Sleeping Beauty and Self-location: A Hybrid Model

Synthese (2007) 157:59–78
DOI 10.1007/s11229-006-9010-7
ORIGINAL PAPER
Sleeping beauty and self-location: A hybrid model
Nick Bostrom
Received: 8 April 2005 / Accepted: 5 April 2006 / Published online: 1 June 2006
© Springer Science+Business Media B.V . 2006
Abstract The Sleeping Beauty problem is test stone for theories about self-
locating belief, i.e. theories about how we should reason when data or theories con-tain indexical information. Opinion on this problem is split between two camps, thosewho defend the “1/2 view” and those who advocate the “1/3 view”. I argue thatboth these positions are mistaken. Instead, I propose a new “hybrid” model, whichavoids the faults of the standard views while retaining their attractive properties. Thismodel appears to violate Bayesian conditionalization, but I argue that this is not the
case. By paying close attention to the details of conditionalization in contexts whereindexical information is relevant, we discover that the hybrid model is in fact consis-tent with Bayesian kinematics. If the proposed model is correct, there are importantlessons for the study of self-location, observation selection theory, and anthropicreasoning.
Keywords Self-locating belief ·Probability indexical information ·Epistemology
sleeping beauty problem ·Anthropic principle
Introduction
Sleeping Beauty
On Sunday, Beauty is put to sleep. She is awakened once on Monday, and put
to sleep again after being administered a memory-erasing drug that causes herto forget her awakening. A fair coin is tossed. If and only if the coin falls tails,Beauty is awakened again on Tuesday. She knows all this. When she awakes onMonday, what should her credence be that the coin will fall heads?
N. Bostrom ( B)
Faculty of Philosophy, University of Oxford,Suite 7, Littlegate House,16/17 St Ebbes st,Oxford Ox11PT, Uke-mail: nick@nickbostrom.com
60 Synthese (2007) 157:59–78
The Sleeping Beauty problem is a variation of some very similar problems of
“imperfect recall” that have been discussed for some time in the game theoreticliterature.
1It was named by Robert Stalnaker, who had learnt about similar cases
from Arnold Zuboff. The problem was brought to the attention of the philosophicalcommunity through an exchange between Adam Elga, who argued for the answer1/3 (hereafter the “1/3 view”), and David Lewis, who defended the 1/2 view. The lastfew years have seen a burst of publications advocating one or the other of the twocompeting doctrines. To date, neither side seems to have gained a decisive advantage.
Sleeping Beauty is an example of a problem involving self-locating beliefs, i.e.,
beliefs that an agent, or a temporal part of an agent, might have about its own loca-tion. An agent-part that knew exactly which possible world is actual can still beignorant about its own location in that world. That can happen if the world containstwo or more agent-parts whose evidential states are subjectively indistinguishable.These agent-parts would then be unable to determine with certainty their own spa-tiotemporal location. (Even if Beauty knew that the outcome of the coin toss wouldbe tails, she could not know whether it was currently Monday or Tuesday.) Anotherway of expressing this is by saying that the agent-parts would be ignorant about whichcentered possible world they are in even though they know which possible world they
are in. Yet another formulation is that agent-parts, or “observer-moments”, possessall non-indexical information about the world but lack some indexical information.Here, we shall use these expressions interchangeably.
The Sleeping Beauty problem is but one piece of the larger puzzle of how to relate
indexical to non-indexical information in our reasoning. If one studies the problemin isolation from this wider context, one risks coming up with answers and principlesthat do not ﬁt with the other parts of the puzzle. I will ﬁrst argue that both the 1/3view and the 1/2 view suffer from such a misﬁt. That done, I will propose a new“hybrid” model which incorporates aspects of both the 1/3- and the 1/2-view but isidentical to neither. The hybrid view overcomes the problems associated with thepurebred answers. It also suggests an explanation for why the 1/3- and the 1/2-viewhave both managed to appeal to their fan bases: they each encapsulate a part of thetruth.
The 1/3 view
The 1/3 view is that upon awakening, Beauty should assign credence 1/3 to HEADS.
Elga’s argument for this view is as follows.
2
When Beauty wakes up, she knows that she is in one of three situations:
H1 HEADS and it is Monday,
T1 TAILS and it is Monday,T2 TAILS and it is Tuesday.
1Robert Stalnaker gave the problem its name, after having of examples of a similar kind in unpub-
lished work by Arnold Zuboff. Closely related problems have also been discussed in the game theoryliterature; see volume 20 of the journal Games and Economic Behavior (1997).
