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De Se Belief and Rational Choice Draft, please don't cite Several philosophers have argued that de se beliefs—one's thoughts about one-self in a characteristically ‘first-personal' way—have special features that setthem apart from other kinds of belief. Frege famously seemed to argue thateveryone's thoughts about themselves are distinct, and unshareable.1 Otherphilosophers have argued that some de se beliefs require a refinement of attitu-dinal content or severing antecedently plausible connections between the objectsof belief and belief states.2 These claims raise further questions about whether the peculiarities of de se belief require special adjustments to theories in which such beliefs may play arole: for example, in the compositional semantics of attitude reports, accountsof assertoric content, and theories of rational belief change. The Sleeping Beautypuzzle raised in Elga (2000) has been used to argue that the latter theories ofrational belief change do require such adjustments.
Sleeping Beauty. In an experiment, Beauty is put to sleep Sundaynight by scientists until Wednesday. She will be woken up for certainon Monday and administered a drug to forget that waking. If andonly if a fair coin tossed during the experiment lands tails she willbe woken up again on Tuesday.
Beauty's predicament raises the following question: given that she knows howthe experiment works, what ought she think is the likelihood of the coin's land-ing heads after she wakes at some point during the experiment? Two camps,"halfers" and "thirders", take the answer to be 1 and 1 respectively. A guiding halfer intuition is that before the experiment, Beauty's belief that a fair coinlands heads should be 1 and that upon waking, no new information of relevance to the toss has been obtained. A guiding thirder intuition is that in repeatediterations of the experiment only one third of the wakings would be wakings inwhich the coin landed heads. I won't rehearse the array of more sophisticatedarguments for each side here.3 What's important is that the ways we use toflesh out either of these answers tend to have dramatic implications for howwe should adjust our frameworks for rational belief change, or the principlesgoverning them, to accommodate de se beliefs.
In a similar spirit, I'm going to use some variants of Sleeping Beauty to argue that theories of rational choice, as supplied in decision theory, require specialchanges to accommodate de se beliefs over and above those required from ourtheories of rational belief change. Questions about rational choice are, of course,closely connected to questions about rational belief since what it is rational todo (at least on one construal) depends not simply on what one believes, butwhat one ought to believe. I'll eventually be exploiting this connection betweenrational belief and rational choice in examining how de se beliefs may complicatedecision theoretic frameworks.
For now, though, I want to provisionally assume the applicability of some standard frameworks for rational choice because such frameworks hold the promiseof supplying us with an easy method for determining answers to the questionoriginally posed by Sleeping Beauty: how our beliefs should change in cases ofde se ignorance. Decision theory tells us how to get from rational beliefs andvalues to rational choices. Consequently, if we can present a case where it isclear what values an agent has, and what choices they should make, we shouldbe able to ‘work backwards' to determine what beliefs they should hold.
The following embellishment of the Sleeping Beauty scenario is designed to allow us to do just this. We suppose that Beauty gets various payoffs dependingon what actions she performs during her experiment, and adjust those payoffsso that different answers to the question "what ought Beauty believe" supplyus, though decision theory, with different actions she should perform. If we haveintuitions about what Beauty should do in these cases, these will privilege oneof the competing views about what Beauty ought to believe. Here's a genericform such an elaboration of the case might take.
The Waking Game. Scientists put Beauty in an empty white roomon Sunday with two buttons labeled "Left" and "Right". Beauty isput to sleep Sunday night for n days. If the toss of a fair coin earlyin the experiment lands tails she will woken up for n days beginningMonday. Otherwise she will be woken up only Monday and putback to sleep for n − 1 days. Each time she is woken, she is giventhe opportunity to push the left or right buttons, and then will beadministered a drug to forget the waking before being put back tosleep. Beauty is given certain payoffs in dollars depending on whichbuttons she pushes at which times, and is broached of the payoffstructure at the outset of the experiment.
In part to simplify the application of decision theoretic frameworks, and in partto get the case to draw apart halfer and thirder views, I want to add two featuresto the scenario Beauty now faces.
Randomizing Prohibited : Mixed strategies—where, for example, Beautyflips a coin to decide whether to push Left or Right—will undulycomplicate the cases I want to consider.
So suppose that scien- tists have prohibited such randomizing. They will allow Beauty's memories to return after the experiment at which point they willadminister a polygraph and ask her if she has attempted to random-ize her choices. If she fails the test scientists will kill Beauty's petdog—a consequence with boundless negative utility. Beauty knowsshe is very likely to fail the test if she tries to randomize her choices.
Previous Runs: Suppose Beauty has seen the experiment performedinnumerable times before on late night reality television. In virtuallyall trials, when the coin landed tails, test subjects pushed the samebutton every day. Scientists have hypothesized that this is the casebecause erasing people's memories of previous wakings ensures thatthey are in the same state relevant to the determination of theirchoice of button-pushings each day.
