More decision theory stuff

Suppose there are two opaque boxes, A and B, of which I can choose one. A nearly perfect predictor of my actions put $100 in the box that they thought I would choose. Suppose I find myself with evidence that it’s 75% likely that I will choose box A (maybe in 75% of cases like this, people like me choose A). I then reason: “So, probably, the money is in box A”, and I take box A.

This reasoning is supported by causal decision theory. There are two causal hypotheses: that there is money in box A and that there is money in box B. Evidence that it’s 75% likely that I will choose box A provides me with evidence that it’s close to 75% likely that the predictor put the money in box A. The causal expected value of my choosing box A is thus around $75 and the causal expected value of my choosing box B is around $25.

On evidential decision theory, it’s a near toss-up what to do: the expected news value of my choosing A is close to $100 and so is that of my choosing B.

Thus, on causal decision theory, if I have to pay a $10 fee for choosing box A, while choosing box B is free, I should still go for box A. But on evidential decision theory, since it’s nearly certain that I’ll get a prize no matter what I do, it’s pointless to pay any fee. And that seems to be the right answer to me here. But evidential decision theory gives the clearly wrong answer in some other cases, such as that infamous counterfactual case where an undetected cancer would make you likely to smoke, with no causation in the other direction, and so on evidential decision theory you refrain from smoking to make sure you didn’t get the cancer.

In recent posts, I’ve been groping towards an alternative to both theories. The alternative depends on the idea of imagining looking at the options from the standpoint of causal decision theory after updating on the hypothesis that one has made a specific choice. In current my predictor cases, if you were to learn that you chose A, you would think: Very likely the money is in box A, so choosing box A was a good choice, while if you chose B, you would think: Very likely the money is in box B, so choosing box B was a good choice. As a result, it’s tempting to say that both choices are fine—they both ratify themselves, or something like that. But that misses out the plausible claim that if there is a $10 fee for choosing A, you should choose B. I don’t know how best to get that claim. Evidential decision theory gets it, but evidential decision theory has other problems.

Here’s something gerrymandered that might work for some binary choices. For options X and Y, which may or may not be the same, let e_X(Y) be the causal expected value of Y with respect to the credences for the causal hypotheses updated with respect to your having chosen X. Now, say that the differential restrospective causal expectation d(X) of option X equals e_X(X) − e_X(Y). This measures how much you would think you gained, from the standpoint of causal decision theory, in choosing X rather than Y by the lights of having updated on choosing X. Then you should the option that provides a bigger d(X).

In the case where there is a $10 fee for choosing box A, d(B) is approximately $100 while d(A) is approximately $90, so you should go for box B, as per my intuition. So you end up agreeing with evidential decision theory here.

You avoid the conclusion you should smoke to make sure you don’t have cancer in the hypothetical case where cancer causes smoking but not conversely, because the differential retrospective causal expectation of smoking is positive while the differential retrospective causal expectation of not smoking is negative, assuming smoking is fun (is it?). So here you agree with causal decision theory.

What about Newcomb’s paradox? If the clear box has a thousand dollars and the opaque box has a million or nothing (depending on whether you are predicted to take just the opaque box or to take both), then the differential retrospective causal expectation of two-boxing is a thousand dollars (when you learned you two-box, you learn that the opaque box was likely empty) and the differential retrospective causal expectation of one-boxing is minus a thousand dollars.

So the differential retrospective causal expectation theory agrees with causal decision theory in the clear case (cancer-causes-smoking), the difficult case (Newcomb), but agrees with evidential decision theory in the $10 fee variant of my two-box scenario, and the last seems plausible.

But (a) it’s gerrymandered and (b) I don’t know how to generalize it to cases with more than two options. I feel lost.

Maybe I should stop worrying about this stuff, because maybe there just is no good general way of making rational decisions in cases where there is probabilistic information available to you about how you will make your choice.