2Elga (2000). For some other defenses of the 1/3 view (see Dorr 2002; Monton 2002; Weintraub
2004).
Synthese (2007) 157:59–78 61
Given that the Monday and the Tuesday awakening would be evidentially indistin-
guishable, we have
P(T1)=P(T2)[By an indifference principle].
If Beauty were to learn that it is Monday, her credence in HEADS should be 1/2
because this is then just the credence that a coin that is about to be tossed, and is
known to be fair, will fall heads. Hence,
P(H1|H1∨T1)=1/2[By appeal to intuition ].
Since P(H1|H1∨T1) can be rewritten as P(H1)/[ P(H1)+P(T1)], we thus have
P(H1)=P(T1)=P(T2).
These credences sum to 1, so it follows that P(H1)=1/3.
The argument replies on an appeal to intuition in the middle step. One could try
to support this step by invoking the Principal Principle . This principle says, roughly,
that one’s credences should accord with one’s estimates of objective chances. As for-mulated by David Lewis, the Principal Principle came with a proviso: one’s credencesare constrained by one’s beliefs about objective chances according to the principleonly if one does not have relevant “inadmissible” information.
3Lewis’s reason for
introducing the proviso was to bracket off cases involving oracles, time travel, andthe like, where one might get information about the future that is not mediated byinformation about current chances. It is possible to maintain, however, that someproblems involving self-location also include inadmissible information and shouldhence be exempted from the Principal Principle’s domain of applicability. In light ofthe positive argument against the 1/3 view, which I will present below, we do in fact
have strong grounds for regarding Beauty’s indexical information as inadmissible.
Given Lewis’s own “best-system” analysis of chance, it would be unsurprising to
ﬁnd that indexical information can be sometimes inadmissible. According to the best-system analysis, chances are a kind of concise partial summary of patterns of local,non-modal, occurrent facts. Since chances, deﬁned in this way, do not even purportto summarize indexical information, there is no reason to suppose that all relevantinformation about future events is always implied by knowledge of the chances—notif we have reason for thinking that indexical information might also be relevant. (Thisis, in fact, the stance that Lewis adopted in his critique of Elga’s argument.
4)
Since our intuitions about what information is admissible in this type of case are no
more secure than our direct intuitions about what credence Beauty should assign toHEADS, the Principal Principle fails to settle the Sleeping Beauty controversy. Whatcounts as admissible information in the Sleeping Beauty problem must emerge fromits solution—which needs to be independently justiﬁed—rather than assumed at theoutset.
Another way in which one may attempt to support the middle step in Elga’s argu-
ment is by invoking Bas van Fraassen’s Reﬂection Principle .
5(The Reﬂection Principle,
in its simplest form, postulates that your credence at a time tis constrained by your
credence at a later time t’ according to Pt(X|Pt/prime(X)=x)=x.)Here we encounter the
same problem again: the applicability of the principle to Sleeping Beauty is at least as
3(see Lewis 1980, 1994). A precursor to the Principal Principle was formulated by Mellor (1971).
4(see Lewis 2001). A similar point was made, independently, in Bostrom (2001).
5(see van Fraassen 1984).
62 Synthese (2007) 157:59–78
problematic as the 1/3 view itself. The Sleeping Beauty problem postulates a break-
down of rationality. Beauty faces the possibility of drug-induced amnesia. Even if weask about Beauty’s credence on Monday, before the drug has been administered, shewill not at that point know whether or not her memory has already been tamperedwith. It is independently known, from other cases, that the Reﬂection Principle shouldnot be followed where forgetting takes place or is suspected. For example, if I knewthat 1 year from now I will have credence 1/2 in the proposition that it rained today,that plainly does not imply that I should assign the same credence now—when I stillvividly remember spending the day reading in the garden. The Reﬂection Principle,therefore, is of no more avail to the 1/3 view than is the Principal Principle.
We are left with the unsupported appeal to intuition in the step of the argument
that assumes that P(H1|H1∨T1)=1/2. How are we to evaluate this intuition? One
way is to examine what else we are led to accept if we adopt the 1/3 view. If the conse-quences are unacceptable, we should revise our intuition. Let us, therefore, considersome variations of the original Sleeping Beauty to explore the wider ramiﬁcations ofthe 1/3 view.
