Sure, I don’t necessarily blame the AIXI equation when it’s facing a relatively stupid Omega in that kind of situation.
However, consider “More Intelligent Finite Omega”, who pulls the agent’s source code and uses an approximate theorem-proving approach until it determines, with high confidence, what AIXI is going to do. Assuming that AIXI has received sufficient evidence to be reasonably confident in its model of MIFO, MIFO can reason like this:
AIXI will be able to accurately simulate me, therefore it will either have determined that the box is already empty, or already full.
Given either of those two models, AIXI will calculate that the best action is to two-box.
Consequently, AIXI will two-box. and then MIFO will leave the opaque box empty, and its prediction will have been correct. Moreover, MIFO had no other choice; if it were put the money in the opaque box, AIXI would still have two-boxed, and MIFO’s prediction would have been incorrect.
If you’re allowed to make the assumption that AIXI is confident in its model of CFO and CFO knows this, then I can make the same assumption about MIFO.
I think you’re right. At first I was worried (here and previously in the thread) that the proof that AIXI would two-box was circular, but I think it works out if you fill in the language about terminating turing machines and stuff. I was going to write up my formalization, but once I went through it in my head your proof suddenly looked too obviously correct to be worth expanding.
Sure, I don’t necessarily blame the AIXI equation when it’s facing a relatively stupid Omega in that kind of situation.
However, consider “More Intelligent Finite Omega”, who pulls the agent’s source code and uses an approximate theorem-proving approach until it determines, with high confidence, what AIXI is going to do. Assuming that AIXI has received sufficient evidence to be reasonably confident in its model of MIFO, MIFO can reason like this:
AIXI will be able to accurately simulate me, therefore it will either have determined that the box is already empty, or already full.
Given either of those two models, AIXI will calculate that the best action is to two-box.
Consequently, AIXI will two-box.
and then MIFO will leave the opaque box empty, and its prediction will have been correct. Moreover, MIFO had no other choice; if it were put the money in the opaque box, AIXI would still have two-boxed, and MIFO’s prediction would have been incorrect.
If you’re allowed to make the assumption that AIXI is confident in its model of CFO and CFO knows this, then I can make the same assumption about MIFO.
I think you’re right. At first I was worried (here and previously in the thread) that the proof that AIXI would two-box was circular, but I think it works out if you fill in the language about terminating turing machines and stuff. I was going to write up my formalization, but once I went through it in my head your proof suddenly looked too obviously correct to be worth expanding.