The Causal Decision Theorist is a reprobate, utterly beyond the grasp of salvation. For the Evidential Decision Theorist, however, there is hope.
Causal Decision Theorists (CDTers) have the view on instrumental rationality that says you should take the action that causes the best outcome. Evidential Decision Theory (EDT) says you should take the action that’s evidence for the best outcome. Usually, these will give the same recommendation, since usually, the evidence we have as to how good things will be if we make a certain choice just comes from what we expect our choices to cause. But in cases like Newcomb’s Problem, the two come apart. The CDTer two-boxes, since the prediction is already made and you can’t cause there to be more money in the box, so two-boxing causes you to get an extra $1,000. EDTers one-box, because if they do so they’ll almost certainly get a million, and if they don’t they’ll almost certainly get a thousand.
EDTers want to be rich, which is why there is hope for them. Now, we’d prefer that they would just always do the thing that makes them rich, whereas they only do so if they get to keep their eyes closed while they do it. But they are not wholly divorced from the desire for riches, so they have hope.
One of the most powerful tools in our arsenal to get the EDTer to see the light is called XOR Blackmail. The case runs as follows. As always, Omega is a near-perfect predictor, who can predict humans’ actions conditional on finding themselves in whatever circumstances, with only a one-in-a-trillion error rate.
XOR Blackmail:
You’re worried your house is infested with termites (current credence is 0.5). If you do, it’ll cost a million dollars in damages to deal with the situation. Now, you get an email:
“Hello! I scanned your home for termites and also predicted how you’d respond to this email. I’m sending this to you to inform you that exactly one of the following is true:
(i) You have termites
(ii) You’ll give me $1,000 in response to this email
Were neither or both of the above true, I would have simply not sent you an email. My bank info is [REDACTED]. Have a nice day!
Love,
Omega”
Your decision will not affect whether you have termites or not; Omega based his judgment on whether you already have them. Do you give Omega the $1,000?
It seems like EDT recommends paying Omega. If you pay, Omega almost certainly predicted you would, meaning Omega almost certainly judged you’d pay and that you don’t have termites, so you almost certainly don’t have termites; you’re out just $1,000. If you don’t pay, Omega almost certainly predicted you wouldn’t, so you almost certainly have termites, so you’re out $1,000,000. So, it seems like paying is evidence for the best outcome, so the EDTer should pay up.
But that’s crazy. Don’t pay up. Being the kind of agent who pays up doesn’t contribute to whether you get termites or not, it just means you’ll happen to get an email in the event that you have no termites. Agents who pay up are just as termite-prone as agents who don’t; the only difference is that the former lose $1,000. It’s not like Newcomb’s Problem, where EDTers get a bunch of money that CDTers wouldn’t, because of the kinds of agents they are.
Just imagine a world with malicious superpredictors like Omega, XOR Blackmailing whoever they can. CDTers would get emails from Omega saying “Hey, your house is on fire right now xor you will give me $1,000,” and then be thankful that they were so quickly alerted to such an urgent matter. EDTers, in contrast, might be not-infrequently hassled with emails saying “A guy just broke into your car xor you’ll give me $100,” paying up every time. The EDTer’s wife pulls her phone out to give him proof she’s not cheating, as the EDTer refuses to look until he’s able to complete his payment. The EDTer ends up in financial ruin, one averted catastrophe at a time.
Or does he? According to Arif Ahmed, the EDTer might be able to avoid blackmail. He says:
Whether [EDT] does recommend [paying up] will depend on your initial beliefs. Suppose you start out—before getting the letter—determined to pay no attention to such things. On receiving the letter you remain confident that you’d never fall for this trick, and so—since you are highly confident that either you have termites or that you will pay $1,000 – you also become highly confident that you do have a termite infestation. (Compare: if some reliable forecaster of the weather tells you, in the northern hemisphere in July, that tomorrow it will either snow or be the hottest day of the year so far, you are likely to believe it.)
In this situation EDT does not recommend paying, because you don’t think that paying is evidence that the rumour [sic] is false. Paying is evidence that the letter is false, though you are in fact highly confident that the letter is true and that you won’t pay. So for this reason EDT does not recommend paying, even if you get the letter and even if you believe it. And there will be recipients of the letter who are confident that the letter is true, confident that they are not going to fall for it, confident that they are following the advice of EDT, and right on all three counts. Of course, if you start out with this belief and your house is termite-free, you would never have got the letter in the first place: that, presumably, is the position that most sensible followers of EDT are actually in.1
In a footnote, he gives an example of a consistent credence state on which the above is true. Letting P be the proposition that the EDTer pays up, and T the proposition he has termites. We may suppose, for example, that after getting the email, the EDTers credences are:
In this case, the EDTer is relatively confident that Omega is right (his credence is 0.82 that exactly one of P and T is true), but paying isn’t evidence of no termites, since we have Cr(~T | P)/Cr(~T | ~P) = 1. That is to say, the likelihood of termites is the same no matter what you do, if these are your credences.