If we set P(H1)=P(T1)=P(T2), it follows that Beauty should, upon awakening,
assign P(HEADS) =1/3a n d P(TAILS) =2/3. The only relevant difference between
HEADS and TAILS is that there would be more awakenings of Beauty on the latterhypothesis. Since the structure of the situation does not depend on the particularnumbers involved, we can test our intuitions by considering a more extreme versionof the problem.
Extreme Sleeping Beauty
This is like the original problem, except that here, if the coin falls tails, Beauty
will be awakened on a million subsequent days. As before, she will be given
an amnesia drug each time she is put to sleep that makes her forget any previ-ous awakenings. When she awakes on Monday, what should be her credence inHEADS?
By reasoning exactly parallel to that which Elga used to support the 1/3 view in the
original version, we obtain
P(H1)=P(T1)=P(T2)=P(T3)=···= P(T1, 000, 001 ).
That is, upon awakening, Beauty should assign P(HEADS) =1/1,000,002. The result
can be generalized: following this line of reasoning, Beauty takes her observation “Iam awake now” as evidence in favor of hypotheses that imply that there are manysuch awakenings of her. The degree of support is proportional to the number of awak-enings postulated by the hypotheses. This consequence in Extreme Sleeping Beautyis counterintuitive. It seems like a rather excessive conﬁdence in the proposition thata fair coin, yet to be tossed, will fall tails.
We can bring out even worse implications if we consider cases in which more
than one agent is involved. Suppose that a possible world contains several agent-parts{a
i}i∈Rthat are subjectively indistinguishable, that is, these agent-parts are in such sim-
ilar evidential situations that individual agent-parts cannot tell which one they are in.Elga has argued for an indifference principle stating that one should assign the samecredence to each centered proposition of the form “My current agent-part is a
i”, for
i∈R, and this is supposed to hold whether or not the agent-parts in {ai}i∈Rall belong
Synthese (2007) 157:59–78 63
to the same agent or to different agents.6This highly restricted indifference principle
follows as a special case from a somewhat stronger indifference principle that appearsto be needed to make sense of many seemingly legitimate scientiﬁc inferences.
7The
principle is thus well supported. But if we use it together with the reasoning in the 1/3view, we obtain implausible results.
Beauty and Doppelganger
This is like the original Sleeping Beauty problem, except here Beauty is never
woken up after being put to sleep on Monday. Instead, if the coin falls tails,another person is created and awoken on Tuesday. This new person will spend herTuesday wakening in a state that is subjectively indistinguishable from Beauty’sMonday state (she will have the same apparent memories and have experiencesthat feel just the same as Beauty’s). When Beauty awakes on Monday, whatshould be her credence in HEADS?
By Elga’s weak indifference principle, Beauty should have the same conditional cre-
dence, given TAILS, in her being Beauty and Doppelganger. And by the same rea-soning that the 1/3 view relies on in the original version of the problem, Beauty shouldhave the same conditional credence, given MONDA Y, in being Beauty as in beingDoppelganger. From this we can derive, just as before, that the awakened Beautyshould assign an equal credence in three centered propositions:
P(“I am Beauty and HEADS” )
=P(“I am Beauty and TAILS” )
=P(“I am Doppelganger and TAILS” ).
Since the credence given to these three centered propositions sum to 1, it follows that
upon awakening, Beauty should hold P(HEADS) =1/3.
Now consider the extreme version of Sleeping Beauty and Her Doppelganger,
where on tails there will be a million different doppelgangers, each having an awaken-ing on some subsequent day that will be subjectively indistinguishable from Beauty’sMonday awakening. It is easy to show, by the same steps as before, that Beauty shouldupon awaking have credence P(HEADS) =1/1,000,002.
The argument does not depend on whether the coin is tossed before the exper-
iment starts or only after Beauty has been put back to sleep on Monday. It is, infact, irrelevant what the objective chance of HEADS is when Beauty is making herassessment.
8The relevant factor is Beauty’s prior credence in HEADS (relative to her
non-indexical background information). For example, Beauty’s posterior credence inHEADS would thus be unaffected if the existence of additional awakenings by herdoppelgangers were made to depend, not on the outcome of a coin toss, but insteadon whether the trillionth digit in the decimal expansion of πis even—provided only
that Beauty’s prior credence in this proposition is 1/2. We can therefore generalizethe foregoing thought experiment:
6(see Elga 2004).
7(see Bostrom 2002a, b).
8If the coin were tossed before she woke up, the chance of HEADS would not be 1/2, but either 1
or 0.
