I agree with you that the reason why you can’t use the 90⁄10 prior is because the decision never depends on a person in a red room.
In Eliezer’s description of the problem above, he tells each green roomer that he asks all the green roomers if they want him to go ahead with a money distribution scheme, and they must be unanimous or there is a penalty.
I think this is a nice pedogogical component that helps a person understand the dilemma, but I would like to emphasize here (even if you’re aware of it) that it is completely superfluous to the mechanics of the problem. It doesn’t make any difference if Eliezer bases his action on the answer of one green roomer or all of them.
For one thing, all green roomer answers will be unanimous because they all have the same information and are asked the same complicated question.
And, more to the point, even if just one green roomer is asked, the dilemma still exists that he can’t use his prior that heads was probably flipped.
[EDIT:] Although I would be a bit more general: regardless of red rooms: if you have several actors, even if they necessarily make the same decision they have to analyze the global picture. The only situation when the agent should be allowed to make the simplified subjective Bayesian decision table analysis if he is the only actor (no copies, etc. It is easy to construct simple decision problems without “red rooms”: Where each of the actors have some control over the outcome and none of them can make the analysis for itself only but have to buid a model of the whole situation to make the globally optimal decision.)
However, I did not imply in any way that the penalty matters. (At least, as long as the agents are sane and don’t start to flip non-logical coins) The global analysis of the payoff may clearly disregard the penalty case if it’s impossible for that specific protocol. The only requirement is that the expected value calculation must be made protocol by protocol basis.
My intuition says that this is qualitatively different. If the agent knows that only one green roomer will be asked the question, then upon waking up in a green room the agent thinks “with 90% probability, there are 18 of me in green rooms and 2 of me in red rooms.” But then, if the agent is asked whether to take the bet, this new information (“I am the unique one being asked”) changes the probability back to 50-50.
I agree with you that the reason why you can’t use the 90⁄10 prior is because the decision never depends on a person in a red room.
In Eliezer’s description of the problem above, he tells each green roomer that he asks all the green roomers if they want him to go ahead with a money distribution scheme, and they must be unanimous or there is a penalty.
I think this is a nice pedogogical component that helps a person understand the dilemma, but I would like to emphasize here (even if you’re aware of it) that it is completely superfluous to the mechanics of the problem. It doesn’t make any difference if Eliezer bases his action on the answer of one green roomer or all of them.
For one thing, all green roomer answers will be unanimous because they all have the same information and are asked the same complicated question.
And, more to the point, even if just one green roomer is asked, the dilemma still exists that he can’t use his prior that heads was probably flipped.
Agreed 100%.
[EDIT:] Although I would be a bit more general: regardless of red rooms: if you have several actors, even if they necessarily make the same decision they have to analyze the global picture. The only situation when the agent should be allowed to make the simplified subjective Bayesian decision table analysis if he is the only actor (no copies, etc. It is easy to construct simple decision problems without “red rooms”: Where each of the actors have some control over the outcome and none of them can make the analysis for itself only but have to buid a model of the whole situation to make the globally optimal decision.)
However, I did not imply in any way that the penalty matters. (At least, as long as the agents are sane and don’t start to flip non-logical coins) The global analysis of the payoff may clearly disregard the penalty case if it’s impossible for that specific protocol. The only requirement is that the expected value calculation must be made protocol by protocol basis.
My intuition says that this is qualitatively different. If the agent knows that only one green roomer will be asked the question, then upon waking up in a green room the agent thinks “with 90% probability, there are 18 of me in green rooms and 2 of me in red rooms.” But then, if the agent is asked whether to take the bet, this new information (“I am the unique one being asked”) changes the probability back to 50-50.