A Devastating Example for the Halfer Rule

A Dev astating Example for the Halfer Rule ∗ Vincen t Conitzer Duk e Univ ersit y Abstract How should w e up date de dicto b elie fs in th e face of de s e evidence? The Sleeping Beauty problem divides philosophers into t wo ca mps, halfers and thir ders . But there is some disagreemen t among halfers ab out how their p os ition sh ould generalize to other examples. A full generalization is not alwa ys giv en; one notable excep t io n is the Halfer Rule , under which the agen t up dates her uncentered b eli efs based on only the uncentered part of her evidence. In this brief article, I provide a simple example for whic h the H al fer Rule prescrib es credences th at, I argue, cannot b e reasonably held by any one. In particular, these credences constitute an egregious violation of t he Reflection Principle. I then d is cu ss the consequ ences for halfing in general. Keywords: Sleeping Beaut y problem, Halfer Rule, Reflection Principle, evidential selection proced u res 1 In tro duction It is far from a se tt led matter how de dicto beliefs should b e up dated when we obtain de se informatio n. The Sleeping Bea ut y proble m is particular ly ef- fective at bringing o ut conflicting intuitions. In it, Beaut y participates in an exp erimen t. She will go to sleep on Sunday . The exp erimenters will then toss a fair coin. If it comes up Heads, Beauty will be aw oken briefly on Monday , and then put back to sleep. If it co mes up T ails, she will b e aw oken briefly on Monday , put bac k to s leep, aga in awok en briefly on T uesday , and ag ain put ba c k to sleep. Essential to the problem is that Beauty will b e unable to disting uis h any of these three p ossible aw a kenings (Monday in a Heads world, Monday in a T ails world, and T uesday in a T ails world). In pa rticular, when b eing put back to sleep after a Monda y aw akening, Beauty will be administer ed a drug that preven ts her from remembering this aw akening, but other wise leav es her brain unaffected. The exp eriment will e nd on W ednesday , when Beauty will be finally aw oken in a noticeably different ro om, so that there is no risk of her mistaking ∗ This paper appears in Philosophic al Studies , V olume 172, Issue 8, pp, 1985-1992, August 2015. The final publicat i on is av ailable at Springer via h ttp://dx.doi.org/10.1007/s11098-014 - 03 84-y 1 this even t for one o f the br ie f awak enings . B e a ut y is at all times fully informed of these r ules of the exp eriment . Now, when Beauty finds hers elf in o ne of the brief aw akening even ts, what should b e her c r edence (sub jective probability) that the co in has come up Heads? Thirders b eliev e that the corre ct a nsw er is 1 / 3, whic h would be the long -run fraction of Heads awak enings if the ex p eriment were to b e repea ted many times. Halfers, on the other hand, b eliev e that Beaut y’s cr edence s hould be unchanged from Sunday , when it should clear ly b e 1 / 2. One b enefit o f b eing a halfer is that being a thirder (or supp orting a n y fr action other than 1 / 2 ) s eems to violate the Refle ction Principle [v an F ra assen , 1984, 1995]: if on Sunday y ou are certa in that tomor ro w, on Monday , you will ha ve credence (say) 1 / 3 in some even t, then you should have cr e dence 1 / 3 in that ev ent now already . But (applying what is known as the Princip al Princip le ) clear ly on Sunday the credence in Heads should b e 1 / 2 , b ecause the c o in is fair. Elga [2000] alr eady notes the conflict betw ee n thir ding and the Reflection Principle, attributing this o bserv atio n to Ned Hall, a nd consider s the Sleeping Be a ut y pr oblem a counterexample to the Reflection Principle. Even if we were cer ta in o f the correct answer to the Sleeping Beauty problem – pr esumably , 1 / 3 o r 1 / 2 – this w o uld fall s ho rt of knowing how de dicto b eliefs should b e formed in the face of de se evidence in ge neral. All it would do is place a constr ain t on how they should b e fo rmed. Indeed, ha lfers disag r ee on how the 1 / 2 ans w er sho uld generalize to other ex amples. But one natural gen- eralization that ha s been dis c ussed in several articles [Halp e rn, 2006, Meacham, 2008, Brig gs , 2 010 ] is