A Dutch Book against Sleeping Beauties Who Are Evidential Decision Theorists
aa r X i v : . [ s t a t . O T ] M a y A Dutch Book against Sleeping Beauties WhoAre Evidential Decision Theorists ∗ Vincent ConitzerDuke University
Abstract
In the context of the Sleeping Beauty problem, it has been argued thatso-called “halfers” can avoid Dutch book arguments by adopting eviden-tial decision theory. I introduce a Dutch book for a variant of the SleepingBeauty problem and argue that evidential decision theorists fall prey toit, whether they are halfers or thirders. The argument crucially requiresthat an action can provide evidence for what the agent would do not onlyat other decision points where she has exactly the same information, butalso at decision points where she has different but “symmetric” informa-tion.
Keywords: self-locating beliefs, Sleeping Beauty problem, evidential de-cision theory, Dutch books. ∗ This paper appears in
Synthese , Volume 192, Issue 9, pp. 2887-2899, October 2015. Thefinal publication is available at Springer via http://dx.doi.org/10.1007/s11229-015-0691-7 /
2) Tails (1 / The Sleeping Beauty problem (reviewed below) has attracted much attentionbecause it relates to a variety of unresolved philosophical problems. It is, inthe first place, a puzzle about beliefs. The question of what Beauty shouldbelieve upon awakening divides philosophers and others into multiple camps,mainly “halfers” and “thirders” (though finer distinctions can be made). Butdecision theory has also been pulled into the debate. This is because one naturalstrategy for adjudicating between the halfer and thirder positions is to evaluatethe effects of the candidate beliefs on Beauty’s decisions; if one position resultsin clearly irrational decisions, this would appear to settle the matter. As itturns out, the success of such arguments appears to hinge on which version ofdecision theory – causal or evidential – Beauty adopts. A natural reaction is tofeel disappointment at the fact that a clean resolution to the original questionappears elusive. But it is also possible to see opportunity: perhaps variantsof the Sleeping Beauty problem actually allow us to adjudicate between causaland evidential decision theory instead. This is the aim of this paper. Of course,this may also help us resolve the original question, insofar as the argumentsagainst halfing or thirding become decisive once the debate between causal andevidential decision theory has been settled in this context.Let us recall the standard variant of the Sleeping Beauty problem [Elga,2000]. Beauty is put to sleep on Sunday, and a fair coin is tossed. If it landsHeads, she will be briefly awoken on Monday only. If it lands Tails, she will bebriefly awoken on Monday, and then again on Tuesday. The table in Figure 1summarizes this. Whenever she is awoken, she does not remember any previousawakenings, nor does anything in the room indicate to her what day it is. Beautyknows all this throughout. Now let us put ourselves in the shoes of Beauty whenshe has just been awoken in the experiment. What should be her credence thatthe coin came up Heads? Halfers believe that the answer is 1 /
2; after all, thefact that she has been awoken should not tell her anything about the possible(uncentered) world that she is in, because the awakening event is consistent witheach of the two possible worlds. In contrast, thirders believe that the answer is1 /
3; after all, only 1 / Titelbaum [2013] gives a useful overview of these connections. /
2) Tails (1 / Dutch book to which one of the two positions is vulnerable. A Dutchbook is a set of bets that the agent in question would all accept individually, butthat together ensure that the agent incurs a strict loss overall. In a diachronic
