Michael Titelbaum has written an excellent survey on Sleeping Beauty, called Ten Reasons to Care About the Sleeping Beauty Problem. Check it out:
It impressed me, though there are some things I don’t like about it. A few, anyway. Would be nice to get ten. Ten? Maybe. It’s worth a try:
1. Titelbaum gives us the following thought experiment:
“Some scientists will flip a fair coin: If it comes up heads, they’ll put nine black balls in a bag and one white; if it comes up tails, they’ll put in one black and nine white. The bag will then be passed around the room, and each person will draw one ball without replacement. Everyone in the room is informed of the experimental protocol, but no one is allowed to see the outcome of the coin flip or the ball anyone else has drawn.You draw your ball and see that it’s black. This should increase your confidence in heads from 1/2 to 9/10. But now suppose you have no non-indexical way of picking yourself out from among the ten people. You all look the same (so you can’t uniquely describe yourself by appearance), the room is cylindrical (so you can’t pick yourself out by absolute position), etc. Then, the only way to describe your new evidence is ‘I picked a black ball.’ But that’s centered evidence, and it has increased your rational credence in the uncentered proposition that the coin came up heads.”
One may ask…why does your confidence in heads increase from 1/2 to 9/10? Conditionalization, of course. You conditionalize on “I got a black ball.” But as Titelbaum notes, this is centered evidence…you can’t tell the difference between you and the others. But if one commits to parsing I rigidly, everything works out. Here is the idea. Number the participants 1 to 10. I don’t know what number I am. But if I am committed to a rigid parsing of I, then “I got a black ball” means either “1 got a black ball” or “2 got a black ball” or … or “10 got a black ball.” I don’t know which it means, but it doesn’t really matter. It means some one of those ten uncentered propositions, and conditionalization on any one of the ten yields a credence of 9/10 in heads. It’s not far from here to the correct thirder argument from evidence, where Beauty conditions on “I am awakened today.” Here the crucial indexical is today. Commit to parsing today rigidly, and everything works out. For thirders, anyway. On parsing, “I am awakened today” means either “I am awakened Monday,” which is uninformative, or “I am awakened Tuesday“, which is very informative indeed. (It implies tails.) It would have been nice if this survey could have gleaned some of that.
2. Titelbaum writes “Conditionalization systematically fails for cases involving self-locating degrees of belief. So does the Reflection Principle…”. This is a terrible, terrible mischaracterization. The so-called Reflection Principle is just the bounded martigale stopping theorem, as is well understood by philosophers who understand stochastic analysis. (See Stopping to Reflect by Schervish et. al.) There are cases where the hypotheses are not met, surely. But there aren’t “violations” of reflection. Not of correct formulations of it, anyway. Take for example the fact that Beauty has credence 1/3 on Monday morning. She did know that this would be the case Sunday night, surely, yet did not update then. Is this a violation of reflection? No, because “On Monday morning” is not a stopping time. That’s one of the hypotheses of the theorem. Reflection between now and later requires, in general, that the agent be able to recognize when “later” has arrived. Beauty does not recognize that it’s Monday morning on Monday morning, so it isn’t a stopping time. As for the claim that conditionalization fails, see #1 above. (Commit to rigid parsing of your indexicals and average. It doesn’t fail!)
3. Titelbaum writes:
“A Dutch Strategy can be assembled against thirders in the Sleeping Beauty Problem. Hitchcock (2004) proposes that on Sunday night, the bookie sells Beauty a bet that costs $15 and pays $30 if the coin comes up heads. Since Beauty is 1/2 confident in heads on Sunday night, she will accept this bet as fair. The bookie also tells Beauty on Sunday night that when she awakens Monday morning, he will sell her a bet for $20 that pays $30 if the coin comes up tails. If Beauty plans on being a thirder, she is certain that she will accept this bet as fair on Monday morning. (Notice that the bookie places this bet only once – on Monday morning – however the coin flip comes out.)”
Not really. Beauty will not accept the $20 for $30 bet on Monday as the mere fact that it is offered tells her it is Monday and changes her credence to 1/2. Titelbaum thinks he means something here, but I don’t know what. What I do know is that this Dutch Book example should not appear in his paper. Or anywhere. (Except of course on this blog, where it’s thematically appropriate.)
4. This isn’t really about Sleeping Beauty, but we don’t “quit certainties”. Well, okay, we do. It’s called forgetting. That’s the only way, though. My certainty that it’s midnight now and my certainty one minute from now that it was midnight a minute ago are the same certainty. I didn’t quit the one and adopt the other. I mean you can do your accounting that way if you want to, but it’s hopelessly inefficient and gains you nothing. In short…madness.