Note, Randomizing Prohibited does not forbid Beauty from simply "choosingarbitrarily." We can suppose such arbitrary choices to be stable across tailswakings for prior test subjects in Previous Runs.
Now, to fill out the details of the Waking Game, suppose the number of days Beauty will wake if the coin lands tails is four and payoffs are given as follows.
Payoffs on Heads: Payoffs on Tails: if Left every day if Right every day if Left on Monday and Right another day if Right on Monday and Left another day The payoffs for heads are straightforward, while the payoffs on tails are a littlemore complex. Assuming the coin lands tails, then if Beauty pushes Left everyday she makes $100, and if she pushes Right every day she gets $200. If, however,she ever changes which button she pushes, then the payoff is determined by herpush on the first day. If she pushed Left on that day then she makes $200, andif she pushed Right then she makes $100.
First, let's ask a relatively simple question: what ought Beauty to plan to do Sunday? It should be clear given Randomizing Prohibited that Beauty hasonly two options. Since Beauty can't randomize and since all the wakings areindistinguishable, she can only plan to push Left upon waking, or plan to pushRight upon waking.
It should be clear that if Beauty plans to push Left and will succeed in doing so upon waking, she can expect an average payoff of $250: about half of thetime she'll get $400, and about half of the time she'll get $100. On the otherhand, if she plans to, and succeeds in, pushing Right when she wakes up, she will always get $200, whether heads or tails. I take it to be uncontroversial thatif Beauty can reliably plan to push either Left or Right, she ought to plan topush Left.4 But there is a distinct question about what Beauty should do: what should she actually do upon waking? It turns out that different views about whatBeauty's de se beliefs ought to be, when combined with standard decision the-oretic frameworks, yield different answers.
When Beauty wakes, there are five scenarios she might be in: either the coin landed heads and it is Monday, or the coin landed tails and it is either Monday,Tuesday, Wednesday, or Thursday. I'll note these options by MonH, MonT,TueT, WedT, and ThuT respectively. Standard thirder arguments support theview that Beauty's credence should be distributed as follows: MonH = MonT = TueT = WedT = ThuT = 15 On the other hand, standard halfer arguments support the alternative credencedistribution: Thirders think that upon waking Beauty should believe that she is most likelyin a tails scenario. But if that is the case, Beauty should probably push Right:she most often stands to gain $100 by doing so. Halfers think that Beautyshould give equal credence to being in a Tails and in a Heads scenario. Butthen Beauty should push Left: she stands to gain $200 half of the time by doingso, instead of gaining $100 the other half of the time by going right.
This plays out slightly differently depending on which version of decision theory one actually endorses, but the outcome is the same: thirders should pushRight, halfers should push Left. To see this, I'll go through the calculations forhalfers and thirders in the context of both causal and evidential decision theory(CDT and EDT respectively).5,6 Causal decision theorists think it is important to separate the outcome of one's choice and its causal effects from states of the world which are evidentiallyrelated to, but not causal outcomes of one's choice. This stance is relevant tothe version of the Waking Game I have set up. To see this, suppose upon wakingBeauty is considering what to do assuming that it is Tuesday. This raises thequestion: how should she see the relationship between her choice today, herchoice the day before, and her choice on the subsequent two days? CDT stresses that that even if Beauty's choice, say, to push Left is evidence that she has and will push Left on other days, this information shouldn't befactored into her decision as determined by her choice to push Left unless shetakes her choice on Tuesday to cause these outcomes. On the present descrip-tion, that seems unlikely (especially for choices in the past). Consequently CDTinstructs Beauty to fix her beliefs about her performance on other days, anddecide what to do on the basis of those credences.
Beauty's belief about what she did do and will do on other days should be constrained by Previous Runs. She should assign a very high credence to the claim that she did do and will do the same things on every day other than ‘today.'So, roughly, she should distribute most of her credence between two possibilities:between being someone who chose and will choose Left on other days—a ‘lefty'—on the one hand, and being someone who chose and will choose Right on otherdays—a ‘righty'. How should she allot her credence between these two claims?It turns out not to matter, since there is a dominance argument for Beauty topush Right on waking. This can be seen from the following computations ofexpected value of choosing Right ‘today' (noted RT ) and choosing Left today(LT ), on the assumption that Beauty is a righty or a lefty and her utilities arelinear in dollars.
Version 1, Computations for Thirder + CDT Let me go through the second of the four computations in a little more detailto spell out the reasoning. Assuming one is a lefty and a thirder, then there isa 1 chance that the coin has landed heads, in which case pushing Right will get Beauty $200. There is a 1 chance that it is Monday and the coin landed tails, in which case pushing Right today will get Beauty $100 (since, being a lefty,she will push Left tomorrow). In the remaining 3 , it is Tuesday, Wednesday, or Thursday and Beauty (again, being a lefty) has already pushed Left on Monday,and so Beauty stands to get $200 if she pushes Right.
Note that whether Beauty thinks she is a lefty or a righty, she stands to gain more by pushing Right than pushing Left. Thus, regardless of how muchcredence she assigns to being either, she ought to push Right.