Dutch book, the bets are offered at different times; all of the Dutch booksdiscussed in this paper are diachronic. Hitchcock [2004] describes a Dutch bookargument against the halfer position. In this Dutch book, the bookie first –before Beauty goes to sleep – sells her a bet that costs 15 and pays out 30 ifthe coin lands Tails. Then, each time that Beauty awakens, he sells her a betthat costs 10 and pays out 20 if the coin has landed Heads. The idea is thatif Beauty is a halfer, then she will always be willing to accept these bets. (Infact, she will be indifferent between accepting them and not accepting them; itis straightforward to slightly modify the payoffs so that she will strictly preferto accept them.) Now, if the coin lands Heads, she will buy bet 1 once and bet2 once, at a total cost of 25, and bet two will pay out 20 – so she will run a netloss of 5. On the other hand, if the coin lands Tails, she will buy bet 1 onceand bet 2 twice , at a total cost of 35, and bet 1 will pay out 30 – so again shewill run a net loss of 5. Thus the Dutch book succeeds. The table in Figure 2summarizes Hitchcock’s Dutch book.However, Draper and Pust [2008] point out that this Dutch book does notsucceed against a halfer that accepts evidential decision theory, because, as Arntzenius[2002] pointed out earlier, such an agent would calculate the expected utilityof her options differently in the Sleeping Beauty problem. Specifically, considerthe situation where Beauty accepts evidential decision theory, has just beenawoken, and is now calculating the expected value of accepting bet 2. For thiscalculation, by EDT, she assumes that, if the coin has come up Tails, she alsohas accepted or will accept bet 2 on the other day. Thus, in this case, acceptingbet 2 (on both days) comes at a cost of 20, and pays out nothing, for a net lossof 20; whereas in the Heads case, accepting bet 2 (once) comes at a cost of 10and pays out 20, for a net gain of only 10. So even if her credence in Heads isas high as 1 /
2, she will not accept bet 2, and the Dutch book fails. To minimize clutter, I will not specify currency units such as dollars or euros. Draper and Pust do propose a modified Dutch book that involves telling Beauty that itis Monday and then offering a bet; this Dutch book works against some halfers, whether theyare causal or evidential decision theorists, but not against so-called “double-halfers” who holdthat the correct credence remains 1 / Briggs’ proof implicitly assumes that Beauty, upon an awakening, always hasthe same information available to her. This rules out variants such as “Techni-color Beauty” [Titelbaum, 2008] where Beauty, upon awakening, sees a piece ofpaper whose color is not always the same. The Technicolor Beauty variant illus-trates that irrelevant additional information, such as the paper’s color – which isuncorrelated with the variable of interest – can change the halfer’s credence forthe variable of interest (at least under a standard interpretation of halfing – butsee Footnote 5). Indeed, Briggs does discuss Technicolor Beauty and concludesthat this instability of the halfer’s credence is a mark against halfing, but thisis only after she has completed her discussion of Dutch books. What has notyet been appreciated, to my knowledge, is the following: if Beauty is an evi-dential decision theorist, then in variants such as Technicolor Beauty where shedoes not always have the same information available to her upon waking, sheis vulnerable to Dutch books, regardless of whether she is a halfer or a thirder.This is what I will demonstrate in what follows. I first introduce a variant ofthe Sleeping Beauty problem.
Beauty will be awoken twice, once on Monday and once on Tuesday. As usual,when she is awoken on Tuesday, she has no memory of the previous awakening.The only information available to her upon awakening (besides the informationthat was available to her at the start of the experiment) is the color of the roomin which she is awoken. Two distinct fair coins are tossed to determine the colorof the room in which she is awoken on each of the two days. Coin 1 has a whiteside and a black side; this coin will be used to determine the color of the roomin which she is awoken on Monday. Coin 2 has a grey side and a side with theword “opposite” written on it; this coin will be used to determine the color of The idea that thirding goes hand in hand with CDT, and halfing with EDT, also findssupport elsewhere, for example in the context of the absentminded driver problem [Schwarz,2014b].
4G (1 /
4) WO (1 /
4) BO (1 /
4) BG (1 / / / /
3, 1 /
3, 1 / then all A standard interpretation of how the halfer assigns credences in general (e.g., Halpern[2006], Meacham [2008], Briggs [2010]) would indeed, in the Technicolor Beauty variant, assigncredences of 1 /
3, 1 /
3, 1 / / Beauty will be offered the following bets. • Bet 1.
This bet will be offered once, right before the experiment. It costs20 and pays out 42 if coin 2 comes up Grey. • Bet 2.