the following, called the “Halfer Rule” by Briggs . The Halfer Rule. Determine which p ossible (uncentered) worlds ar e ruled out by the centered evidence; set their proba bilities to zero. F or those that are no t ruled out, renormalize the probabilities, so that they again sum to one while keeping the ratios the same. 1 If Bea ut y adopts the Halfer Rule, she indeed pla ces credence 1 / 2 in Heads after b eing aw oken, b ecause no p ossible w orlds are ruled o ut. Again, not all halfers agree with the Halfer Rule in g eneral. F or example, the Halfer Rule prescrib es that, if Bea ut y is alwa ys told a t some p oint during her Monday aw akening that it is Monday , her credence in Heads at that p oin t s hould still be 1 / 2, be cause still no p o ssible world is ruled out. But Lewis [200 1 ] a dvo cates a version of halfing that re sults in a cr edence of 2 / 3 in Heads after b eing told it is Monday . This is a violatio n of the Reflection Principle – B eaut y knows that she will change her credence to 2 / 3 o n Monday , reg ardless of how the coin came up, and yet stic ks with 1 / 2 on Sunday – and ar guably o ne that is mor e ser ious than the thirder’s alleged violatio n o f it, because in this case Beauty knows where 1 F or my purp oses, it is not necesssary to s p ecify how credences i n cent ered wo r lds are determined, i.e., ho w the total credenc e i n a p ossible world is divi de d across its cen ters. This is b ecause I will only consider credences in uncen tered ev ents in what follows. Titelbaum [2012] gives an example where halfers obtain an implausible credence in a cent ered even t, if a certain condition on ho w the halfer distributes credence across cen ters holds. 2 in time s he is when her credence is 2 / 3. Indeed, Drape r and Pust [2008] have po in ted o ut tha t this credence of 2 / 3 would make Beauty susceptible to a very simple dia c hronic Dutc h b o ok, whe r e she is sold one b et on Sunday when her credence is 1 / 2 and another on Monday when her credence is 2 / 3 , resulting in a sure lo ss ov erall. 2 More recently , Pittard [2 015] has also a rgued against the Halfer Rule. As he p o in ts out, his own in terpr e ta tion of halfing ca n lead to a disagreement paradox where tw o participa nts in an exp erimen t o btain different credences in spite o f having the same informa tion. (The Halfer Rule do es not lead to this disagreement par ado x in his example.) It should b e noted that it would b e trivial to turn thes e disag reeing participants into a money pump by a rbitrage of their differen t credences. 3 In summary , the Halfer Rule is no t universally a greed to constitute the correct gener alization of halfing. On the other ha nd, it is a very natura l gener - alization, it has attra cted significant supp ort, and it avoids problems that other int er pretations of halfing encounter. How ever, I will now pro ceed to show that it is fata lly flaw ed. 2 A V aria nt with T wo Coins The Slee ping Bea ut y v ariant that I need is very simple. Beauty will b e put to sleep on Sunday , a nd b e aw oken once on Monday a nd once on T uesday . As alwa ys, she will b e unable to remember her Monday aw a k ening o n T uesday . Two fair coins, ca lled “ one” and “ t wo,” will b e tos sed on Sunday . When s he wak es up on Monday , B e a ut y will b e shown the outcome of co in toss one. When she wak es up o n T uesday , she will b e shown the outcome of coin toss tw o. Bea ut y cannot distinguis h the t wo coins, so seeing the outcome of the coin toss still do es not tell her whic h day it is. She only learns that the coin cor responding to to day came up (say) Hea ds . Figure 1 illustrates the example. 