Consider $6 iff (Green ∧ Heads) - $6 iff (Green ∧ Tails) + $4 iff Tails. This bet is equivalent to $0 + $2 = $2, so you would be willing to pay $2 for this bet.
If the coin comes out heads, the bet will become $6 iff Green, with a value of $1. If the coin comes out tails, the bet will become $4 - $6 iff Green = $4 - $3 = $1. Therefore, assuming that the outcome of the coin is revealed first, you will, with certainty, regret having payed any amount over $1 for this bet. This is not a rational decision procedure.
Consider $6 iff ((Green and Heads) or (Blue and Tails)). This is a known bet (1/3) so worth $2. But if the coin is flipped first, and comes up Heads, it becomes $6 iff Green, and if it comes up tails, it becomes $6 iff Blue, in either case worth $1. And that’s silly.
Very well done! I concede. Now that I see it, this is actually quite general.
My point wasn’t just that I had a decision procedure, but an explanation for it. And it seems that, no matter what, I would have to explain
A) Why ((Green and Heads) or (Blue and Tails)) is not a known bet, equiprobable with Red, or
B) Why I change my mind about the urn after a coin flip.
Earlier, some others suggested non-causal/magical explanations. These are still intact. If the coin is subject to the Force, then (A), and if not, then (B). I rejected that sort of thing. I thought I had an intuitive non-magical explanation. But, it doesn’t explain (B). So, FAIL.
Look at it another way.
Consider $6 iff (Green ∧ Heads) - $6 iff (Green ∧ Tails) + $4 iff Tails. This bet is equivalent to $0 + $2 = $2, so you would be willing to pay $2 for this bet.
If the coin comes out heads, the bet will become $6 iff Green, with a value of $1. If the coin comes out tails, the bet will become $4 - $6 iff Green = $4 - $3 = $1. Therefore, assuming that the outcome of the coin is revealed first, you will, with certainty, regret having payed any amount over $1 for this bet. This is not a rational decision procedure.
How about:
Consider $6 iff ((Green and Heads) or (Blue and Tails)). This is a known bet (1/3) so worth $2. But if the coin is flipped first, and comes up Heads, it becomes $6 iff Green, and if it comes up tails, it becomes $6 iff Blue, in either case worth $1. And that’s silly.
Is that the same as your objection?
Yes, that is equivalent.
Very well done! I concede. Now that I see it, this is actually quite general.
My point wasn’t just that I had a decision procedure, but an explanation for it. And it seems that, no matter what, I would have to explain
A) Why ((Green and Heads) or (Blue and Tails)) is not a known bet, equiprobable with Red, or
B) Why I change my mind about the urn after a coin flip.
Earlier, some others suggested non-causal/magical explanations. These are still intact. If the coin is subject to the Force, then (A), and if not, then (B). I rejected that sort of thing. I thought I had an intuitive non-magical explanation. But, it doesn’t explain (B). So, FAIL.