This bet will be offered once each time that Beauty awakens inthe white or the black room, but never in the grey room. Thus, it willbe offered once overall if coin 2 comes up Grey, but twice overall if coin2 comes up Opposite. It costs 24 and pays out 33 if coin 2 comes upOpposite.It should be noted that these bets are legitimate in the sense that the bookieis not exploiting any information that Beauty does not have available to her. Iwill revisit this point in Subsection 4.3 below. found that keeps the credence in Heads at 1 / P (see red | HR) = 1 (because HR has only one awakening) but P (see red | TR) = P (see red | TB) = 1 / P (HR | see red) = P (see red | HR) P (HR) P (see red | HR) P (HR) + P (see red | TR) P (TR) + P (see red | TB) P (TB)= 1 · (1 / · (1 /
4) + (1 / · (1 /
4) + (1 / · (1 /
4) = 1 / P (WG | see white) = P (see white | WG) P (WG) P (see white | WG) P (WG) + P (see white | WO) P (WO) + P (see white | BO) P (BO)= (1 / · (1 / / · (1 /
4) + (1 / · (1 /
4) + (1 / · (1 /
4) = 1 / P (see white | WG) = 1 / /
3, 1 /
3, 1 /
6G (1 /
4) WO (1 /
4) BO (1 /
4) BG (1 / ·
24 = 68, and receives a payout of 2 ·
33 = 66 from thetwo iterations of bet 2 – so again she runs a loss of 2. The table in Figure 4summarizes the Dutch book.But who will actually accept these bets? Both causal and evidential decisiontheorists will accept bet 1, because before the experiment there is a 50% chancethat coin 2 comes up Grey, so that the expected payout from bet 1 is 21, whichis greater than 20. Will a causal decision theorist accept bet 2? No: given thatthe room is (say) white, she believes that there is a probability of 2 / / ·
33 = 22,which is less than the cost of the bet, 24. So the causal decision theorist is notvulnerable to this Dutch book.All that remains to show is that the evidential decision theorist will acceptbet 2 whenever it is offered to her. Here, then, is the crux of the argument.Suppose the room is white. Then, accepting the bet is strong evidence thatshe also would also accept the bet in the black room. After all, the situation(including the bets) is entirely symmetric between white and black, so it ishard to see why Beauty would accept the bet in the white room but not in theblack room. Similarly, not accepting the bet is strong evidence that she wouldalso not accept it in the black room. Now, her credence is 2 / / / · − (1 / ·
24 = 12 − >
0. So she will accept the bet in the whiteroom! Of course, by the symmetry between white and black, this means thatshe will also accept the bet in the black room. Hence, the evidential decision7heorist falls for the Dutch book.Some intuition for what makes this Dutch book work is as follows. Fromthe perspective of maximizing expected net gain, clearly bet 1 is a good one toaccept, resulting in an ex ante expected net gain of (2 / · − (2 / ·
20 = 1.This suggests that the evidential decision theorist’s mistake is in accepting bet 2.Always accepting bet 2 results in an ex ante expected net loss of (2 / · − (2 / ·
18 = 3. So what makes the evidential decision theorist accept this bet? Supposeshe is in a white room. She will reason that if she accepts, then she would alsoaccept in a black room. There are three possible worlds where she is in a blackroom at some point: WO, BO, and BG. But BG is ruled out by the evidence ofcurrently being in a white room and therefore does not factor into her currentexpected payoff calculation. Moreover, this is precisely the one world whereaccepting the bet in a black room comes at a cost! Therefore, she evaluates thequality of the bet based on a biased selection of the centered worlds in a blackroom, making the bet look better than it is. The causal decision theorist, onthe other hand, ignores bets in black rooms altogether when making a decisionin a white room, and thereby avoids being affected by this selection bias.
