Yes, experiments at the frontier of science are often unrepeatable, but that's just a selection effect, no? Those problems are interesting precisely because we have not nailed down all the cause-and-effect relationships yet. An enormous number of cause-and-effect rules are so well understood that they are not considered scientifically interesting anymore, and it is those rules that allow us to navigate the world, get to places on time, stay out of mortal danger, and so on. Of course there is random error as you say, but the error is not infinitely large.

Anyway, I keep picking this nit because the Russel quote says that our causal models of the world are not just imperfect but actually "obsolete and misleading," which sounds like an exaggeration.

I am not sure where our disagreement lies at the moment.

I'm not entirely sure either. I was just saying that a causal decision theorist will not be moved by Wildberger's reasoning, because he'll say that Wildberger is plugging in the wrong probabilities: when calculating an expectation, CDT uses not conditional probability distributions but surgically altered probability distributions. You can make that result in one-boxing if you assume backwards causation.

I think the point we're actually talking about (or around) might be the question of how CDT reasoning relates to you (a). I'm not sure that the causal decision theorist has to grant that he is in fact interpreting the problem as "not (a) but (b)". The problem specification only contains the information that so far, Omega has always made correct predictions. But the causal decision theorist is now in a position to spoil Omega's record, if you will. Omega has already made a prediction, and whatever the causal decision theorist does now isn't going to change that prediction. The fact that Omega's predictions have been absolutely correct so far doesn't enter into the picture. It just means that for all agents x that are not the causal decision theorist, P(x does A|Omega predicts that x does A) = 1 (and the same for B, and whatever value than 1 you might want for an imperfect predictor Omega).

About the way you intend (a), the causal decision theorist would probably say that's backward causation and refuse to accept it.

One way of putting it might be that the causal decision theorist simply has no way of reasoning with the information that his choice is predetermined, which is what I think you intend to convey with (a). Therefore, he has no way of (hypothetically) inferring Omega's prediction from his own (hypothetical) action (because he's only allowed to do surgery, not conditionalization).

Are you using choice to signify strongly free will?

No, actually. Just the occurrence of a deliberation process whose outcome is not immediately obvious. In both your examples, that doesn't happen: John's behavior simply depends on the arrival of the cab or his feeling of thirst, respectively. He doesn't, in a substantial sense, make a decision.

I do not agree that a CDT must conclude that P(A)+P(B) = 1. The argument only holds if you assume the agent's decision is perfectly unpredictable, i.e. that there can be no correlation between the prediction and the decision. This contradicts one of the premises of Newcomb's Paradox, which assumes an entity with exactly the power to predict the agent's choice. Incidentally, this reduces to the (b) but not (a) from above.

By adopting my (a) but not (b) from above, i.e. Omega as a programmer and the agent as predictable code, you can easily see that P(A)+P(B) = 2, which means one-boxing code will perform the best.

But that's not CDT reasoning. CDT uses surgery instead of conditionalization, that's the whole point. So it doesn't look at P(prediction = A|A), but at P(prediction = A|do(A)) = P(prediction = A).

Your example with the cab doesn't really involve a choice at all, because John's going to work is effectively determined completely by the arrival of the cab.

I agree; wherever there is paradox and endless debate, I have always found ambiguity in the initial posing of the question. An unorthodox mathematician named Norman Wildberger just released a new solution by unambiguously specifying what we know about Omega's predictive powers.

I seems to me that what he gives is not so much a new solution as a neat generalized formulation. His formula gives you different results depending on whether you're a causal decision theorist or not.

The causal decision theorist will say that his pA should be considered to be P(prediction = A|do(A)) and pB is P(prediction = B|do(B)), which will, unless you assume backward causation, just be P(prediction = A) and P(prediction = B) and thus sum to 1, hence the inequality at the end doesn't hold and you should two-box.

I don't really think Newcomb's problem or any of its variations belong in here. Newcomb's problem is not a decision theory problem, the real difficulty is translating the underspecified English into a payoff matrix.

The ambiguity comes from the the combination of the two claims, (a) Omega being a perfect predictor and (b) the subject being allowed to choose after Omega has made its prediction. Either these two are inconsistent, or they necessitate further unstated assumptions such as backwards causality.

First, let us assume (a) but not (b), which can be formulated as follows: Omega, a computer engineer, can read your code and test run it as many times as he would like in advance. You must submit (simple, unobfuscated) code which either chooses to one- or two-box. The contents of the boxes will depend on Omega's prediction of your code's choice. Do you submit one- or two-boxing code?

Second, let us assume (b) but not (a), which can be formulated as follows: Omega has subjected you to the Newcomb's setup, but because of a bug in its code, its prediction is based on someone else's choice than yours, which has no correlation with your choice whatsoever. Do you one- or two-box?

Both of these formulations translate straightforwardly into payoff matrices and any sort of sensible decision theory you throw at them give the correct solution. The paradox disappears when the ambiguity between the two above possibilities are removed. As far as I can see, all disagreement between one-boxers and two-boxers are simply a matter of one-boxers choosing the first and two-boxers choosing the second interpretation. If so, Newcomb's paradox is not as much interesting as poorly specified. The supposed superiority of TDT over CDT either relies on the paradox not reducing to either of the above or by fiat forcing CDT to work with the wrong payoff matrices.

I would be interested to see an unambiguous and nontrivial formulation of the paradox.

Some quick and messy addenda:

• Allowing Omega to do its prediction by time travel directly contradicts box B contains either $0 or $1,000,000 before the game begins, and once the game begins even the Predictor is powerless to change the contents of the boxes. Also, this obviously make one-boxing the correct choice.
• Allowing Omega to accurately simulate the subject reduces to problem to submit code for Omega to evaluate; this is not exactly paradoxical, but then the player is called upon to choose which boxes to take actually means the code then runs and returns its expected value, which clearly reduces to one-boxing.
• Making Omega an imperfect predictor, with an accuracy of p<1.0 simply creates a superposition of the first and second case above, which still allows for straightforward analysis.
• Allowing unpredictable, probabilistic strategies violates the supposed predictive power of Omega, but again cleanly reduces to payoff matrices.
• Finally, the number of variations such as the psychopath button are completely transparent, once you decide between choice is magical and free will and stuff which leads to pressing the button, and the supposed choice is deterministic and there is no choice to make, but code which does not press the button is clearly the most healthy.