A similar verdict is reached by EDT. EDT, unlike CDT, allows evidential relations between the outcome of one's choice and states of the world to factorinto one's decision as to what to do. In the present circumstances, for example,EDT might allow Beauty, on the assumption that the coin landed tails, toconceive of her choice ‘today' as effectively settling her choices on other days.
So beauty can think of herself, ‘today', as choosing whether she is a lefty or arighty. Thus we have the following.
Version 1, Computations for Thirder + EDT Again, the thirder thinks Beauty should push Right.
As I said before, halfers give a different response, regardless of whether they are causal or evidential decision theorists. CDT yields the following values.
Version 1, Computations for Halfer + CDT Regardless of whether one is a righty or a lefty, one stands to gain by pushingLeft. The EDT thirder gets the same result.
Version 1, Computations for Thirder + EDT Recall that the original motivation for examining this version of the Waking Game was to ‘work backwards' within decision theory from values and rationalchoices to credences. The idea was that if we had strong intuitions about whatwe ought to do in a case where thirders and halfers diverged in their recommen-dations, we could use our intuitions to arbitrate between those views. I thinkVersion 1 of the Waking Game is a case with a fairly intuitive response: the ra-tional thing to do upon waking in the game is to push Left, for the simple reasonthat it seems one stands to gain by doing so. Before defending this claim, letme set up the argument which reveals how the supposition that Beauty oughtto push Left constrains our alternatives.
Consider the following three premises governing the first version of the Wak- (P1) The rational thing for Beauty to do in Version 1 of the Waking Game is to push Left upon waking.
(P2) There is a particular credence distribution over MonH, MonT, . . , ThuT that it is rational for Beauty to have in the Waking Game (and hence thecase of Sleeping Beauty) upon waking.
(P3) If it is rational in the Waking Game to have a particular credence distribu- tion over MonH, MonT, . . , ThuT and the rational thing to do in Version1 of the Waking Game upon waking is to do A, then standard decisiontheory yields the verdict that one should do A given the credences it isrational for Beauty to have upon waking.
I'm using the term "standard decision theory" as a blanket term to cover viewslike CDT and EDT, and reasonable variants of them which yield results suchas I've shown above. Given (P1), (P2), and (P3), we can derive (C1) Beauty should not have degree of belief 1 that the coin lands tails in the Waking Game (hence in Sleeping Beauty).
Indeed, if we consider very similar situations with the numbers slightly adjusted,it is easy to see we can strengthen this to (C2) Beauty should have degree of belief 1 that the coin lands tails in the Waking Game (hence in Sleeping Beauty).
This is a result which arguably puts pressure on the thirder. The thirder accepts(P2) and rejects (C1). Consequently the thirder must deny that it is rational inthe Waking game to push Left, or jettison standard decision theory.
Though I think each of the premises in the above argument are plausible, there are grounds for doubting each. Consequently I'd like to briefly examinesome issues relevant to each premise in turn.
(P1): Beauty thinks she ought to aim to be a lefty on Sunday. This is becauseif she succeeds she'll likely be wealthier for her efforts. But as we've seen shemight change her mind when she wakes up since she may, if she is a thirder,think it very likely the coin toss landed tails, which favors pushing Right. Toalter her plan, however, seems like a bad idea. The reason for this is not justthat it seems rational for Beauty to plan on Sunday to push Left since, afterall, plans to act and the acts themselves might diverge in terms of rationality,especially in cases involving de se ignorance.7 Rather, the reason it is rationalfor Beauty to push Left is simply because pushing Left seems the most reliableway for Beauty to get the most of what she wants. To sharpen intuitions let mealter the case in three ways: by increasing the number of tails cases, changingpayoffs, and by iterating the game.
Suppose that scientists have designed an extremely efficient robot to run the experiment which can do multiple wakings per day—up to 1 every few minutes.
And suppose Beauty is going into the experiment for an extended period—sayabout two months. Then there is the possibility that Beauty will wake up to10,000 times. Suppose this is the case.
Version 2Number of days woken up if coin lands tails: 9,999 Payoffs on Heads: Payoffs on Tails: if Left every day if Right every day if Left on Monday and Right another day if Right on Monday and Left another day Suppose further, that Beauty will get to go through the experiment about 10times, where between experiments her memories are restored (so that she alwaysknows which run of the experiment she is in). Halfers who, upon waking, pushLeft will average about $25,000. Thirders who, upon waking, push Right willalmost always end up with $20. Why would they push Right? Their calculationslook roughly as follows.
Version 2, Computations for Thirder + CDT The computations for EDT yield a similar verdict: the choice the thirder thinksthey face is essentially a choice between $2 and $1.50.8 This seems untrue to thecase. Following the halfer who pushes Right on these grounds would, I think,be highly irrational.
The main point of the foregoing discussion has not been to further contrast the views of the thirder and halfer. Rather, I am merely trying to give the casefor thinking that (P1) holds: that when Beauty wakes up in Version 1 of theWaking Game, the rational thing for her to do is push Left.
