aa r X i v : . [ s t a t . O T ] O c t A Devastating Example for the Halfer Rule ∗ Vincent ConitzerDuke University
Abstract
How should we update de dicto beliefs in the face of de se evidence?The Sleeping Beauty problem divides philosophers into two camps, halfers and thirders . But there is some disagreement among halfers about howtheir position should generalize to other examples. A full generalizationis not always given; one notable exception is the
Halfer Rule , under whichthe agent updates her uncentered beliefs based on only the uncentered partof her evidence. In this brief article, I provide a simple example for whichthe Halfer Rule prescribes credences that, I argue, cannot be reasonablyheld by anyone. In particular, these credences constitute an egregiousviolation of the Reflection Principle. I then discuss the consequences forhalfing in general.
Keywords:
Sleeping Beauty problem, Halfer Rule, Reflection Principle,evidential selection procedures
It is far from a settled matter how de dicto beliefs should be updated whenwe obtain de se information. The Sleeping Beauty problem is particularly ef-fective at bringing out conflicting intuitions. In it, Beauty participates in anexperiment. She will go to sleep on Sunday. The experimenters will then tossa fair coin. If it comes up Heads, Beauty will be awoken briefly on Monday,and then put back to sleep. If it comes up Tails, she will be awoken briefly onMonday, put back to sleep, again awoken briefly on Tuesday, and again put backto sleep. Essential to the problem is that Beauty will be unable to distinguishany of these three possible awakenings (Monday in a Heads world, Monday in aTails world, and Tuesday in a Tails world). In particular, when being put backto sleep after a Monday awakening, Beauty will be administered a drug thatprevents her from remembering this awakening, but otherwise leaves her brainunaffected. The experiment will end on Wednesday, when Beauty will be finallyawoken in a noticeably different room, so that there is no risk of her mistaking ∗ This paper appears in
Philosophical Studies , Volume 172, Issue 8, pp,1985-1992, August 2015. The final publication is available at Springer viahttp://dx.doi.org/10.1007/s11098-014-0384-y /
3, which would be the long-runfraction of Heads awakenings if the experiment were to be repeated many times.Halfers, on the other hand, believe that Beauty’s credence should be unchangedfrom Sunday, when it should clearly be 1 /
2. One benefit of being a halfer isthat being a thirder (or supporting any fraction other than 1 /
2) seems to violatethe
Reflection Principle [van Fraassen, 1984, 1995]: if on Sunday you are certainthat tomorrow, on Monday, you will have credence (say) 1 / / Principal Principle ) clearly on Sunday the credence in Headsshould be 1 /
2, because the coin is fair. Elga [2000] already notes the conflictbetween thirding and the Reflection Principle, attributing this observation toNed Hall, and considers the Sleeping Beauty problem a counterexample to theReflection Principle.Even if we were certain of the correct answer to the Sleeping Beauty problem– presumably, 1 / / de dicto beliefsshould be formed in the face of de se evidence in general. All it would do isplace a constraint on how they should be formed. Indeed, halfers disagree onhow the 1 / The Halfer Rule.
Determine which possible (uncentered) worlds are ruled outby the centered evidence; set their probabilities to zero. For those that are notruled out, renormalize the probabilities, so that they again sum to one whilekeeping the ratios the same. If Beauty adopts the Halfer Rule, she indeed places credence 1 / /
2, because still no possible world is ruled out. But Lewis [2001] advocates aversion of halfing that results in a credence of 2 / / / For my purposes, it is not necesssary to specify how credences in centered worlds aredetermined, i.e., how the total credence in a possible world is divided across its centers. Thisis because I will only consider credences in uncentered events in what follows. Titelbaum[2012] gives an example where halfers obtain an implausible credence in a centered event, if acertain condition on how the halfer distributes credence across centers holds.
