Found a couple of rather dated SB papers today. The better of the two was *A Devastating Example for the Halfer Rule* by Vincent Conitzer. Conitzer rediscovers (again) what’s so bad about double halfing. (It violates reflection.) Of course, this has been known for well over a decade, since the outstanding paper *A Challenge for Halfers* by Cian Dorr. (The title of Conitzer’s paper is rather funny to me as I have been referring to Dorr’s paper as a “devastating” blow to double halfing for some time. It’s bad enough that Titelbaum and others are ripping Dorr off here. I don’t think we need another devastating shot! Double halfing is quite dead by now.) Conitzer’s paper is definitely clever and he reinvents the wheel more than once. I particularly like the following passage, in which Conitzer is discussing a variant of the problem:

“…it is now the Halfer Rule that runs afoul of the Reflection Principle: if Beauty is certain that her credence on Monday (or, for that matter, Tuesday) will be 1/3, then why is it not 1/3 already on Sunday? In fact, it seems to me that this violation of the Reflection Principle is more serious than the thirder’s alleged violation of it in the original Sleeping Beauty problem, for the following reason. In the original problem, it would be unreasonable to say that the fact that the thirder will end up having a credence of 1/3 on Tuesday implies that she should already have a credence of 1/3 on Sunday. After all, she does not always wake up on Tuesday, and if she were capable of, in her sleep, recognizing that she has not been awoken, she would assign credence 1 in Heads then. That is why the purported violation focuses on the Monday credence in Heads, not the Tuesday one. But it seems illegitimate to consider Monday separately from Tuesday, because Beauty cannot distinguish them. Thus, it seems debatable whether the thirder really violates the Reflection Principle (more precisely, whether she violates any version of this principle by which we would care to abide).”

This is of course another case of reinventing (or at least starting to reinvent). If Conitzer had read *Stopping to Reflect* by Seidenfeld, Schervish and Kadane he would know precisely which “version of this principle by which we would care to abide”…namely that in which refection between *now* and *later* requires (in particular) that *later* be a *stopping time*. For the thirder, *Monday* is not a stopping time. (The thirder cannot conspire to say *stop!* on and only on Monday!) Of course if he has read the aforementioned paper (or any of my unpublished ones) then he’s merely guilty of forgetting to cite someone. Somehow though I doubt this. Conitzer is not a philosopher, and I suspect that he just isn’t very familiar with the literature. If that is so, and I suspect it is, then he’s really quite cleverer than 98% of the philosophers writing about Sleeping Beauty. (Of course that’s high crime, and probably implies that his paper will never see the light of day.)

The other paper, *When Beauties Disagree: Why Halfers Should Affirm Robust Perspectivalism*, by John Pittard, rediscovers another old fact, namely that Lewisian halfers can disagree with each other about credences even when they’re in the same room, trust each other, can communicate, had equal priors at some point in their pasts and have the same uncentered evidence. In my survey I put it this way:

“…consider fellow Lewisian Sleeping Gorgeous, who gets awakened once if tails, twice if heads. Gorgeous has credence 1/3 in heads upon learning Monday. She and Beauty, who we can take to have been awakened in the same room, agree about how to determine credences, can talk to each other about their evidence, trust each others’ judgments and yet find themselves on opposite sides of objective chance concerning a future toss of a fair coin.”

Conitzer writes: “It should be noted that it would be trivial to turn these disagreeing participants into a money pump by arbitrage of their different credences.” That is worth mentioning, but it belies an ignorance as to how Lewisian halfing works. It’s sample weight dilution, a standard technique for correction of bias…in this case oversampling of the *tails* world. From the perspective of Beauty, her *tails* money is diluted by a factor of two. So if we’re to regard betting as a zero sum game, money pumping Conitzer needs also to dilute his enjoyment of her stake by a factor of two if *tails*. Similarly for Gorgeous if heads–and the money pump disappears. Is that odd? Yes! No one ever said that Lewisian halfing isn’t that. It’s a useful fiction for sampling bias correction, it isn’t meant to be taken literally, as a “guide to life”, as Lewis characterizes credences elsewhere. At any rate you can read all about it in my survey–it’s probably the only good reading of Lewis. Everyone else trashes his position without tact, insight or mercy. Or insight. (O, it’s only Da’id-Lew-is whose self is beastly dead. An impossible person!)