How could one argue that the intuitions I have been trying to draw out are illusory? One response comes from consideration of the contrasting verdictsof CDT and EDT as regards Newcomb's puzzle. I don't want to get into thedetails of this case, since this would take us too far afield. I just want to notethat Newcomb's puzzle presents a case where endorsing CDT leaves one less welloff than endorsing EDT. This prompts a kind of question addressed to causaldecision theorists, a version of which I am raising here for detractors of (P1). Itwas succinctly put by David Lewis: "If you're so smart, why ain'cha rich?"9 A standard answer on behalf of CDT is to claim that the Newcomb situation is one in which ‘rationality is being punished.' It is a controversial matter whether this idea is fruitfully appealed to by a defender of CDT. But it shouldbe uncontroversial that this strategy is of no use in defending a rejection of(P1). The charge that rationality is being punished in Newcomb's puzzle is madeplausible by considering that the payoffs in that choice situation are restructured based on one's dispositions to act. This might open up the possibility that theway the payoffs are restructured systematically penalizes persons disposed to aparticular type of choice—perhaps those disposed to make the rational choiceamong them. In the Waking Game there is no similar restructuring of options.
Payoffs are fixed in the scenario with Beauty's full awareness of them. Sinceher choice won't alter the payoff structure, but only (by normal means) herpayoffs themselves, it seems implausible to suppose that the losses Beauty suffersby pushing Right are the outcome of a circumstance which penalizes rationalchoices. Rational people, so the story goes, capitalize on payoff structures andprobabilities to secure the most of what they want. If the payoffs aren't beingrestructured, it is hard to see how Beauty's failure to secure more money ispurely due to an unfair or unfortunate structure of the game.
There are perhaps other ways to defend the rejection of (P1), but the stan- dard way of coping with the "why ain'cha rich?" objection seems particularlyunmotivated here.
(P2): Though it might be difficult to arbitrate between the halfer and thirderviews, it can feel obvious that at least these parties are debating a genuine ques-tion with a unique answer. That is, it can seem that whatever the case is, thereis some unique credence distribution that Beauty ought to have upon wakingin the case of Sleeping Beauty and the Waking Games.10 A challenge to this assumption, however, is furnished by Arntzenius (2002).
Arntzenius stresses that when the coin lands tails, Beauty will have her beliefsartificially reset on Tuesday to conform to those she had Monday and that thisis a highly relevant kind of cognitive mishap, in that it ensures that Beautyviolates Bayesian conditionalization. Beauty is aware that she is going to bethe subject of such a cognitive malfunction. Consequently, the main questionBeauty faces is not what she ought to believe, but how she ought to behave tominimize the negative effects brought on by that malfunction.
Some evidence for Arntzenius' position comes from considering situations in which Beauty, on the assumption that she is a thirder (say) and endorses aparticular decision theory, ought to accept bets at odds which apparently violateher credences. Similar problems afflict the halfer view. Arntzenius claims wecan explain these situations as the upshot of the view that Beauty's credencesin particular propositions in Sleeping Beauty are irrevocably corrupted by thecognitive malfunction she knows she either has or may yet suffer. The bestBeauty can do is to consider herself to be somewhere at some point during theexperiment, and consider what someone in that situation stands to gain or loseby adopting various plans.
Arntzenius sums up his position as follows: "[For Beauty not] to have a def- inite degree of belief in heads might be strange, but it might be the best thatshe can do given the forced irrationality that is inflicted upon her. . The mainmoral of [Sleeping Beauty] is that in the face of forced irrational changes inone's degrees of belief one might do best simply to jettison them altogether."11It is unclear whether the examples Arntzenius supplies are enough to establishhis position, but the idea is certainly one that can look more appealing after considering variants of the Waking Game. I'll return to consider Arntzenius'suggestion again in §3.
(P3): The idea that standard decision theories ought to be abandoned is onethat has been advocated recently by Egan (2007) in response to a raft of exam-ples where the seemingly rational things to do are not systematically reflectedin the exclusive application of either EDT or CDT. The problem raised by theWaking Game for standard decision theory, however, is of a very particular va-riety. Egan's puzzles, if his analyses are accepted, seem to show that in somecases EDT wins out while in others CDT does. This seems to point to some-thing like a hybrid view, or at least something in the neighborhood of standarddecision theory. The first version of the Waking Game, however, seems to showthat if one is a thirder, both EDT and CDT, and anything suitably similar tothem, will have to go by the board. Consequently, the way in which (P3) fails, ifit is rejected here, will arguably be in a more dramatic way than Egan proposes.
There is more to be said about these premises, but let me recapitulate what I take to be some morals so far. The thirder accepts (P2) and rejects (C1). Conse-quently, she must reject either (P1) or (P3). Barring an account which overturnsintuitions about maximizing gains that I have drawn on, it is extremely diffi-cult to reject (P1). Thus, without such an account, the thirder should seriouslyconsider rejecting (P3), and hence abandoning standard decision theory as acompletely general account of what it is rational to do given what one believes.
