As I understand it, the purpose of bothering to advocate TDT is that it beats CDT in the hypothetical case of dealing with Omega (who does not exist), and is therefore more robust
See section 7 of the TDT paper (you’ll probably have to read from the beginning to familiarize yourself with concepts). It doesn’t take Omega to demonstrate that CDT errs, it takes mere ability to predict dispositions of agents to any small extent to get out of CDT’s domain, and humans do that all the time. From the paper:
The argument under consideration is that I should adopt a decision theory in which my decision takes general account of dilemmas whose mechanism is influenced by “the sort of decision I make, being the person that I am” and not just the direct causal effects of my action. It should be clear that any dispositional influence on the dilemma’s mechanism is sufficient to carry the force of this argument. There is no minimum influence, no threshold value.
See section 7 of the TDT paper (you’ll probably have to read from the beginning to familiarize yourself with concepts). It doesn’t take Omega to demonstrate that CDT errs, it takes mere ability to predict dispositions of agents to any small extent to get out of CDT’s domain, and humans do that all the time. From the paper: