I believe both of your computations are correct, and the fallacy lies in mixing up the payoff for the group with the payoff for the individual—which the frame of the problem as posed does suggest, with multiple identities that are actually the same person. More precisely, the probabilities for the individual are 90⁄10 , but the probabilities for the groups are 50⁄50, and if you compute payoffs for the group (+$12/-$52), you need to use the group probabilities. (It would be different if the narrator (“I”) offered the guinea pig (“you”) the $12/$52 odds individually.)
byrnema looked at the result from the group viewpoint; you get the same result when you approach it from the individual viewpoint, if done correctly, as follows:
For a single person, the correct payoff is not $12 vs. -$52, but rather ($1 minus $6/18 to reimburse the reds, making $0.67) 90% and ($1 minus $54/2 = -$26) 10%, so each of the copies of the guinea pig is going to be out of pocket by 2⁄3 0.9 + (-26) 0.1 = 0.6 − 2.6 = −2, on average.
The fallacy of Eliezer’s guinea pigs is that each of them thinks they get the $18 each time, which means that the 18 goes into his computation twice (squared) for their winnings (18 * 18⁄20). This is not a problem with antropic reasoning, but with statistics.
A distrustful individual would ask themselves, “what is the narrator getting out of it”, and realize that the narrator will see the -$12 / + $52 outcome, not the guinea pig—and that to the narrator, the 50⁄50 probability applies. Don’t mix them up!
I believe both of your computations are correct, and the fallacy lies in mixing up the payoff for the group with the payoff for the individual—which the frame of the problem as posed does suggest, with multiple identities that are actually the same person. More precisely, the probabilities for the individual are 90⁄10 , but the probabilities for the groups are 50⁄50, and if you compute payoffs for the group (+$12/-$52), you need to use the group probabilities. (It would be different if the narrator (“I”) offered the guinea pig (“you”) the $12/$52 odds individually.)
byrnema looked at the result from the group viewpoint; you get the same result when you approach it from the individual viewpoint, if done correctly, as follows:
For a single person, the correct payoff is not $12 vs. -$52, but rather ($1 minus $6/18 to reimburse the reds, making $0.67) 90% and ($1 minus $54/2 = -$26) 10%, so each of the copies of the guinea pig is going to be out of pocket by 2⁄3 0.9 + (-26) 0.1 = 0.6 − 2.6 = −2, on average.
The fallacy of Eliezer’s guinea pigs is that each of them thinks they get the $18 each time, which means that the 18 goes into his computation twice (squared) for their winnings (18 * 18⁄20). This is not a problem with antropic reasoning, but with statistics.
A distrustful individual would ask themselves, “what is the narrator getting out of it”, and realize that the narrator will see the -$12 / + $52 outcome, not the guinea pig—and that to the narrator, the 50⁄50 probability applies. Don’t mix them up!