This might in turn seem to apply a great deal of pressure to the thirder view.
After all, standard decision theory is not merely a theory with ‘good fit' to theset of data given by other uncontroversially rational choices. It is also a theorywhose structure seems intuitively tailored to track rational decision making.
What's more, the discussion so far may lead one to believe that thirders aloneare in a bind. I suspect this apperance is illusory. It turns out that the halfer,and indeed any theorist who claims Beauty ought to have a particular credencedistribution in the case of Sleeping Beauty, may ultimately face a challengesimilar to the thirder. To see this, I'll have to introduce some new complicationsinto the Waking Game.
Subjectively Distinguishable States It is an important assumption of Sleeping Beauty, and my original WakingGames, that Beauty's wakings are subjectively indistinguishable. If things wereotherwise, Beauty's rational beliefs—de se and otherwise—might change in dra-matic ways. Sometimes theorists consider cases in which Beauty is capable ofdistinguishing her wakings, and in which it might be more clear what she oughtto believe. The idea is that these situations can be used to try to glean in-formation about what Beauty ought to believe in the original Sleeping Beautycase by way of analogy. Something like this strategy is adopted, for example,in Titelbaum (2008).
The cases I'd presently like to examine are variants of this kind. The exam- ples will get quite complicated, but I believe this might be necessary to relievethirders of the burdens of §1.
Modified Waking Game. As in the original Waking Game, exceptthat each day Beauty will be placed in a different colored roomfrom among n options. The colors of the rooms are very easy todistinguish (i.e. red, green, etc.). She'll be placed in one of the nrooms each day. At the time scientists flip the coin to decide howmany times beauty will be awakened, they will also roll an n!-sideddie. Each 0 < i ≤ n! corresponds to a permutation of the rooms thatBeauty may be placed in. Thus, on tails, beauty is sure to wake toeach of the n rooms at least once, whereas on heads, there is only a1/n chance of her waking in any given room. Beauty is broached ofthese details and the colors in advance.
Let the number of days and payoffs be as before.
Modified Waking Game, Version 1Number of days woken up if coin lands tails: 4Room Colors: Red, White, Green, Blue Payoffs on Heads: Payoffs on Tails: if Left every day if Right every day if Left on Monday and Right another day if Right on Monday and Left another day What should Beauty plan to do on Sunday, again provided she cannot randomizeand is sure to execute her plan? The availability of colors to coordinate herdecisions now allow for five equivalence classes of plans (equivalent under therelation of equal expected payoff) based on how many colored rooms she choosesto push Left in upon waking. The optimal plan is to always push Left excepton one color.
Approx. Expected Payoffs Left on all colors Left on 3 colors, Right on 1 Left on 2 colors, Right on 2 Left on 1 colors, Right on 3 Right on all colors Thus, adding subjectively distinguishable states allows for more elaborate, co-ordinated strategies with higher payoffs than in cases with subjective indistin-guishability.
I want to contrast two different scenarios where these more elaborate strate- gies may or may not be available to Beauty, not because she is unaware of whatcircumstance she is in, but because of facts about her own psychology. To bringout this contrast we'll need to prevent Beauty from forming plans on Sunday.
In-Game Explanations: Beauty knows she is in some Waking Game onSunday, but doesn't know the exact rules, number of wakings, payoffs,and so forth. She'll be told them every day that she wakes by a recordingover a loud speaker right after she gets up.
Again, let's suppose that Randomizing Prohibited and something analogous toPrevious Runs hold. Consider the following circumstance.
Case 1: Beauty wakes and hears the rules of Version 1 of the ModifiedWaking Game (with the colors of the rooms specified). She opens hereyes to find herself in a red room. She reasons as follows: "It would beideal if I could get myself in a position to push Left in all rooms but one.
Unfortunately, if the coin landed tails I'll have to coordinate with myselfin other wakings—but I can't. Perhaps I could effectively coordinate byjust picking an arbitrary color to be the "Right pushing" room right now,executing the corresponding plan, and hoping that I will adopt the sameplan on other days. I'd probably pick red if that was what I ought to do.
But the problem is I've already seen that I'm in a red room, and I'm avery suggestible person. Seeing the red room will doubtless systematicallyinfluence my decision as to which "arbitrary" color I choose in detrimentalways: it will make me very likely to pick the color of the room I'm in.
This makes it highly likely that if I pick a color now and execute thecorresponding plan (and the coin landed tails) I won't coordinate withmyself in the right way: I'll be liable to choose to push Right every day.
And I can't force myself to choose a color other than red now—then I'lljust think the same thing every other day and always end up pushingLeft. No, I can't capitalize on the different colors of the room to pryapart my choices on different days. I'm better off just making a decisionindependently of color considerations." In this case Beauty has some very sophisticated views about her own psycho-logical states. She thinks facts about those states put her in a bad positionto coordinate her choices in the right way by capitalizing on subjective distin-guishability. We can suppose, for the sake of the example, that Beauty has goodevidence for this, and is in fact right. Having woken up and seen the color ofthe room she is in, planning to push Right on some one color is not a decisionwhich will generally lead to her having coordinated her choices in the right way.
