If Joe believes that his precommitment is inviolable, or even that it affects the probability of him breaking up later, then it appears to him that he is confronted with a causal loop. His decision-making program, at that moment, addresses Newcomb's problem, even if it's wrong in believing in the causal loop.

But I think this only proves that flawed reasoners may face Newcomb's problem. (It might even turn out that finding yourself facing Newcomb's problem proves your reasoning is flawed.)

It's still interesting enough to up-vote.