My thoughts: 1) The failure of CDT is its modeling of the decision process as ineffable ‘free will’ upon which things in the past cannot depend. Deviation from CDT is justified only when such dependencies exist. 2) The assumption that your decision is predictable requires the existence of such a dependency. 3) If we postulate that no such dependency exists, either CDT wins or our postulates are contradictory.
In particular, in Newcomb’s Soda, the assumptions that the soda flavor predicts the ice-cream flavor with high probability and that the assignment of soda (and the choice of subjects) is uncorrelated with subjects’ decision theory require that we are exceptional in employing decision theory. If lots of subjects were using CDT or EDT, they would all be choosing ice cream independently of their soda, and we wouldn’t see that correlation (except maybe by coincidence). So it doesn’t have to be stated in the problem that other subjects aren’t using evidential reasoning—it can be seen plainly from the axioms! To assume that they are reasoning as you are is to assume a contradiction.
The AB game is confusing because it flirts with contradiction. You act as if you’re free to choose A or B according to your decision theory while simultaneously assuming that Omega can predict your choice perfectly. But in fact, the only way Omega can predict perfectly is by somehow interacting with your decision theory. He can either administer the game only to people whose decision theory matches their genes, or manipulate people’s answers, or manipulate their genes. In the first case, EDTers will get a free gene test if they have G_A, but will not be miraculously healed if they have G_B. In the second case, you’ll find yourself pressing ‘B’ if you have G_B no matter what you try to precommit to. In the third case only, you have legitimate reason to commit to ‘A’, because your predetermined decision has causal influence on your genes.
You might try to counter with the case where Omega ensures that all children who are born will answer in a way consistent with their genes, and both things are determined at conception. But if this is the case, then if you have G_B, you can’t commit yourself to EDT no matter how hard you think about decision theory. This follows from the assumptions, and the only reason to think otherwise is if you still count free will among your premises.
To illustrate: Suppose you’re checking the lottery results online, and you see that you won, and you’re on your laptop at the house of a friend who knows what lottery numbers you buy and who has used his wi-fi to play pranks on guests in past. Suddenly the evidence doesn’t fare so well against that million-to-one prior.
This reminds me of reading about the Miracle of the Sun (http://en.wikipedia.org/wiki/Miracle_of_the_Sun) in The God Delusion and in a theist’s response. I found Dawkins fairly unpersuasive; the many agreeing testimonials weren’t enough to overcome the enormous prior improbability, but they were still disconcertingly strong evidence. The theists’ response cleared this up by giving historical background that Dawkins omitted. Apparently, the miracle was predicted in advance by three children and had become a focal point in the tensions between the devout and the secular. Suddenly, it was not at all surprising that the gathered crowd witnessed a miracle.
So I’d agree that miracles often have probability of under one in a million, but it’s also vitally important to understand the effect of motivation on the likelihood of the evidence. If I thought every testimony to every reported miracle was based on unbiased reporting of fact, I’d have to conclude that many of them happened (caused by aliens messing with us or something).