Though lamentable, it appears it is best for Beauty to push Left in such cir-cumstances. Her psychology prevents her from capitalizing on the benefits ofsubjective distinguishability, effectively putting her in the circumstance of theunmodified Waking Game.
In a contrasting case, though, we can suppose things had gone ever so slightly Case 2: Just as in Case 1, except that before Beauty opens her eyes shechecks herself: "What would be ideal is if I could get myself in a positionto push Left in all rooms but one. If I open my eyes now, though, I mightbe forgoing a great option: pick a single arbitrary color before openingmy eyes to single out a room in which I'll push Right. Since I'm likelyto keep my eyes closed and reason this way on other days, and since (byPrevious Runs) I'm liable then to pick the same arbitrary color, it will beas if I was able to form a plan on Sunday to push Right only once. Great!I choose red." Beauty opens her eyes to find herself in the red room.
Beauty's reasoning appears sound. Now, it seems, Beauty is in a great positionto push Right.
Let me articulate another argument, analogous in structure to the one I gave in §1, which is suggested by the foregoing examples. It begins by tugging onthe same intuitions concerning what it is rational for Beauty to do.
(P0 ) The rational thing for Beauty to do in Case 1 is push Left, and the rational thing for her to do in Case 2 is push Right.
As with the corresponding premise of §1, we can support (P0 ) by changing payoffs, increasing the duration of the experiment, and iterating games (withBeauty becoming aware of which trail she is in by having her memories ofpreceding trials return). As before, detractors from (P0 ) will systematically face high losses with apparently no explanation for why this is compatible withtheir choices being rational.
A second premise concerns what sort of credence distribution Beauty ought to have upon waking. Unlike before, it will be helpful to constrain the credencedistributions which seem reasonable. In addition to the propositions MonH,MonT, TueT, WedT, and ThuT it will be useful to consider two (de se) propo-sitions stating that Beauty will, if she wakes on several other days during theexperiment, choose the same thing each day. Let Righty be the proposition thatBeauty chooses Right every other day (if she is given the option), and Lefty bethe proposition that Beauty chooses Left every other day. Then we can definethe following notion of a reasonable credence distribution for my variants of theModified Waking Game.
A credence distribution C is respectable if the following three conditionshold.
(i) C(MonH )∈ { 1 , 1 } (ii) C(Righty ) + C(Lefty) ≈ 1.
(iii) C(MonT )≈ C(TueT ) ≈ C(WedT ) ≈ C(ThuT ).
Respectable credences are the ones it seems rational for Beauty to have in myvariants of the Modified Waking Game.
(P0 ) There is some respectable credence distribution which it is rational in Case 1 for Beauty to have. Likewise for Case 2.
Let's go over the reasoning for each clause. (i) should hold because somethinglike halfer or thirder reasoning should apply in the cases. Note that this does notmean, for example, that halfers are committed to the claim that Beauty oughtto believe to degree 1 that the coin landed tails in these cases. Halfers may claim that the changes brought about by introducing subjectively distinguish-able wakings are relevant to what Beauty ought to believe. Ditto for thirders.
Nonetheless, if one is a halfer or a thirder, one will likely expect it be rationalfor Beauty to have some credence in heads in the cases given, and that 1 and 1 are the best options. It seems hard to motivate other values.
(ii) should hold in Case 1 because Beauty essentially takes herself to be in the unmodified Waking Game, where the same assumption seems rational.
Even if Beauty decides to act based on an attempt to use colors to coordinateher choices, I have assumed that she knows she would end up either pushingLeft every day or Right every day. In Case 2, Beauty again may either ignorethe coloring, in which case she should have the same beliefs as the unmodifiedWaking Game, but it is more likely that she will follow through on her plan,in which case she will push Right, but only in the red room—i.e. only ‘today'.
Thus she will push Left every other day and (ii) holds.
(iii) could be slightly more controversial. It is an application of a kind of indifference principle, which says that Beauty doesn't think that it is much morelikely, say, for it to be Monday while the coin landed Tails, than for it to beTuesday while the coin landed Tails. General indifference principles are notalways easy to defend, but this particular application seems to be justified onintuitive grounds.12 Not only should Beauty's credence distribution in Cases 1 and 2 be re- spectable, but it also seems they should be identical, perhaps admitting fordifferences in how she proportions her beliefs between Lefty and Righty, anddifferences in her beliefs about her psychology, say. I'll ignore the latter diver-gences since they are irrelevant to the argument to follow.
(P0 ) Beauty should have the same respectable credence distribution in both Case 1 and 2, up to divergent relative credences between C(Righty ) andC(Lefty).
