Ah, I see. The agent described in "A model of UDT with a halting oracle" is more symmetric, because it has a choice of three actions: "Cooperate", "Defect", and "Break down and cry" (if we count failing to act as an action).

I think I see a way to prove the proposition without using Löb's theorem, and I don't even need to change the definition of A. I'll post here whether it works out.

Thanks for the comment!