2n time she is when her credence is 2 /
3. Indeed, Draper and Pust [2008] havepointed out that this credence of 2 / / /
3, resulting ina sure loss overall. More recently, Pittard [2015] has also argued against the Halfer Rule. Ashe points out, his own interpretation of halfing can lead to a disagreementparadox where two participants in an experiment obtain different credences inspite of having the same information. (The Halfer Rule does not lead to thisdisagreement paradox in his example.) It should be noted that it would betrivial to turn these disagreeing participants into a money pump by arbitrageof their different credences. In summary, the Halfer Rule is not universally agreed to constitute thecorrect generalization of halfing. On the other hand, it is a very natural gener-alization, it has attracted significant support, and it avoids problems that otherinterpretations of halfing encounter. However, I will now proceed to show thatit is fatally flawed.
The Sleeping Beauty variant that I need is very simple. Beauty will be putto sleep on Sunday, and be awoken once on Monday and once on Tuesday. Asalways, she will be unable to remember her Monday awakening on Tuesday. Twofair coins, called “one” and “two,” will be tossed on Sunday. When she wakesup on Monday, Beauty will be shown the outcome of coin toss one. When shewakes up on Tuesday, she will be shown the outcome of coin toss two. Beautycannot distinguish the two coins, so seeing the outcome of the coin toss stilldoes not tell her which day it is. She only learns that the coin corresponding to today came up (say) Heads. Figure 1 illustrates the example. One may wonder whether, similarly, we could set up a Dutch book against the thirderbased on her alleged violation of the Reflection Principle. But this would involve her beingoffered bets on Monday awakenings, without being told that it is Monday, but not on Tuesdayawakenings, and it has been argued that this does not constitute a fair Dutch book becausethe bookie is exploiting information that Beauty does not have [Hitchcock, 2004]. (Also, frombeing offered the bet Beauty might infer that it is Monday and thereby change her credencesand decline the bet.) Pittard nevertheless defends these credences, arguing that it may be reasonable to con-sider this a robustly perspectival context, one in which two disputants should end up havingdifferent beliefs in spite of them having the same evidence, being able to communicate withoutrestriction, etc. This may be reminiscent of the perspectival realism described by Hare [2010](see also Hare [2007, 2009]). Hare [2009] goes into some detail discussing what conclusion twointerlocutors, each of whom takes herself to be “the one with present experiences,” shouldreach. If indeed they should not be able to reach complete agreement, as seems likely, thenthis would appear to be a robustly perspectival context. However, in this case it does notseem possible to turn the situation into a money pump, because it does not seem possible tosettle any bets made in a satisfactory way; we cannot adjudicate from a neutral perspective.Indeed, Hare concludes that the interlocutors should agree that the other is correct from theother’s point of view . In contrast, bets made by the participants in Pittard’s experiment could
3H (1 /
4) HT (1 /
4) TH (1/4) TT (1/4)Monday see Heads see Heads see Tails see TailsTuesday see Heads see Tails see Heads see TailsFigure 1: A two-coins variant of the Sleeping Beauty problem with four possibleworlds, each with probability 1 /
4. Note that Beauty is always awoken on bothdays in this variant, but her information upon awakening is not always the same.Now consider the following question. When Beauty is awoken and observes a(say) Heads outcome, what should be her credence that the coin tosses came up the same ? That is, what should be her credence in the event “(both coins cameup Heads) or (both coins came up Tails)”? It seems exceedingly obvious that theanswer should be 1 /
2. Clearly this was the correct credence on Sunday beforelearning anything (by the Principal Principle), and intuitively, the outcome ofthe coin toss today – whatever it is – tells Beauty absolutely nothing aboutwhether the coins came up the same. This requires that the coins are fair; ifeach coin had, say, a 2 / / correct answer shortly. But, for thereader who is already convinced of that, let me get to the point and show whichcredences result from applying the Halfer Rule. The possible worlds that areconsistent with a Heads observation are HT (coin one came up Heads and cointwo came up Tails), TH, and HH. Because each of these three worlds has thesame probability ex ante , applying the Halfer Rule results in placing credence1 / / / What is so wrong about the Halfer Rule suggesting that the correct credenceis 1 / /