This is because although Beauty has different evidence in Case 1 and Case 2about what she is presently doing, what plans will succeed, and perhaps whatshe has already or may yet do, none of these differences are plausibly pertinentto C(MonH ) or to any other claims about what day it is or whether the coinlanded tails.
So the premises (P0 )–(P0 ) seem just as plausible as (P they yield the following surprising conclusion.
(C0) Supplied with values and rational credences, standard decision theory will not always compute the rational thing to do.
The argument is simple. Suppose (P0 ), (P0 ), and (C0) hold. Suppose further that C(MonH )= 1 in both Case 1 and Case 2. If the former holds, then both EDT and CDT yield the result that Beauty should push Left in Case 2. Indeed,choosing Left dominates choosing Right in expected value: there is a 1 chance of Beauty getting $400 over $200 by pushing Left, but (in the ‘best case' scenario)only 1 chance of getting $200 over $100 if she pushes Right. The resulting verdict contradicts (P0 ).
Suppose, instead, that C(MonH )= 1 in both Case 1 and Case 2. Then both EDT and CDT direct Beauty to push Right in Case 1.
Again, the choice dominates: at least 3 of the time she stands to get $200 over $100 by pushing Right and faces a mere 1 chance of forgoing $400 for $200. Again this contradicts On the assumption of the premises, standard decision theory systematically produces the wrong results.
My argument so far has depended on crucial alterations to the original Sleep- ing Beauty case which animated the halfer and thirder positions—in particular,it has depended on the introduction of subjectively distinguishable wakings.
However, it is easy to see how the new argument is indirectly relevant to bothof their positions. We have a pair of cases where, on plausible assumptionsabout how credences should be alloted in those cases, decision theory falters.
This shows that the result of §1 is not really a special problem for the thirder.
Let me elaborate a little on what conclusions I think we should draw from theforegoing arguments. First, let's return to the tension between (P0 )–(P0 ) and standard decision theories.
(P0 ) The rational thing for Beauty to do in Case 1 is push Left, and the rational thing for her to do in Case 2 is push Right.
(P0 ) There is some respectable credence distribution which it is rational in Case 1 for Beauty to have. Likewise for Case 2.
(P0 ) Beauty should have the same respectable credence distribution in both Case 1 and 2, up to divergent relative credences between C(Righty ) andC(Lefty).
If (P0 ) is true, it is hard to imagine that (P0 ) could be false. So barring grounds for rejecting (P0 ), the examples of §2 point to a tension between the idea that Beauty ought to have specific credences in Sleeping Beauty-like cases and stan-dard decision theory.
It would, however, be very difficult to reject (P0 ) to rescue universal application of standard decision theories. The reason is that ifwe want to preserve the intuition—given in (P0 )—that it is the rational thing to do for Beauty to perform certain actions, we will want standard decision theoryto never direct Beauty to do otherwise. But if we reject (P0 ), it seems like it should be rationally permissible for Beauty to have any credence distributionover the relevant propositions. Thus Beauty will not be rationally criticizable if she selects a credence distribution on which standard decision theory directsher to, say, push Right in Case 1. This violates (P0 ), at least on its strongest reading. Even if Beauty is not permitted to rationally select her credences, thenshe will have no credences. Thus standard decision theory will not fault her formaking either choice in Cases 1 or 2. Again this violates (P0 ).
Consequently, the cases of §2, and in retrospect the cases of §1, most strongly suggest accepting (C0)—that is, they suggest jettisoning standard decision the-ory as the theory which uniformly yields results about what it is most rationalto do in cases of de se ignorance.13 This conclusion is a conclusion just aboutthe implications of integrating de se beliefs into our theories of rational choice.
Exactly how our frameworks for rational choice should be adjusted is a compli-cated issue I'll say a little bit more about shortly. In the interim, this conclusionabout models of rational choice has some important additional implications fortheories of rational belief update. Let me say why.
I originally presented the motivation for examining the Waking Game as a way of indirectly getting at the beliefs Beauty ought to have in her variouspredicaments: we could work backward from intuitions about what Beautyought to do, along with knowledge of what her values are through standarddecision theory to figure out what she ought to believe. That strategy has nowbeen shown unreliable, because we've seen that standard decision theory candirect Beauty to do intuitively irrational things in various choice situations,regardless of which reasonable credence distribution she had.
This is significant, because we would normally expect that what one ought to believe in certain scenarios is conceptually tied to what one ought to do inthem, and hence exploring rational choice should be a sure-fire way of uncover-ing rational beliefs. Consider, for example, the strategy of constructing ‘dutchbook' arguments to undermine particular credence distributions by showing thatbetting along the lines of those credences may lead one to systematically losemoney. Such a strategy is merely a special case of the strategy of trying to figureout what Beauty ought to believe by seeing what she ought to do. Indeed, itturns out to be a relatively special case given that it is possible that sometimesone ought not to bet along one's credences, as emphasized by Arntzenius (2002).