2 One m ay wonder whether, si milarly , we could set up a Dutch b ook against the thirder based on her alleged violation of the Reflection Pr inciple. But this w ould inv olve her being offered b ets on Monday aw akenings, without being told that it is Monda y , but not on T uesday a wak enings, and it has b een argued that this does not constitut e a fair Dutc h b ook because the b ookie is exploiting information that Beaut y do es not hav e [Hitc hco c k , 2004]. (Also, from being offered the b et Beau ty migh t infer that i t is Monda y and thereb y c hange her credences and decline the b et.) 3 Pittard nev ertheless defends these credences, arguing that it m ay be r ea sonable to con- sider this a r obustly p ersp e ctival con text, one in which t wo disputan ts should end up ha ving differen t b eliefs in spite of them having the same evidence, b eing able to communica te without restriction, etc. Thi s may be reminis cent of the p ersp e ctival r ea lism described by Hare [2010] (see also Hare [2007, 2009]). H ar e [2009] goes into some detail discussing what conclusion t wo int erl ocutors, each of whom tak es herself to b e “the one with present exp eriences,” should reac h. If indeed they should not be able to reach complete agreement, as seems lik ely , then this w ould app ear to b e a r obust l y p erspectiv al conte xt. Ho we ver, i n this case it do es not seem poss i ble to turn the situation in to a m oney pump, because i t does not seem p ossible to settle any bets m ade in a satisfactory wa y; w e cannot adjudicate from a neutral persp ectiv e. Indeed, Hare concludes that the in terlo cutors should agree that the other is correct fr om the other’s p oint of view . In con trast, bets made by the participants in Pittard’s exp erimen t could 3 HH (1 / 4) HT (1 / 4 ) TH (1/ 4) TT (1/ 4) Monday see Heads see Heads see T ails see T ails T uesday see Heads see T a ils see Heads see T ails Figure 1: A t wo-coins v aria n t of the Sleeping Beauty pro blem with four po ssible worlds, each with probability 1 / 4. Note that Bea ut y is alwa ys aw oken on b oth days in this v ar ian t, but her information upon awak ening is not alw ays the same. Now co nsider the follo wing question. When Beaut y is aw o k en and observes a (say) Hea ds o utcome, what should b e her credence that the coin tosses came up the same ? That is, what s ho uld be her credence in the ev ent “(b oth coins came up Heads) or (b oth coins came up T a ils)”? It seems exceeding ly obvious that the answer should b e 1 / 2. Clearly this was the correct cre dence on Sunday b efore learning anything (by the Pr incipal Pr inciple), and intuitiv ely , the o utcome of the coin toss to da y – wha tev er it is – tells Beauty abs olutely nothing abo ut whether the coins c ame up the same. This requir es that the coins are fa ir; if each co in ha d, say , a 2 / 3 c hance of co ming up Heads, then learning that to day’s coin has come up T ails would give Beauty evidence that the coins are less likely to hav e co me up the sa me. But we explicitly a ssume that the co ins are fair. I will a rgue in more detail that 1 / 2 is the c orr e ct answer shortly . But, for the reader who is a lready convinced of that, let me get to the p oin t a nd sho w which credences r esult from applying the Halfer Rule. The p ossible worlds that are consistent with a Heads obse rv ation ar e HT (coin one came up Heads and coin t wo c a me up T ails), TH, and HH. Because ea c h o f these thre e worlds has the same pro babilit y ex ante , applying the Halfer Rule r esults in placing cr edence 1 / 3 in each of these worlds. But this implies pla cing only 1 / 3 c r edence in the even t that b oth co ins came up the sa me, b ecause o f the three remaining worlds only HH ha s them coming up the same. By symmetry b et ween Heads and T ails, the Halfer rule a lso pre s cribes 1 / 3 credence in the even t that b oth coins came up the same if T a ils is observed. 4 3 The Halfer Rule and the Reflection Principle What is so wrong ab out the Halfer Rule suggesting that the correc t cre de nc e is 1 / 3 in the ab o ve example? W ell, it is now the Halfer Rule that runs afoul of the Refle c tion Principle: if Beauty is certain that her credence on Monday (or, for that matter , T ues da y) will b e 1 / 3, then why is it no t 1 / 3 already on easily be settled f rom a neut r al persp ectiv e. 4 Inciden tally , applying the Thir de r R ul e does give the right answ er: of all Heads aw ak- enings, t wo are in the HH world, in whi c h the coins come up the same, and the remaining t wo are in the HT and TH w orl ds, in which the coins do not come up the same. So if we use the Thirder Rule, the r esulting credence in the ev ent that both coins came up the same is 2 / 4 = 1 / 2. (I ap ologize f or any confusion caused b y the unfortunate coincidence that the Halfer Rule prescrib es 1 / 3 in this con text, and the Thir de r Rul e 1 / 2.) 