‘yea’ solution in the previous formulation. In this case, however, the mistake appears even more obvious. So I could understand people making a bad decision on the previous formulation but a better decision this time. (If they switch here but stay ‘nay’ on the previous one then I was my hands of them and let their flawed thinking remain opaque.)
I agree. The challenge is articulating why it’s a bad idea, rather than just recognizing it as such, and having an articulation that survives the transition back to the other formulation.