5. Titelbaum screws up self-indication. (All philosophers do. It’s frustrating.) He writes, for example:
“If the Self-Indication Assumption is correct, our very existence is evidence for hypotheses positing many (populated) universes over hypotheses that posit just one.”
Not true. Here is the way self-indication works. I am sampled, according to self indication, uniformly at random from the pool of like observers in the space of metaphysically possible worlds where the worlds occur in proportion to their objective chance. The emphasized phrase is what is important here. Self indication does not say that I am sampled from a pool of like observers in a hypothetical space of epistemically possible worlds where worlds occur in proportion to their epistemic probability. You can look at the situation this way. Suppose that the universe expands and contracts in perpetuity. Each cycle is a “metaphysically possible world”. Indeed, those just are the metaphysically possible worlds. Now…when I am wondering whether they come with one or zillions of branches at a time, I am not thinking that some cycles have one and some have zillions. On the contrary, I think (and you do too, I hope) that either they always come with one branch or they always come with zillions. Whether there are or aren’t zillions of parallel branches (or “universes”) in the world is not a matter of contingency. When I say I am indifferent to the proposition that there are zillions of parallel universes I am not saying that I think half of the expansions involve zillions of parallel universes and half don’t. What I am saying is that either they all do or they all don’t, and I am indifferent between those choices. Remember…a credence is an expected frequency. I have credence one half in the proposition that, in the sequence of observers like me, the frequency of observers populating single-branch worlds is 1, and credence one half in the proposition that, in the sequence of observers like me, the frequency of observers populating single-branch worlds is 0. Therefore my credence in the proposition that I populate a single branch world is the expectation of this frequency…name 1/2.
6. Titelbaum mishandles quantum mechanics. The correct model if that when there is a branching then, yes, both branches exist in some multi-verse, but the probability of consciousness taking the one branch is 70% (in his example). Think of it as the “least effort branch” or whatnot. At any rate this isn’t really that important. I mean it is, but really it’s not about Sleeping Beauty, it’s about finding a place for consciousness without violating the causal closure of the physical or lapsing into epiphenomenalism. So maybe the comment doesn’t belong here. But, my point, I guess, is that Titelbaum’s section on quantum mechanics didn’t belong in his paper, either, because it doesn’t say anything that is particularly sensible.
7. What about Cian Dorr’s beautiful paper A Challenge for Halfers? The marginalization of that awesome paper is one of the great travesties of Sleeping Beauty lore. Titelbaum has seventy-nine papers in his bibliography. Dorr’s paper is better and more central to the discussion than at least seventy-five of them.
8. Nothing good about Jacob Ross? Too bad. Difficult terrain, though. Oh well.
9. Nothing good about Pust’s attack on conditionalization on centered evidence? Damn.
10. I’ll close with one of the best passages in Titelbaum’s paper:
“So it looks like the halfer must maintain that upon learning it’s Monday, Beauty should be more than 1/2 confident in heads. That’s the unwelcome consequence. After all, it’s Monday evening, the coin hasn’t been flipped yet (because of Elga’s first modification), the coin is fair, and Beauty is certain of all this. So how can her rational credence in heads be other than 1/2? David Lewis recognized this unwelcome consequence of his own halfer view. He also was famously committed to the Principal Principle, which requires an agent to assign credences equal to chance values of which she is certain unless she has what Lewis called ‘inadmissible’ evidence. An obvious example of inadmissible evidence is information causally downstream from the outcome of the chance process in question. For example, if you watch a fair die roll come up 3, you’re allowed to change your credence in that outcome while remaining certain that its objective chance was 1/6. But Beauty doesn’t seem to have any information causally downstream from the coin flip on Monday evening. So Lewis cast about for a ‘novel and surprising’ kind of inadmissible evidence that would authorize a greater-than-1/2 Monday-evening credence in heads.”
I would love to have written that myself. This not-so-subtle diss of the greatest philosopher of the second half of the 20th century may well be dead on. We don’t know, because, alas, the greatest philosopher of the second half of the 20th century is dead. I have preferred to treat Lewis with more charity, imagining that he came to his position not for having “cast about” for an unlikely save, but for coherent reasons involving “sample weight dilution”. My reading may be too generous: Titelbaum may be more on the right track. Certainly it at least needs to be said, but (possibly because my admiration for Lewis gets in my way) I do find myself resisting….
At any rate those are ten things I would have done differently, had I written the survey.