So if the AI were building a robot [...] how would it bet in each?
That would depend on what the AI hoped to achieve by building the robot. It seems to me that specifying that clearly should determine what approach the AI would want the robot to take in such situations.
(More generally, it seems to me that a lot of anthropic puzzles go away if one eschews indexicals. Whether that's any use depends on how well one can do without indexicals. I can't help suspecting that the Right Answer may be "perfectly well, and things that can only be said with indexicals aren't really coherent", which would be ... interesting. In case it's not obvious, everything in this paragraph is quarter-baked at best and may not actually make any sense. I think I recall that someone else posted something in LW-discussion recently with a similar flavour.)
I think I agree with everything written here, so at least your naive decision theory looks like it can handle anthropic problems.
That would depend on what the AI hoped to achieve by building the robot.
Which side of the presumptuous philosopher's bet would you take?
I am posting this is because I'm interested in self-modifying agent decision theory but I'm too lazy to read up on existing posts. I want to see a concise justification as to why a sophisticated decision theory would be needed for the implementation of an AGI. So I'll present a 'naive' decision theory, and I want to know why it is unsatisfactory.
The one condition in the naive decision theory is that the decision-maker is the only agent in the universe who is capable of self-modification. This will probably suffice for production of the first Artificial General Intelligence (since humans aren't actually all that good at self-modification.)
Suppose that our AGI has a probability model for predicting the 'state of the universe in time T (e.g. T= 10 billion years)' conditional on what it knows, and conditional on one decision it has to make. This one decision is how should it rewrite its code at time zero. We suppose it can rewrite its code instantly, and the code is limited to X bytes. So the AGI has to maximize utility at time T over all programs with X bytes. Supposing it can simulate its utility at the 'end state of the universe' conditional on which program it chooses, why can't it just choose the program with the highest utility? Implicit in our set-up is that the program it chooses may (and very likely) will have the capacity to self-modify again, but we're assuming that our AGI's probability model accounts for when and how it is likely to self-modify. Difficulties with infinite recursion loops should be avoidable if our AGI backtracks from the end of time.
Of course our AGI will need a probability model for predicting what a program for its behavior will do without having to simulate or even completely specify the program. To me, that seems like the hard part. If this is possible, I don't see why it's necessary to develop a specific theory for dealing with convoluted Newcomb-like problems, since the above seems to take care of those issues automatically.