Pittard meanwhile appears to spend upwards of 25 pages attempting to defend the literal, guide-to-life view of Lewisian halfing–without crediting Lewis! Which seems a far greater offense to his memory than trashing his position. At any rate, I confess that I jumped to the end of Pittard’s paper, hoping to find the serious mistake that would justify not reading carefully…and found it. I will now explain what it is.

Pittard thinks he has a “hard case” for Lewisians. It relates to the following generalization of the so-called “reflection principle”:

**EXPERT REFLECTION:**If I know that

*S*is an expert on

*p*relative to myself, then my credence for

*p*conditional on

*S*’s credence for

*p*being

*x*should also be

*x*.

Here, to be an “expert” on *p* relative to me,* S* needs to know all I do about *p* and perhaps more. (I condone S’s priors and know that any worlds I have eliminated, *S* too has eliminated–perhaps more.) Pittard claims that many philosophers have endorsed **EXPERT REFLECTION**. There is something already quite odd about this. **EXPERT REFLECTION** clearly has naive reflection as a consequence, and there are known counterexamples to naive reflection. Future time slices of me, in particular, are experts relative to me about everything, so any instance in which reflection between *now* and *later* fails will be an instance in which **EXPERT REFLECTION** fails. On the other hand, there is, as Conitzer puts it, “a version of this principle by which we would care to abide”. And it’s called (queue drumroll) **The Bounded Martingale Stopping Theorem**! (I love repeating myself. And I have to do it, apparently, as philosophers seldom read memos–or emails, I have discovered–from sarcastic mathematicians.) This theorem fixes naive reflection by adding a condition to the antecedent…namely that *later* be a “stopping time”; in particular the agent needs to be able to recognize when *later* has arrived. (So the agent could say *stop!* at that time; i.e. could recognize that her location in time matched that described by her former self.)

So, if **EXPERT REFLECTION** wants to be true, it’s wanting for a corresponding condition in the antecedent. Namely, if we want reflection between *me* and *expert *to hold, then *expert* has to recognize that his identity matches that described by* me*. (I don’t have a catchy name for such an expert…*stopping expert* doesn’t seem to work very well, though, arguably, neither does *stopping time–*philosophers have, after all, steadfastly refused to recognize the significance of the latter*.*) If we make this change, then **EXPERT REFLECTION** is saved. There is no need to discredit it. Indeed, to discredit it is misleading. It also reflects ignorance of the literature. Yet this is exactly what Pittard does.

Pittard is worried about a case where Beauty (here a Lewisian) tries to figure out a bystander’s expected credence in *heads* during one of her awakenings. The bystander knows what day it is and knows whether Beauty is awake. So for Beauty, bystander’s expected credence in heads is 3/8; if* Monday* (which Beauty assigns probability 3/4) bystander has no evidence bearing on the toss and if *Tuesday* (which Beauty assigns probability 1/4) bystander knows* tails*. Pittard thinks that bystander is an expert relative to Beauty about heads, but wants not for Beauty to adopt his credences. So he thinks he needs to attack **EXPERT REFLECTION**. He does this by constructing an unrelated (and disanalogous) counterexample in which an expert fails to recognize that his identity matches that described by me.

This is hopelessly misguided. With the requisite added hypothesis, **EXPERT REFLECTION** cannot fail (it’s just a theorem), and if bystander were an expert here then certainly he’d be the sort of expert that satisfies the hypothesis (he can obviously self-identify under the sort of description Beauty has in mind), so reflection between *Beauty* and *bystander* would be valid. *If*, that is, bystander were an expert relative to Beauty! He is not! (His priors don’t match hers.) Indeed, this is already in Lewis, which Pittard clearly did not read closely enough. Lewis says, quite clearly, that upon learning *Monday* Beauty learns something relevant to *heads*…namely that she is not in the future. In other words, she eliminates something upon learning *Monday*. What she eliminates is *half of the* *tails* *world*. (This is what “robust perspectivalism” *really* is…the view that one can, contra strict Bayesianism, dilute worlds in SB situations without eliminating them. Pittard has proposed a rather clever name for what Lewis does, but his paper is not a good exposition of it.) I’ve been calling this *dilution*. Bystander doesn’t do this upon waking Monday–so from Beauty’s perspective, bystander knows less, conditional on *Monday*, than she herself does. As much is already reflected in bystander’s faulty (from Beauty’s perspective) priors, before learning *Monday*. So the real reason that reflection between *Beauty* and *bystander* fails is that *bystander* is not an expert on heads relative to *Beauty*. It’s not because there’s some deep problem with **EXPERT REFLECTION**. That principle is easily fixed, as I have indicated. Nor is any of this really new.

