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JGWeissman comments on Outlawing Anthropics: An Updateless Dilemma - Less Wrong
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What if, instead of requiring agreement of all copies in a green room, one copy in a green room was chosen at random to make the choice?
In this case the chosen copy in the green room should update on the anthropic evidence of being chosen to make the choice. That copy had a 1/18 probability of being chosen if the coin flip came up heads, and a 1/2 probability of being chosen if the coin flip came up tails, so the odds of heads:tails should be updated from 9:1 to 1:1. This exactly canceled the anthropic evidence of being in a green room.
They are not equivalent. If one green room copy is chosen at random, then the game will be played exactly once whether the coin flip resulted in heads or tails. But if every green room copy plays, the the game will be played 18 times if the coin came up heads and 2 times if the coin came up tails.