64 Synthese (2007) 157:59–78
Beauty and Doppelganger (generalized)
Letϕbe a (non-indexical) proposition, to which Beauty assigns a prior
credence of 1/2. Beauty is never woken up again after being put to sleep onMonday. If ϕis true then there will be a total of N>0 awakenings of doppel-
gangers in states that are subjectively indistinguishable from Beauty’s Mondayawakening.
As before, we get P(HEADS) =1/(N+2). Let us consider what this recommen-
dation amounts too. Each (awake) agent-part, merely by taking into account theindexical fact that “I am currently an agent-part of this kind”, should give a greatercredence to hypotheses in proportion as they imply that there is a greater number ofsubjectively indistinguishable agent-parts of that kind. At this point, those familiarwith the literature on observation selection theory may notice an disturbing simi-larity between the reasoning behind this position and the so-called “Self-IndicationAssumption”. That assumption states that each observer should regard her own exis-tence as evidence supporting hypotheses that imply the existence of a greater totalpopulation of observers in the world, the degree of support being proportional to theimplied (expected) number of observers.
The Self-Indication Assumption was originally introduced in discussions about the
Doomsday argument, as an attempt to neutralize that argument. It turns out, how-ever, that the assumption has implications of its own that are perhaps even morecounterintuitive than those of the Doomsday argument. It seems that the followingthought experiment, in particular, gives us fairly strong grounds for rejecting theSelf-Indication Assumption.
Presumptuous Philosopher
It is the year 2100 and physicists have narrowed down the search for a theory
of everything to only two remaining plausible candidate theories, T
1and T 2
(using considerations from super-duper symmetry). According to T 1the world
is very, very big but ﬁnite and there are a total of a trillion trillion observersin the cosmos. According to T
2, the world is very, very, very big but ﬁnite and
there are a trillion trillion trillion observers. The super-duper symmetry consid-erations are indifferent between these two theories. Physicists are preparing asimple experiment that will falsify one of the theories. Enter the presumptuousphilosopher: “Hey guys, it is completely unnecessary for you to do the experi-ment, because I can already show to you that T
2is about a trillion times more
l i k e l yt ob et r u et h a nT 1!” (Whereupon the presumptuous philosopher explains
the Self-Indication Assumption.)9
By modifying this example slightly, we can substitute the reasoning embodied in
the 1/3 view for that of the presumptuous philosopher. To do this, suppose that thetwo theories that the physicists have come up with differ not only in regard to howmany observers there are but also in regard to how many agent-parts there are thatare subjectively indistinguishable from your own current one. Elga’s 1/3 view can nowtake the place of the Self-indication Assumption.
It is worth noting that the situation described in this modiﬁed version of Presump-
tuous Philosopher is by no means a farfetched possibility. Contemporary cosmologistsface essentially that predicament. They are trying to determine whether the universe
9(see Bostrom 2002a, b; 2003)
Synthese (2007) 157:59–78 65
is ﬁnite or inﬁnite. Given the standard Big Bang model and the assumption that
spacetime is singly connected, the universe is inﬁnite if and only if it is either openor ﬂat. Whether it is open or ﬂat, or closed, depends on whether the cosmic energydensity, /Omega1, exceeds a certain threshold value. Current measurements indicate that
the actual density is very close to the critical value, /Omega1≈1. It is an important open
empirical question whether the actual value is above, below, or exactly at the criticallevel. Measurements are being conducted to obtain a better estimate of the cosmicenergy density. If the universe is inﬁnite then with probability one there are an inﬁ-nite number of agent-moments in states subjectively indistinguishable to your currentone.
10Therefore, if Elga’s 1/3 view is correct, we could conclude that we already have
“inﬁnitely strong” evidence that the universe is inﬁnite. The consequence that it wouldbe a waste of money to carry out the planned experiments because we can predict theoutcome from our armchair (with probability 1), is extremely implausible. Somebodywishing to toe this line should be willing to bet at practically anyodds on the outcome
of these future experiments.
11
We have seen that the original argument given for the 1/3 view is inconclusive,
that neither the Principal Principle nor the Reﬂection Principle could be successfullyinvoked to buttress it, and that when we unfold the reasoning embedded in the 1/3view, we ﬁnd that highly counterintuitive consequences follow. We have good reasonto reject the 1/3 view.
12
We have not yet considered another argument in favor of the 1/3 view, one that is
based on long-run frequency or betting considerations. We will discuss this argumentin a later section. But ﬁrst, let us turn our gaze to the 1/2 view. We shall argue that thisview, too, should be rejected.