3, then why is it not 1 / easily be settled from a neutral perspective. Incidentally, applying the Thirder Rule does give the right answer: of all Heads awak-enings, two are in the HH world, in which the coins come up the same, and the remainingtwo are in the HT and TH worlds, in which the coins do not come up the same. So if weuse the Thirder Rule, the resulting credence in the event that both coins came up the sameis 2 / /
2. (I apologize for any confusion caused by the unfortunate coincidence that theHalfer Rule prescribes 1 / / / on Tuesday implies that she should already have a credence of 1 / /
3. But I leave formalizing the sense in which theviolation is more serious for another day.To make matters yet worse for the Halfer Rule, consider the following twistto the two-coins example. On both Monday and Tuesday, after Beauty hasobserved the coin toss outcome and been awake for a little while longer, theexperimenter tells her what day it is. Say she observed Heads and was thentold (a bit later) that it is Monday. Now only two worlds survive elimination:HH and HT. The Halfer Rule will assign each of them credence 1 /
2, resultingin a credence of 1 / But this is yet anotherviolation of the Reflection Principle: after seeing the outcome of the coin tossbut before learning what day it is, Beauty, if she follows the Halfer Rule, placescredence 1 / /
2. This is perhaps the most egregious violation of the Reflection Principlethat we have encountered, because in this case she is not put to sleep and doesnot have memories erased as she transitions from one credence to another. Again, I leave formalizing the sense in which the violation is more serious than The Thirder Rule still gives 1 / On the face of it, the same happens in the Shangri La example given by Arntzenius [2003].(I thank an anonymous reviewer for Philosophical Studies for calling my attention to this.)In this example, someone experiences A or B according to the outcome of a coin toss. Heknows, though, that at a certain point in time after the experience, any memories of B willbe replaced by false memories of A, while any memories of A will be left intact, so that hewill not be able to tell the two cases apart. Then, while experiencing A, he has credence 1 inHeads, in spite of knowing full well that he will later have credence 1 / never compromised, andshe never loses information.
5H (1 /
4) HT (1 /
4) TH (1/4) TT (1/4)Monday see Heads see Heads asleep asleepTuesday see Heads asleep see Heads asleepFigure 2: A cost-cutting variant of the two-coins example in Figure 1, the onlymodification being that Beauty is no longer awoken on Tails.the other violations for another day.
If the Halfer Rule is untenable, then is there another full generalization of halfingthat is more defensible? I have already mentioned a few interpretations of halfingthat do not always agree with the Halfer Rule and get into their own brands oftrouble as a result. In this final section, I hope to assess a bit more systematicallyhow halfing may be generalized in a trouble-free way.One helpful example to consider is a variant of the two-coins example in-troduced earlier. The only modification that is needed to obtain this variantis the following. To cut down on the cost of the various drugs involved in theawakenings, the experimenter has decided to only awaken Beauty when the coincorresponding to the current day has come up Heads. On Tails days, the exper-imenter just lets her sleep. Beauty is of course informed of this modification atthe outset. As a result, on a Heads awakening it is no longer necessary to showher that the coin has come up Heads, because this is already implied by the factthat she was awoken at all. On the other hand, nothing is lost by showing herthe Heads outcome anyway. Figure 2 illustrates the modified example.Now what should Beauty believe upon awakening (with Heads)? It appearsto me that in this variant, any reasonable generalization of halfing must placecredence 1 / / P (HH | see H) = P (see H | HH) P (HH) P (see H | HH) P (HH) + P (see H | HT) P (HT) + P (see H | TH) P (TH)= 1 · (1 / · (1 /
4) + (1 / · (1 /
4) + (1 / · (1 /
4) = 1 / P (HH | see H) = P (see H | HH) P (HH) P (see H | HH) P (HH) + P (see H | HT) P (HT) + P (see H | TH) P (TH)= 1 · (1 / · (1 /
4) + 1 · (1 /
4) + 1 · (1 /
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