Psy-Kosh: They don't have to believe they have such causal powers over each other. Simply that they are in certain ways similar to each other.

I agree that this is definitely related to Newcomb's Problem.

Simpleton: I earlier dismissed your idea, but you might be on to something. My apologies. If they were genuinely perfectly rational, or both irrational in precisely the same way, and could verify that fact in each other...

Then they might be able to know that they will both do the same thing. Hmm.

Anyway, my 3 comments are up. Nothing more from me for a while.