4 Sunday? In fact, it seems to me that this violatio n o f the Reflectio n P rinciple is more serio us than the thirde r ’s alleg ed violation of it in the o riginal Sleeping Beauty problem, for the following reason. In the o riginal problem, it would b e unreasona ble to sa y that the fact tha t the thirder will end up having a cr edence of 1 / 3 on T uesday implies that she sho uld alrea dy have a credenc e of 1 / 3 o n Sunday . After all, s he do es not a lw ays wak e up on T ues day , and if she w er e capable of, in her sleep, recog nizing that she has not b een awok en, she would assign credence 1 in Heads then. Tha t is why the purpor ted violatio n focuse s on the Monday credence in Heads, not the T uesday one. But it s eems illegitimate to consider Monday separately from T uesday , b ecause Bea ut y canno t distinguish them. Th us, it see ms debata ble whether the thirder really v iolates the Reflection Principle – more precisely , whether she violates any version of this principle by which we would care to a bide. By con tra st, in the tw o-coins example considered here, it do es not seem that the argument that Monday and T uesday should be considered to gether c an b e o f much help to the supp orter o f the Halfer Rule, bec ause Beaut y is alwa ys a woken and, a ccording to the Halfer Rule, a lw ays ends up with a credence o f 1 / 3. But I leav e for ma lizing the sens e in which the violation is more serio us for another day . T o make matters yet worse for the Halfer Rule, conside r the following twist to the tw o-coins example. O n both Monday and T uesday , after Beauty has observed the coin toss outcome and been a wake for a little while longer, the exp erimen ter tells her wha t day it is. Say she observed Hea ds and was then told (a bit later) that it is Monday . No w only tw o worlds s urviv e elimination: HH and HT. The Halfer Rule will as s ign each of them credence 1 / 2, resulting in a credence of 1 / 2 that bo th coins came up the same. 5 But this is yet another violation of the Reflection Principle: after seeing the o utcome o f the co in toss but before learning what day it is, Be a ut y , if she follows the Halfer Rule, places credence 1 / 3 in the even t that the co ins came up the sa me, but she also k nows that once she is told what day it is, in either case, she will shift her credence to 1 / 2. This is p erhaps the most egregious violatio n of the Reflec tio n P rinciple that we hav e encountered, beca us e in this case she is not put to sleep and do es not hav e memorie s era sed a s she transitions from one c redence to another. 6 Again, I leave formalizing the sense in which the violation is mor e serious than 5 The Thirder Rule still gives 1 / 2 as we l l , b ec ause there are only tw o p ossible cen tered wo r lds r emaining, namely M onda y in HH and Monday i n HT. 6 On the face of it, the same happens in the Shangri La example given by Arntzenius [2003]. (I thank an anon ymous reviewer for Philosophical Studies for calling my atten tion to this. ) In this example, someone exp erience s A or B according to the outcome of a coin toss. He kno ws, though, that at a certain point in time after the experience, any memories of B will be replaced by false memories of A, whil e any memories of A will b e left inta ct, so that he will not b e able to tell the tw o cases apart. Then, while exp eriencing A, he has credence 1 in Heads, in spite of knowing f ull well that he will later hav e credence 1 / 2 in Heads, without his memory being compromised in this particular case. Of course, this is en tirely due to the fact that in a parallel case, his memory w ould be compromised to b e indistinguishable from what he curr en tly kno ws wi ll be his (true) memory of A. Thereb y , he will lose a piece of inf or mation that he current ly has. But nothing simi lar happ ens in the enriched tw o-coins example. At the point i n time when Beaut y is told what day i t is, her memory is neve r compromised, and she never loses information. 