Robin, you're jumping into the middle of a big extended discussion. We're not only blaming anthropic updating, we're blaming Bayesian updating in general, and proposing a decision theory without it (Updateless Decision Theory, or UDT). The application to anthropic reasoning is just that, an application.

UDT seems to solve all cases of time inconsistency in decision problems with one agent. What UDT agents do in multi-player games is still an open problem that we're working on. There was an extensive discussion about it in the previous threads if you want to see some of the issues involved. But the key ingredient that is missing is a theory of logical uncertainty, that tells us how different agents (or more generally, computational processes) are logically correlated to each other.