I meant in the second example. I agree that in the first one if she doesn't use the desire as evidence of the gene she'll get a result saying she should smoke. But in the second one even if she does ignore that then the probability of cancer given that she smokes is higher than the probability of cancer given that she doesn't.
If she doesn't have the gene, then she can smoke or not without any change in risk. She doesn't know if she has the gene or not, but if she then smoking makes her more likely to get cancer. So, if she sums across both possibilities, she's more likely to get cancer if she smokes. Not by as large a margin as if she knew she had the gene, or even by as large a margin as she would estimate if she included the urge to smoke as evidence that she does, but it's still more likely.
P(C|S&~G) = P(C|~S&~G) = P(C|~S&G) < P(C|S&G). So P(C|S) > P(C|~S), even without looking at the probability of the gene given that she prefers to smoke.
That's why I said the gene was rare - presumably so rare that her pleasure from smoking overwhelms the expected disutility from cancer.
I stumbled upon this paper by Andy Egan and thought that its main result should be shared. We have the Newcomb problem as counterexample to CDT, but that can be dismissed as being speculative or science-fictiony. In this paper, Andy Egan constructs a smoking lesion counterexample to CDT, and makes the fascinating claim that one can construct counterexamples to CDT by starting from any counterexample to EDT and modifying it systematically.
The "smoking lesion" counterexample to EDT goes like this:
EDT implies that she should not smoke (since the likely outcome in a world where she doesn't smoke is better than the likely outcome in a world where she does). CDT correctly allows her to smoke: she shouldn't care about the information revealed by her preferences.
But we can modify this problem to become a counterexample to CDT, as follows:
Here EDT correctly tells her not to smoke. CDT refuses to use her possible decision as evidence that she has the gene and tells her to smoke. But this makes her very likely to get cancer, as she is very likely to have the gene given that she smokes.
The idea behind this new example is that EDT runs into paradoxes whenever there is a common cause (G) of both some action (S) and some undesirable consequence (C). We then take that problem and modify it so that there is a common cause G of both some action (S) and of a causal relationship between that action and the undesirable consequence (S→C). This is then often a paradox of CDT.
It isn't perfect match - for instance if the gene G were common, then CDT would say not to smoke in the modified smoker's lesion. But it still seems that most EDT paradoxes can be adapted to become paradoxes of CDT.