In fact, it’s getting old. Time to do your homework. The paper is called *Stopping to Reflect*. The theorem is called the *Bounded Martingale Stopping Theorem*. Get your head around it or get out of the way. I’m not kidding. Editors, meanwhile, when you see a paper about “reflection” (or for that matter “expert reflection”), you need to send the paper to an *actual expert*…not merely someone who answers to that description.

Pittard’s example meanwhile, while something of a rehash, is perfectly fine as an indictment of naive **EXPERT REFLECTION**. It goes something (not much, but I’m allowed to rehash too) like this:

Tex tosses a fair coin. If *heads*, he tells his *best friend* the outcome. If *tails* he tells no one. Tex has only two friends. They and I are each indifferent as to which friend (Chip or Dale) is *best friend*. If Chip gets the *heads* message his credence in heads will of course shoot up to 1. Otherwise, it will drop to 1/3. Not zero, since Chip doesn’t know that he is *best friend*. Not 1/2, as he doesn’t know that he isn’t. Now…after all has played out, from my vantage point Chip is clearly an expert. He knows all that I know, and in fact more, as he knows either (a) that he is *best friend* and the coin landed heads or (b) that he is not *best friend* or the coin landed tails.) So I would do well to adopt his expected credence in *heads* as my own. And that’s okay, because it’s 1/2–same as mine. It’s 1/2 because there is a 3/4 probability that he didn’t hear anything, in which case his credence in *heads* is 1/3, and a 1/4 probability that he heard *heads*, in which case his credence in* heads* is 1. And, well, you know it from there. By the same token, Dale is an expert…and his expected credence is 1/2 as well. So no trouble for **EXPERT REFLECTION** yet.

Trouble arises when we consider reflection between *me* and *best friend *(under that description)*. Best friend* has expected credence 2/3 in *heads*. Moreover, *best friend* is an expert relative to me. After all, *best friend* is either Chip or Dale, and we already agreed that they’re both experts. But it’s clearly false that I ought to adopt 2/3 as my credence in *heads*. So reflection between *me* and *best friend* (under that description) fails. How did this happen? We’ve already had this discussion. *Best friend* fails to recognize that his identity matches that of the description. So there was never any reason to think that reflection between *me* and *best friend* (so described) should be valid.

It will perhaps be instructive to see how this is essentially John Collins’s Prisoner example. Recall that the Prisoner will, with probability 1/2, be executed at midnight. (Otherwise he will be left alone in his cell.) He has no clock, so as midnight approaches his credence in *executed* drops. We don’t know how far it drops because we don’t know if his internal clock is running fast or slow. But it will drop at least a little by, say, 11:59, as he won’t be entirely sure at that point whether or not midnight has passed. For example, let’s say his internal clock is indifferent, at 11:59, between “before midnight” and “after midnight”. Then Prisoner’s credence in *executed *will have dropped to 1/3. Now at 6 O’clock the Prisoner christens his 11:59 time slice* best friend* and decides that the expected value of *best friend’*s credence in *executed* is 1/3. (A back-of-envelope calculation suggests that anything bigger than 1 – ln 2 is plausible here, assuming the Prisoner’s internal clock is unbiased.) Moreover, *best friend* is an expert relative to the Prisoner. He knows everything the Prisoner does, plus the fact that he lived long enough for his internal clock to assume whatever state it has. Reflection between *Prisoner* and *best friend* fails, however. Usually we say that this is because 11:59 is not a stopping time. Prisoner has no idea, at 11:59, that it’s 11:59–so he couldn’t, for example, say *stop* at that time. Put another way, *best friend* doesn’t recognize that he fits the description *best friend*…much as in Pittard’s example.

Seen from this vantage, The Prisoner is already a counterexample to naive “expert reflection”; there wasn’t really any need for Pittard to have supplied another. On the other hand, it was probably good than he did. It just would have been better if he had realized what it accomplishes, and why.

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