Thus the foregoing reflections should cast suspicion on every member of a large class of arguments, including those involving dutch books, about what cre-dences Beauty should have which, apparently, are the most fruitful and decisivewe could give. This does not eradicate all hope of finding grounds for Beautyto prefer one credence distribution over another. There is always analogicalreasoning—examining cases similar to those in which Beauty finds herself wherethere is no contest as to what Beauty's credences should be and extrapolatingfrom those results to the more problematic cases. There is also the strategy,originally adopted by Elga (2000), of trying to reason to a conclusion in the caseof Sleeping Beauty from putatively uncontroversial principles governing rationalbelief revision. Though such arguments might be available, the Waking Gamesadd to worries that conclusive results will be very hard won.
Moreover, even if we do find what credences it is rational for Beauty to have, the Waking Games show that we will still be left with a theoretically distinct and perhaps more pressing question about what Beauty ought to do. Typicallywe are interested in what one ought to believe because this should play a guidingrole in action, and not merely for the satisfaction of knowing our credences yieldto evidence in just the right way. The Waking Games seem to show that even ifBeauty ought to have a certain credences in ‘narrow' de se propositions, whatshe ought to do in those cases is computed in a way that swings free of thosecredences. Is there a way of systematizing her choices in such scenarios? Andif so, how? The answer to the second of these questions is clearly the topic for another investigation, but I would like to briefly say some things to help suggest that ananswer to the first question is ‘yes'. If the intuitions motivating (P1) and (P0 ) are reliable, then they already point to a strategy for Beauty to adopt. If Beautyknows what kind of situation she is in, regardless of whether she knows exactlywhere or when she is in it, she can use that information to assess the relativevalues of various choice strategies which might be adopted by persons in hersituation. It seems the most rational choice can be pinpointed as the strategywhich, when adopted, tends to yield the highest utility. In this calculation, it isnot ‘narrow' de se propositions concerning the time, e.g., which guide Beauty'schoice, but ‘broad' de se propositions such as that I am presently in such-and-such a Waking Game.
We've already seen the basic idea here suggested in the discussion of Arntze- nius (2002). When rejecting the claim that Beauty ought to have any particularcredence in heads, Arntzenius writes: ". . [Beauty's] epistemic state upon wak-ing up is best described by saying that she believes she is in the situation de-scribed in the Sleeping Beauty story."14 Whether or not Beauty actually oughtto have beliefs in narrower de se propositions, it is the broader ones Arntzeniusdiscusses which apparently should play the decisive role in her choices. If thisis true, perhaps it matters less in cases of centered uncertainty exactly whereand when you are, than the general nature of your predicament.
1Frege (1956)2See Lewis (1979) and Perry (1979) respectively.
3For a sampling, see and Dorr (2002), Hitchcock (2004), and Titelbaum (2008) in addition to the original Elga (2000) for some thirder arguments and Lewis (2001), White (2006), andMeacham (2008) for halfer views.
4Of course, she may not be able to reliably do this. This might be, for example, because she anticipates that it would be irrational for her to carry out her plan upon waking. Also,I'm assuming in this case that Beauty has no particular aversion to risk.
5Both CDT and EDT have slight variants, so I'm fixing on a particular construal here. I make no claims to exhaustiveness, but I hope the examples chosen are representative.
6This claim, if correct, shows that pace Briggs (2010) it's not clear that one's choice of causal or evidential decision theory specially privileges a halfer or thirder view.
7How could a plan to perform an action in the future be rational, but the future action itself be irrational? Perhaps I now have information which I know I may, or will, lack atthe time of acting. Given what I know now I should wish my future self to do A. But wereI to reason on the basis of only the more limited amount of information had by my futureself, it would be more prudent to wish my future self to perform some distinct action B. If my intention to act in the future is something like a cause of my so-acting, I may plan to Aknowing that it will be subjectively irrational to do A in the future due to a loss of informationat that time. See Elga (2004) for an example of this very kind of phenomenon which arisesas one ‘loses' de se information over time, without obviously suffering any sort of cognitivemalfunction.
8In fact, for any position but the halfer's one can construct such a game.
9See Lewis (1981b) for a discussion of the status of the objection and the standard reply on behalf of CDT.
10More carefully: there is some credence distribution or constrained range of distributions over the various conjunctive propositions governing the outcome of the coin toss and the‘present day' that it is irrational for Beauty to deviate from.
11Arntzenius (2002), p.61.
12In fact, the assumption of (iii) isn't actually needed for the argument to follow, since we can create alternative scenarios where other days than Monday play the special role ofdetermining what payoffs Beauty will get provided she changes her choice on other days.
Appealing to (iii), however, will simplify the argument tremendously.
13In fact, since the case of ignorance in §2 is arguably not irreducibly de se, the arguments may show more: that in complex cases of ignorance like Sleeping Beauty, standard decisiontheoretic frameworks fail, regardless of whether irreducibly de se ignorance is at issue.
14Arntzenius (2002) p.61.
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Source: http://www.jshaw.net/documents/DeSe.pdf