What has gone wrong for the evidential decision theorist, particularly the evi-dential decision theorist who is a halfer and is hence supposed to be immune toDutch books according to Briggs [2010]? In this section, I first discuss the keytechnical problem with attempting to apply Briggs’ proof in the context of theWBG variant. As noted earlier, a key issue is the possibility that knowledgeof a decision in one information state affects beliefs about decisions in a differ-ent information state. One way around the Dutch book, therefore, is to denythe possibility of beliefs being affected in this way. I continue by arguing thatthis escape route is unreasonable. I conclude this section by discussing to whatextent susceptibility to the Dutch book indicates irrationality.
Why does Briggs’ proof of the invulnerability to Dutch books of evidentialdecision theorists who are halfers not apply here? To appreciate this, it willbe helpful to first discuss some essential features of her proof. As noted earlier,it implicitly assumes that the information that Beauty has available to herupon awakening during the experiment is always the same. Briggs uses N W to refer to the number of centers (awakening events within the experiment) inpossible (uncentered) world W . Suppose Beauty is considering a bet whose netpayout (including the initial cost of the bet) is X W in world W . If she is anevidential decision theorist, she will reason that if she accepts (rejects) the betnow, then she also accepts (rejects) it on all other occasions. She concludes that8er net payout is N W X W if she accepts, and 0 if she rejects. Of course shedoes not necessarily know in which possible world she is, so she has to considerthe expected value. Letting Cr u denote halfer credences, an evidential decisiontheorist who is a halfer will accept the bet if P W Cr u ( W ) N W X W > N W − W affect her decision, so that N W X W is replaced by X W in the above. However, if she is a thirder rather thana halfer, then her credence in world W will be proportional to Cr u ( W ) N W ratherthan Cr u ( W ). Hence, again, she accepts the bet if P W Cr u ( W ) N W X W > N W comes from the credence in thiscase. Briggs proves that betting in this way (“betting at thirder odds”) leavesBeauty immune to Dutch books.All of this makes sense when Beauty always has the same information uponawakening. When this is not the case, we should first enrich the notation abit. What is relevant is not the total number of centers N W in a world, butrather the number of centers N IW consistent with the current information I .For example, in the WBG variant, it does not suffice to know that N WG = 2;rather, we need that N whiteWG = 1, N greyWG = 1, and N blackWG = 0. Then, one mightsuppose that with information I , the credence in some world W (that is notyet ruled out by I ) is Cr u ( W ) for the halfer and proportional to Cr u ( W ) N IW for the thirder. (Note that this would be consistent with the credences in theWBG variant.) The causal decision theorist who is a thirder would then acceptthe bet if P W Cr u ( W ) N IW X W >
0. Now, what about the evidential decisiontheorist who is a halfer?
Suppose it were the case that now accepting (rejecting)the bet with information I leads her to believe that she always accepts (rejects)it with information I , but does not influence her beliefs about what she would dogiven any other information . Then, in world W , she believes her net payout ifshe rejects the bet is c (i.e., whatever she expects to get from any bets acceptedwhen she has information other than I ), and her net payout if she accepts thebet is N IW X W + c . The c term cancels out, and hence, again, she will acceptthe bet if P W Cr u ( W ) N IW X W >
0. So the argument would appear to carrythrough. The problem is that, as I have argued (and will argue further inSubsection 4.2), it is unreasonable to suppose that the decision made with thecurrent information does not affect beliefs about decisions made with slightlydifferent information! If it does affect them, then the equivalence argument fallsapart: we can no longer cancel out the c term in the above because it nowdepends on the decision made with information I , and as a result the conditionfor accepting a bet changes in the case of the evidential decision theorist who isa halfer. This is what allows the Dutch book for the WBG variant. (It is worthemphasizing again that in the WBG variant, the credences of 1 / This assumes that she will be offered the same bet upon each awakening, but this is areasonable requirement: see Subsection 4.3. Again, note that she should always be offered the same bet whenever she has information I ; otherwise, the bet offered would in fact give her additional information. See Subsection 4.3for further discussion. C , what happens is that, for each possible world W with multiple centers that are like C , she counts the effects of the actionmultiple times in W . This is a mistake. On the other hand, when a halfer incentered world C assesses the probability of a possible world W that containsmultiple centers that are indistinguishable from C , she fails to give probabilityto W that is proportional to the number of such centers in W . This, too, is amistake. However, the two mistakes happen to cancel each other out exactly, if the only centers that are like