Let E stand for the proposition that the EDTer gets the email. The analogy Ahmed gives with the weather forecaster is important. What it shows—and all that it shows—is that the following claims are compatible:
Cr(P xor T | E) is high
Cr(T | P, E) ≈ Cr(T | ~P, E)
But we should notice something weird in Ahmed’s example. What’s weird is that there is no evidential entanglement between breaking one’s resolve—which, here, one has only a 10% chance of doing—and of having termites. That is, the EDTer is equally likely to break their resolve to not pay regardless of whether they have termites. “Flo, the thing you just called weird is just a restatement of Ahmed’s conclusion that termites are evidentially independent of paying conditional on getting the email. That’s literally the same claim as the one that paying is evidentially independent of termites conditional on getting the email. You can’t refute Ahmed by calling something one Bayes’ Theorem away from his claim weird.” But isn’t it weird now that it’s been restated this way? The EDTer will falter and pay up in, say, 10% of cases where they get the email. Don’t you suppose Omega might be able to predict when he’s in one of the 10% of the cases, and take that into account when deciding whether to send the email?
If we suppose the EDTer knows he’ll falter in 10% of cases, and the EDTer knows that Omega will have probably predicted whether this is one of the 10% of cases, then it seems to follow that, if the EDTer pays up, that will provide evidence that this is one of the cases where he falters. So EDT does recommend paying up. Of course, if the EDTer might not let such thoughts occur to him, and instead stick with his resolve. But the question is what EDT recommends.
But, of course, once the EDTer recognizes that this is what EDT recommends, he’ll recognize it just as well in every case where he gets the email, and he’ll pay up every time, so the 10% figure will have to be raised to about 100%.
Now, additionally, let D be the proposition that the EDTer is disposed to pay up (which disposition both imperfectly predicts whether he actually pays and whether Omega predicts he’ll pay), and let O be the proposition that Omega predicts he’ll pay. The causal diagram is as follows:
And so we have that Cr(T, D, O, E, P) = Cr(T)Cr(D)Cr(O | D)Cr(E | T, O)Cr(P | D, E).2 Let’s assume you’ll definitely get the email if you have termites xor Omega predicts you’ll pay, that your disposition to pay predicts Omega’s prediction with accuracy a ≈ 1, that your disposition predicts perfectly whether you’ll pay, and that your credence that you have a disposition to pay is d ≈ 0 (since you’re very resolute on not paying). And reminder that your credence that you have termites is t = 0.5. By assumption, we have no nonzero terms where E and O have the same truth value, and no nonzero terms where P and D have different truth values. So we have:
Very importantly, note that the d washes out, and if a is close to 1, the quantity above is close to 0. Similarly, we have:
Again, the d washes out, and if a is close to 1, the quantity is about 1. The ratio of the latter to the former is just
which is very big if a is close to 1, i.e. if Omega is super accurate. Indeed, if a is bigger than a half—that is, if Omega is a better than chance as predicting—then the likelihood of termites conditional on not paying vs. paying will be bigger than 1. That is, if Omega is accurate, then having termites conditional on not paying is way more likely than having them conditional on paying. Notably, the value of d is irrelevant. Even if you’re super confident that you’ll stick to your guns and not pay, because Omega is sensitive to your prior disposition, paying is strong evidence that you don’t have termites.
The only assumption I’ve made that’s absent from Ahmed’s discussion, as far as I can tell, is that Omega’s accuracy is independent of whether you have a disposition to pay or not (assuming your high resolve to pay). This assumption seems perfectly kosher; when we’re talking about Newcomb problems, we generally assume that if you change your mind at the last minute, or you succumb to temptation to act other than how you planned, etc., then Omega will have probably predicted that. I don’t feel like doing the math, but I imagine that even if we make Omega less accurate, but still pretty accurate, in the case where the EDTer falters, faltering will still be evidence of no termites.
The conclusion makes sense. No matter how firm your resolve is, still, if you changed your mind, there’s a very good chance Omega would have predicted you’d do that, and so changing your mind is very good evidence that you don’t have termites. It therefore seems to remain that EDT recommends paying in XOR Blackmail.
Evidential Decision Theory, p. 36.
Please check my work. It’s not unlikely I’ve screwed up.
A better version of the “resolve” defense is that the EDT agent should (before receiving any letters) resolve to follow the EDT-optimal policy, which means not choosing actions by applying EDT to updated beliefs. (I.e., they should be updateless/resolute.)
omg about a year ago i came up with an argument for why EDT doesn't pay, and i have since forgotten it. I'll see if I can reproduce. Btw, what do you think EDT would do in the Smoking Lesion problem?