The 1/2 view
According to the 1/2 view presented by David Lewis, Beauty should upon awaken-
ing have credence 1/2 in HEADS, and her conditional credence in HEADS givenMONDA Y should be 2/3.
P(H1)=1/2,
P(H1|H1∨T1)=2/3.
Suppose that Beauty is informed that it is Monday, and let P
+be her new credence
function after she has obtained this information. Lewis claims that P+should be
10(see Bostrom 2002b).
11Even if during the next 200 years we obtained overwhelming empirical evidence that the universe
is ﬁnite, we should, on this view, continue to assign credence 1 to the universe being inﬁnite.
12Kierland and Monton, in a very recent paper (Kierland and Monton 2005), argue that the case
where persons are duplicated (as in Doppelganger) should be treated in a fundamentally different wayfrom cases where awakenings of the same person are duplicated (as in the original Sleeping Beauty).They would therefore likely resist the implications that we have seen follow from Elga’s position; andthey could do this by rejecting the interpersonal version of the Self-Indication Assumption. I will not
discuss this option in detail in this paper, but it is worth noting that the Extreme Sleeping Beauty
thought experiment is unaffected by this issue. Furthermore, one can imagine possible situations inwhich the number of awakenings that an individual should except to experience is strongly corre-lated with the truth of some physical theory. One may therefore be able to derive counterintuitiveimplications from this view similar to the ones described above, albeit perhaps only in less empiricallyrealistic situations.
66 Synthese (2007) 157:59–78
obtained by conditionalizing Pon MONDA Y. Thus,
P+(HEADS )=P(HEADS |MONDA Y )=2/3.
Lewis’s argument for this claim is simple: before the experiment, Beauty should
assign credence 1/2 to the proposition that a fair coin to be tossed in the future will fallheads. She already knows that she will be awakened. Therefore, when she awakes, sheobtains no new relevant information; so her credence in HEADS should remain 1/2.
This argument starts to look peculiar when we compare it to Lewis’s explanation
of why Beauty should increase her credence in HEADS upon being informed that it
is Monday:
Now when Beauty is told during her Monday awakening that it’s Monday, ...
she is getting evidence — centered evidence — about the future: namely that she isnot now in it. That’s new evidence: before she was told that it is Monday, she didnot yet have it ...This new evidence is relevant to HEADS, since it raises her
credence in it by 1/6 (i.e. from 1/2 to 2/3).
13
On this reasoning, it would seem, one could similarly argue that when Beauty awakeson Monday (but before she is informed that it is Monday) she likewise gets relevantevidence — centered evidence — about the future: namely that she isnow in it. Since it
makes no difference whether the coin is tossed before the experiment begins or onMonday evening (a point of agreement between Lewis and Elga), let us suppose thecase where the coin is tossed just before Beauty awakens on Monday. If being in “thefuture” means being in the period after the coin has been tossed, Beauty now hasnew relevant information about her current location relative to this period (namely,that she is in it now). Lewis is thus committed to the view that one’s beliefs about achance event such as a coin toss canbe affected by obtaining evidence that is purely
about one’s own current location. Yet he offers no argument for why only centeredevidence that it is Monday, but not centered evidence that one is currently in the“experimental phase” (i.e. that it is either Monday or Tuesday, rather than, say, thepreceding Sunday) can be relevant to HEADS. Absent such an argument, his claimthat Beauty upon awakening should assign credence 1/2 to HEADS is a completelyunsupported assumption, one which those who disagree with the 1/2 view should feel
free to reject. Opponents of the 1/2 view can simply insist that Beauty does get cen-
tered relevant evidence when she ﬁnds herself awake in the experimental (Mondayor Tuesday) phase. Lewis’s argument for the 1/2 view therefore fails.
If we unpack the implications of accepting the 1/2 view, we ﬁnd that it has implica-
tions no less counterintuitive than those of the 1/3 view. Let us begin by consideringagain the ampliﬁed version of the Sleeping Beauty problem.
Extreme Sleeping Beauty
This is like the original problem, except that here, if the coin falls tails, Beauty
will be awakened on a million subsequent days. As before, she will be given
an amnesia drug each time she is put to sleep that makes her forget any previ-ous awakenings. When she awakes on Monday, what should be her credence inHEADS?
The adherent of the 1/2 view will maintain that Beauty, upon awakening, should retain
her credence of 1/2 in HEADS, but also that, upon being informed that it is Monday,
13(Lewis 2001, p175).