C are the centers that are indistinguishable from C . Typical Sleeping Beauty variants have this feature, giving rise to the ideathat halfers who adopt evidential decision theory avoid Dutch books. However,it is possible for two centered worlds to be alike while simultaneously being dis-tinguishable . This is what is happening in the WBG variant – white centeredworlds and black centered worlds are alike but distinguishable. Because of this,the mistake in assessing actions’ payoffs is still made, but it is not canceled outbecause no mistake is made in assessing the probabilities of possible worlds. Based on the above, one strategy for the evidential decision theorist to avoid theDutch book is to never let decisions in one information state affect beliefs aboutdecisions in different information states. Suppose she takes this approach andwe vary the decision that she makes in the current centered world. As we do so,her beliefs (conditional on this decision) about what she would do in centeredworlds that are indistinguishable from the current one will also vary, but herbeliefs about what she would do in other centered worlds (in particular, onesthat are alike but distinguishable) will not. This indeed avoids the Dutch book.But this approach seems highly unappealing.Of course, one could add details to the case to make this approach seemmore palatable. For example, we may suppose that before the experiment, aneurological examination revealed to Beauty that the part of her brain that isactivated to make decisions in white rooms is entirely disjoint from the partactivated to make decisions in black rooms. With these (or perhaps alternative)additional details, it can perhaps be successfully argued that her beliefs aboutwhat she does in black rooms should not be affected by information about whatshe does in white rooms. But this fails to get the evidential decision theorist outof trouble. The Dutch book does not need to succeed no matter what details areadded to the case. For it to exhibit a problem with evidential decision theory,all that is necessary is that is succeeds for some details. We may just as wellspecify that the neurological examination reveals to Beauty that the part of herbrain involved in decision making is entirely uninfluenced by the color of the10oom. In that case, it seems entirely unreasonable for beliefs about black-roomdecisions to be uninfluenced by knowledge of white-room decisions.Moreover, even though this is in fact not necessary for the argument tosucceed, I would argue that when no details are added to the case – i.e., Beautydoes not have any additional relevant information, such as the results of aneurological examination – by default, beliefs about black-room decisions shouldbe influenced by knowledge of white-room decisions as I have suggested. By wayof analogy, suppose we see Kim treating another person kindly, and this otherperson happens to stand to her left. Clearly, this will increase our credencethat Kim would treat other people who stand to her left kindly. But it wouldbe preposterous to not also increase our credence that Kim would treat peoplewho stand to her right kindly, unless we have reason to believe that there isa fundamental asymmetry between left and right (e.g., if we know Kim doesnot hear well with her right ear and this causes her great frustration). Thesituation is similar in our context: unless we have a particular reason to believethat the color of the room is relevant to the decisions (as in the first example ofa neurological examination), the Dutch book goes through. Does susceptibility to diachronic Dutch books really indicate irrationality? Thisquestion has been discussed at length in earlier work [Hitchcock, 2004, Briggs,2010], and I do not have much to add that is new, but it is worth revisiting thekey points here. Some Dutch book arguments have been made that require thebookie to have information that Beauty does not. For example, consider thefollowing Dutch book argument against a thirder, presented by Hitchcock [2004]precisely in order to highlight this issue. On Sunday, Beauty is offered a bet thatpays out 30 on Heads, which costs 15. Then, on Monday, Beauty is offered a betthat pays out 30 on Tails, which costs 20. The argument is that she is willingto accept both bets – in particular, she is willing to accept the latter becauseshe at that point believes the probability of Tails is 2 / Now, susceptibility to being Dutch-booked by a bookie that has additionalinformation does not seem to indicate a failure of rationality. After all, consider One might also suppose that knowledge of her decision in a white room makes Beautyonly (say) 99% confident in what her decision would be in a black room, where the remaining1% is intended to capture a small probability that the room color is somehow relevant to thedecision. It is easy to see that the Dutch book still goes through under these conditions. As already pointed out by Hitchcock, to be precise, what information the bookie has isnot exactly what is at issue. If the bookie does not know what day it is, but someone elseprevents the bookie from offering the second bet on Tuesday to make the Dutch book work,this is just as problematic. The point is that the process as a whole by which bets are offeredto Beauty cannot use information that is unavailable to Beauty.
