I get the feeling that this is only controversial at all because people hear “Omega has never made an incorrect prediction” and internalize it as “Omega has just been lucky, and this is unlikely to continue”, rather than “Omega is a superintelligence that might as well be infallible, so there’s no point in trying to beat it”. I can see no reason to two box if Omega’s predictive power has been demonstrated and unbeaten a statistically significant number of times. I could try to prove I can outsmart a superintelligence with a high probability of failure, or I could take the one million dollars and pay off my student loans and put more effort into investing the rest to make up the 1000 I lose from box A.
But to the question at hand, if we assume Omega has a success rate of 100⁄100 so far, and decide to give ourselves a slight advantage and say that Cthulhu can cause Omega to fail an as of yet unobserved 1% of the time, we can calculate the value of each decision, as has been shown numerous times. The value of 1boxing would be 0.99x, where x is the maximum value in box B, and 2Boxing would be y+0.01x, where y is the value in box A. At x:y = 1:1, 2boxing has the advantage. At 1.01:1, they’re about equal. At anything higher than 1.01:1, 1boxing wins. There’s probably an elegant formulation, somewhere.
I suspect that there are people who would value beating Omega enough to screw with these numbers, though. If that is enough for the 1000:1 case to swing in favor of 2boxing, though, I’d expect that person to lose a great deal of money on various nigh-impossible challenges. It’s hard to say how this would map into the real world, given the lack of observed nigh-perfect predicters like Omega.
I get the feeling that this is only controversial at all because people hear “Omega has never made an incorrect prediction” and internalize it as “Omega has just been lucky, and this is unlikely to continue”, rather than “Omega is a superintelligence that might as well be infallible, so there’s no point in trying to beat it”. I can see no reason to two box if Omega’s predictive power has been demonstrated and unbeaten a statistically significant number of times. I could try to prove I can outsmart a superintelligence with a high probability of failure, or I could take the one million dollars and pay off my student loans and put more effort into investing the rest to make up the 1000 I lose from box A.
But to the question at hand, if we assume Omega has a success rate of 100⁄100 so far, and decide to give ourselves a slight advantage and say that Cthulhu can cause Omega to fail an as of yet unobserved 1% of the time, we can calculate the value of each decision, as has been shown numerous times. The value of 1boxing would be 0.99x, where x is the maximum value in box B, and 2Boxing would be y+0.01x, where y is the value in box A. At x:y = 1:1, 2boxing has the advantage. At 1.01:1, they’re about equal. At anything higher than 1.01:1, 1boxing wins. There’s probably an elegant formulation, somewhere.
I suspect that there are people who would value beating Omega enough to screw with these numbers, though. If that is enough for the 1000:1 case to swing in favor of 2boxing, though, I’d expect that person to lose a great deal of money on various nigh-impossible challenges. It’s hard to say how this would map into the real world, given the lack of observed nigh-perfect predicters like Omega.