To draw out the analogy to Godelian incompleteness, any computable decision theory is subject to the suggested attack of being given a “Godel problem″ like problem 1, just as any computable set of axioms for arithmetic has a Godel sentence. You can always make a new decision theory TDT’ that is TDT+ do the right thing for the Godel problem. But TDT’ has it’s own Godel problem of course. You can’t make a computable theory that says “do the right thing for all Godel probems”, if you try to do that it would not give you something computable. I’m sure this is all just restating what you had in mind, but I think it’s worth spelling out.
If you have some sort of oracle for the halting problem (i.e. a hypercomputer) and Omega doesn’t, he couldn’t simulate you, so you would presumably be able to always win fair problems. Otherwise the best thing you could hope for is to get the right answer whenever your computation halts, but fail to halt in your computation for some problems, such as your Godel problem. (A decision theory like this can still be given a Godel problem if Omega can solve the halting problem, “I simulated you and if you fail to halt on this problem...”). I wonder if TDT fails to halt for its Godel problem, or if some natural modification of it might have this property, but I don’t understand it well enough to guess.
I am less optimistic about revising “fair” to exclude Godel problems. The analogy would be proving Peano arithmetic is complete “except for things that are like Godel sentences.” I don’t know of any formalizations of the idea of “being a Godel sentence”.
To draw out the analogy to Godelian incompleteness, any computable decision theory is subject to the suggested attack of being given a “Godel problem″ like problem 1, just as any computable set of axioms for arithmetic has a Godel sentence. You can always make a new decision theory TDT’ that is TDT+ do the right thing for the Godel problem. But TDT’ has it’s own Godel problem of course. You can’t make a computable theory that says “do the right thing for all Godel probems”, if you try to do that it would not give you something computable. I’m sure this is all just restating what you had in mind, but I think it’s worth spelling out.
If you have some sort of oracle for the halting problem (i.e. a hypercomputer) and Omega doesn’t, he couldn’t simulate you, so you would presumably be able to always win fair problems. Otherwise the best thing you could hope for is to get the right answer whenever your computation halts, but fail to halt in your computation for some problems, such as your Godel problem. (A decision theory like this can still be given a Godel problem if Omega can solve the halting problem, “I simulated you and if you fail to halt on this problem...”). I wonder if TDT fails to halt for its Godel problem, or if some natural modification of it might have this property, but I don’t understand it well enough to guess.
I am less optimistic about revising “fair” to exclude Godel problems. The analogy would be proving Peano arithmetic is complete “except for things that are like Godel sentences.” I don’t know of any formalizations of the idea of “being a Godel sentence”.