11 (1 /
4) WB (1 /
2) B (1 / /
2, and nottake the bet.However, like Hitchcock’s Dutch book, the Dutch book presented in thispaper does not require the bookie to have superior information. It is sufficientfor the bookie to know what Beauty knows (i.e., the color of the room) in orderto decide which bet to offer her. In Hitchcock’s words, he can “sleep with her” –that is, be put to sleep and awoken and have his memory impaired in exactly thesame manner. As a result, being offered a bet can never provide Beauty withadditional information. This remains true even if she is told the bookie’s entirebetting strategy at the outset . This is perhaps what most strongly suggests thatsusceptibility to such a Dutch book indicates a degree of irrationality: even ifBeauty is completely aware of the game the bookie is playing with her, she stillfalls for the sure loss.
The evidential decision theorist may hope that the type of Dutch book presentedhere is inherently restricted to scenarios where the agent’s memory is impaired.But I believe that the problem runs at least a bit deeper than that. For example,we can easily modify the WBG variant so that there are now two Beauties,one (“White”) who is awoken whenever the original Beauty was awoken in thewhite room, and one (“Black”) who is awoken whenever the original Beautywas awoken in the black room. These Beauties can be awoken simultaneously(in separate rooms) rather than sequentially, thereby combining the WO andBO worlds into a single WB world. The table in Figure 5 summarizes thisvariant. We can then let the two Beauties bet under a joint account whosevalue they are both trying to maximize, and, if they are evidential decisiontheorists, they will fall prey to the same Dutch book, even without memoryimpairment (assuming no communication between them). The table in Figure 6summarizes the Dutch book. For the purpose of symmetry, we split bet 1 intotwo halves, each denoted 1’, one for White and one for Black, with half thecost and half the payout each. Note that in this context, my interpretation12 (1 /
4) WB (1 /
2) B (1 / Specifically, when offeredbet 2, White places credence 2 / · / / · − (1 / ·
24 = 12 − >
0, she accepts the bet (andBlack will do so as well, by symmetry).Does the fact that the two Beauties together are susceptible to a Dutchbook indicate that they are irrational? This is certainly not as well establishedas it is for the case of a Dutch book for a single agent (as summarized inSubsection 4.3), and some skepticism is in order. For example, it is well known ingame theory that rational behavior by multiple agents can result in an outcomethat is strongly Pareto dominated, i.e., there exists another outcome that allagents would strictly prefer. The Prisoner’s Dilemma is the standard example.However, such examples rely on the agents having different preferences. Incontrast, the two Beauties above have the exact same preferences. Also, itseems that the key properties that make a Dutch book convincing, as discussedin Subsection 4.3, still hold here. It is true that the bookie will have moreinformation than either single Beauty alone. However, this is easily fixed bystipulating that there are two bookies, also with a joint account, each of whomis assigned to sleep with and offer bets to one of the Beauties. Then, again, beingoffered a bet does not provide either Beauty with more information, and thisremains true even if the bookies’ joint betting strategy is common knowledgeat the outset.It appears, then, that the Dutch book argument presented in this paperdeals a serious blow to evidential decision theory. Of course, evidential decisiontheory is often applied in settings where a decision provides evidence not aboutpast or future decisions that are similar, or about decisions by another similar Variants of the Sleeping Beauty problem with clones are fairly common – see, e.g., Elga[2004] and Schwarz [2014a].
Acknowledgments
I thank the anonymous reviewers for many useful comments that have helpedto significantly improve the paper.
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