In this case the only reason the money pumping doesn’t work is because Omega is unable to choose its policy based on its prediction of your second decision: If it could, you would want to switch back to b, because if you chose a, Omega would know that and you’d get 0 payoff. This makes the situation after the coinflip different from the original problem where Omega is able to see your decision and make its decision based on that.
In the Allais problem as stated, there’s no particular reason why the situation where you get to choose between $24,000, or $27,000 with 33⁄34 chance, differs depending on whether someone just offered it to you, or if they offered it to you only after you got <=34 on a d100.
Well, Omega doesn’t know which way the coin landed, but it does know that my policy is to choose a if the coin landed heads and b if the coin landed tails. I agree that the situation is different, because Omega’s state of knowledge is different, and that stops money pumping.
It’s just interesting that breaking the independence axiom does not lead to money pumping in this case. What if it doesn’t lead to money pumping in other cases too?
In this case the only reason the money pumping doesn’t work is because Omega is unable to choose its policy based on its prediction of your second decision: If it could, you would want to switch back to b, because if you chose a, Omega would know that and you’d get 0 payoff. This makes the situation after the coinflip different from the original problem where Omega is able to see your decision and make its decision based on that.
In the Allais problem as stated, there’s no particular reason why the situation where you get to choose between $24,000, or $27,000 with 33⁄34 chance, differs depending on whether someone just offered it to you, or if they offered it to you only after you got <=34 on a d100.
Well, Omega doesn’t know which way the coin landed, but it does know that my policy is to choose a if the coin landed heads and b if the coin landed tails. I agree that the situation is different, because Omega’s state of knowledge is different, and that stops money pumping.
It’s just interesting that breaking the independence axiom does not lead to money pumping in this case. What if it doesn’t lead to money pumping in other cases too?