5 HH (1 / 4) HT (1 / 4 ) TH (1/ 4) TT (1/ 4) Monday see Heads see Heads asleep asleep T uesday see Heads asleep see Heads asleep Figure 2: A cost-cutting v ariant of the tw o-coins exa mple in Figure 1, the o nly mo dification being that Be aut y is no longer awok en on T ails. the other violations for a nother da y . 4 What Options Remain for th e Halfer? If the Halfer Rule is untenable, then is there ano ther full gener alization of halfing that is more defensible? I hav e a lready men tioned a few interpretations of ha lfing that do not alwa ys agree with the Ha lfer Rule and get in to their own br ands of trouble as a result. In this final section, I hop e to assess a bit more s ystematically how ha lfing ma y b e ge ne r alized in a trouble- free way . One helpful ex ample to cons ider is a v ariant o f the tw o- c o ins exa mple in- tro duced ea r lier. The only mo dification tha t is needed to o btain this v ariant is the following. T o cut down on the co st of the v ario us dr ugs in volved in the aw akenings, the exp erimen ter has decided to only a waken Beauty when the coin corres p onding to the curr en t day has come up Hea ds. O n T ails days, the exp er- imen ter just lets her sleep. Beauty is o f co urse infor med o f this mo dification at the outset. As a result, on a Heads aw akening it is no longer necessary to show her that the coin has come up Heads, b ecause this is already implied by the fa ct that she was awok en at all. On the o ther hand, nothing is lost b y showing her the Heads o utcome an yway . Figure 2 illustr a tes the mo dified example. Now what should Beauty believe upon awak ening (with Heads)? It a ppears to me that in this v ariant, a n y rea sonable gener alization o f halfing must place credence 1 / 3 in each of the worlds HH, HT, a nd TH. Sp ecifically , TT is ruled out by the evidence, HT and TH should have the same credenc e by symmetry , and it app ears that the only motiv ation one could hav e for g iving HH a hig her credence is that this world has mor e ce n ters – but that is thirder reas oning! If these 1 / 3 credences are rig h t, it leads to the following ques tion. How could the fact that we no lo nger a waken Beauty o n T ails days affect her co rrect c r edence on Heads days? If one answers that, well, in fact, it should not a ffect it, then all is lost for the halfer. It implies that the halfer is stuck with the Halfer Rule’s prescrib ed credences for the origina l tw o-c o ins exa mple, which ar e un tenable . So, the halfer must adopt a p osition that allows for the prescrib ed credence to change when we c ha nge whether Be a ut y is aw oken under o ther conditions – conditions that she hers e lf would b e able to distinguish fro m the curr en t ones. This may seem unappea ling; in particula r, the thirder needs to ma ke no such mov e. Still, reasonable ge neralizations of ha lfing ma y fit the bill. F or example, co nsider the following approach, based o n sp ecifying the eviden tial selection pro cedure. The ha lfer co uld treat her curr e nt w a king exp erience as 6 being ra ndomly sele c ted from her waking exp eriences in the actual world. In the or iginal tw o- coins ex ample, b y Bayes’ rule this results in 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 / 4) 1 · (1 / 4) + (1 / 2) · (1 / 4) + (1 / 2) · (1 / 4) = 1 / 2 thereby escaping the Halfer Rule’s fatal mistake. But in the mo dified (cost- cutting) tw o-co ins v ariant, w e obtain 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 / 4) 1 · (1 / 4) + 1 · (1 / 4) + 1 · (1 / 4) = 1 / 3 so that this is still a sens ible genera lization o f halfing. Still, this g eneralization is not without its o wn tro ubles. F or one, a pplying this ge ne r alization to the scenario desc r ibed by Pittard [201 5] results in the same credences that he ad- vocates, which lead to his disa greemen t para do x. (Indeed, he argues for these credences based o n a similar ev iden tial selection pro cedure.) It seems, then, that g eneralizing to arbitrar y exa mples will require the halfer to adopt a rule that leads to one v ariety or another of unin tuitive consequences. Perhaps a rule ca n b e fo und whose unintuitiv e conse quences are, up on further insp e ction, quite reas onable, or at leas t a bullet worth biting in or der to hold on to halfing. 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