The problem here is that your billion people are for some reason giving the answer most likely to be correct rather than the answer most likely to actually be profitable. If they were a little more savvy, they could reason as follows:
“The scales tell me that there’s $6000 worth of coins in the jar, so it seems like a good idea to buy the jar. However, if I did not receive the largest weight estimate from the scales, my decision is irrelevant; and if I did receive the largest weight estimate, then conditioned on that it seems overwhelmingly likely that there are many fewer coins in the jar than I’d think based on that estimate—and in that case, I ought to say no.”
Ooh, and we can apply similar reasoning to the marble problem if we change it, in a seemingly isomorphic way, so that instead of making the trade based on all the responses of the people who saw a green marble, Psy-Kosh selects one of the green-marble-observers at random and considers that person’s response (this should make no difference to the outcomes, assuming that the green-marblers can’t give different responses due to no-spontaneous-symmetry-breaking and all that).
Then, conditioning on drawing a green marble, person A infers a 9⁄10 probability that the bucket contained 18 green and 2 red marbles. However, if the bucket contains 18 green marbles, person A has a 1⁄18 chance of being randomly selected given that she drew a green marble, whereas if the bucket contains 2 green marbles, she has a 1⁄2 chance of being selected. So, conditioning on her response being the one that matters as well as the green marble itself, she infers a (9:1) * (1/18)/(1/2) = (9:9) odds ratio, that is probability 1⁄2 the bucket contains 18 green marbles.
Which leaves us back at a kind of anthropic updating, except that this time it resolves the problem instead of introducing it!
The problem here is that your billion people are for some reason giving the answer most likely to be correct rather than the answer most likely to actually be profitable. If they were a little more savvy, they could reason as follows:
“The scales tell me that there’s $6000 worth of coins in the jar, so it seems like a good idea to buy the jar. However, if I did not receive the largest weight estimate from the scales, my decision is irrelevant; and if I did receive the largest weight estimate, then conditioned on that it seems overwhelmingly likely that there are many fewer coins in the jar than I’d think based on that estimate—and in that case, I ought to say no.”
Ooh, and we can apply similar reasoning to the marble problem if we change it, in a seemingly isomorphic way, so that instead of making the trade based on all the responses of the people who saw a green marble, Psy-Kosh selects one of the green-marble-observers at random and considers that person’s response (this should make no difference to the outcomes, assuming that the green-marblers can’t give different responses due to no-spontaneous-symmetry-breaking and all that).
Then, conditioning on drawing a green marble, person A infers a 9⁄10 probability that the bucket contained 18 green and 2 red marbles. However, if the bucket contains 18 green marbles, person A has a 1⁄18 chance of being randomly selected given that she drew a green marble, whereas if the bucket contains 2 green marbles, she has a 1⁄2 chance of being selected. So, conditioning on her response being the one that matters as well as the green marble itself, she infers a (9:1) * (1/18)/(1/2) = (9:9) odds ratio, that is probability 1⁄2 the bucket contains 18 green marbles.
Which leaves us back at a kind of anthropic updating, except that this time it resolves the problem instead of introducing it!