Synthese (2007) 157:59–78 67
she should become extremely conﬁdent in HEADS:
P+(HEADS )=1, 000, 001 /1, 000, 002.
This consequence is itself quite implausible. It is, after all, rather gutsy to have credence
0.999999% in the proposition that an unobserved fair coin will fall heads.
We can extract an even more counterintuitive consequence by modifying the exam-
ple slightly. Instead of using a single coin toss, with a prior probability of heads equalto 1/2, we could stipulate a sequence of 10 independent tosses of the same coin. Theprior probability that all of these tosses will come up heads is 2
−10, which is less than
one in a thousand ( ≈0.00098%). Suppose that unless the coin comes up heads all ten
times, Beauty will not be awakened again after the Monday awakening. If, however,the tossing does yield ten heads, then Beauty will be awakened on a million subsequentdays. We can then ask what odds Beauty could reasonably accept if offered to bet onsuch a sequence of coin tosses.
Beauty the High Roller
Beauty is awakened on Monday and after having been awake for an hour she
is offered a bet. She is told that a fair coin will be tossed ten times. If it landsheads all ten times then Beauty wins $1,000. If it lands tails at least once, thenBeauty loses $100,000. But there is a twist: If Beauty wins, the experiment endsat that point. If Beauty loses, she will be put to sleep, given an amnesia drugthat causes her to forget her awakening, and then awoken again the next day;and this procedure will be repeated for a total of one million days. (On each ofthese subsequent awakenings, Beauty will spend an hour in a state of ignoranceabout what day it is before she is put to sleep. No bet is offered after the initialMonday awakening.)
Beauty awakes on Monday and prudently decides to reject the bet that she is
offered. But just as she is about to declare her decision, David Lewis’s ghost appearsin a puff of smoke. The ghost explains the 1/2 view reasoning and argues that Beauty’scredence in the proposition that all ten tosses will come up heads should be very closeto unity. In fact, the ghost calculates that, even taking into account the low prior prob-ability of this proposition, Beauty should nevertheless assign it a posterior credenceof 99.8% after taking into account that she has just learnt that her current awakeningis the initial Monday awakening.
14The expected value of the gamble to Beauty is
therefore positive:
EV≈0.998×$1, 000 +0.002×(−$100, 000 )=$998−$200=$798.
So according to the ghost’s reckoning, Beauty ought to take the bet. But surely it
would be crazy for Beauty to follow the ghost’s advice.15Hence we should reject the
1/2 view.
14Let H∗be the proposition that the coin falls heads all ten times. Let Mbe the centered proposition
stating that Beauty’s is currently awake on the ﬁrst Monday. We then have
P(H∗|M)=P(M|H∗)P(H∗)
P(M|H∗)P(H∗)+P(M|¬H∗)P(¬H∗).
With P(H∗)=2−10,P(¬H∗)=1−P(H∗),P(M|H∗)=1, and P(M|¬H∗)=2/1, 000, 002, it easy to
check the ghost’s calculation.
15We assume that Beauty is risk-neutral and that her utility function is linear in money. If she has a
diminishing marginal utility of money, or is risk-averse, we can simply adjust the stakes or the number
68 Synthese (2007) 157:59–78
A hybrid model?
If the 1/3- and the 1/2-view both have unacceptable consequences, how can we build
a better model for reasoning with indexical information?
Consider again the 1/3 view. The problem with that view was that it led to a bias
in favor of hypotheses entailing that there are many subjectively indistinguishableduplicates of one’s current agent-part. When all the hypotheses under considerationagree on the number of such duplicates, the bias does not manifest itself; but itcauses trouble in cases like Sleeping Beauty. The obvious way to correct this “many-duplicates” bias is to divide one’s credence in being any one particular duplicate withthe total number of duplicates.
Consider the following two possible worlds, in which the only agent-parts are those
in the Sleeping Beauty experiment. (We shall assume throughout that all “agent-parts”are of equal duration.)
w1: h1 [The “heads” world ],
w2: t1 t2 [The “tails” world ].
As before, H1, T1, and T2 are the centered propositions expressing that one is cur-
rently h1, t1, and t2, respectively; HEADS is the proposition that the actual world isw1; and TAILS the proposition that the actual world is w2. The proposal for removingthe many-duplicate bias is that we set
P(H1)=1/2,
P(T1)=P(T2)=1/4.
Of course, we still have
P(H1|HEADS )=1,
P(T1|TAILS )=P(T2|TAILS )=1/2.
It follows that P(HEADS) =P(TAILS) =1/2. