I didn’t fully understand OP’s argument, but I used a different approach that feels simpler and more intuitive to me.
First, I agree that there is no paradox: if, upon waking up in a green room, you are offered to bet $1:$1 on whether the coin came “heads”, then you should take the bet: after pooling your gains and losses, you and your clones will have gained (1/2 * 18 * 1$) - (1/2* 2 * $1) = $8. There is no paradox with this bet.
So what’s different in Eliezer’s bet? That a single bet is made for the entirety of the clones who wake up in a green room. When the coin comes “heads”, there might be 18 clones in green rooms, but there’s not 18 bets, only a single one.
To check my intuition, I kept Eliezer’s version, but simplified the bet structure: if you and your other clones in green rooms unanimously agree, I will give $1 to your collective of clones if the coin came “heads” and take $1 if it came “tails”. Well then it’s a fair bet, the expected value is (1/2 * $1) - (1/2 * $1) = $0.
The way I see it, since all clones will behave the same way, it is no different from asking a single one chosen at random among green rooms to decide. Imagine the following variant: after waking up the clones, I pick one at random among those who are in a green room, and I offer that one to decide whether to pay all clones in green rooms $1 and take $3 from all clones in red rooms. I don’t contact any other clone.
Well now the gain in probability mass from waking up in a green room is exactly counterbalanced by the loss of probability mass of being chosen: you have 9 times more green rooms in the “heads” case than in the “tails” case, but a clone in a green room is 9 times less likely to be picked in the “heads” case than in the “tails” case. Accepting the bet in this variant gives the same general utility as in Eliezer’s.
I didn’t fully understand OP’s argument, but I used a different approach that feels simpler and more intuitive to me.
First, I agree that there is no paradox: if, upon waking up in a green room, you are offered to bet $1:$1 on whether the coin came “heads”, then you should take the bet: after pooling your gains and losses, you and your clones will have gained (1/2 * 18 * 1$) - (1/2* 2 * $1) = $8. There is no paradox with this bet.
So what’s different in Eliezer’s bet? That a single bet is made for the entirety of the clones who wake up in a green room. When the coin comes “heads”, there might be 18 clones in green rooms, but there’s not 18 bets, only a single one.
To check my intuition, I kept Eliezer’s version, but simplified the bet structure: if you and your other clones in green rooms unanimously agree, I will give $1 to your collective of clones if the coin came “heads” and take $1 if it came “tails”. Well then it’s a fair bet, the expected value is (1/2 * $1) - (1/2 * $1) = $0.
The way I see it, since all clones will behave the same way, it is no different from asking a single one chosen at random among green rooms to decide.
Imagine the following variant: after waking up the clones, I pick one at random among those who are in a green room, and I offer that one to decide whether to pay all clones in green rooms $1 and take $3 from all clones in red rooms. I don’t contact any other clone.
Well now the gain in probability mass from waking up in a green room is exactly counterbalanced by the loss of probability mass of being chosen: you have 9 times more green rooms in the “heads” case than in the “tails” case, but a clone in a green room is 9 times less likely to be picked in the “heads” case than in the “tails” case. Accepting the bet in this variant gives the same general utility as in Eliezer’s.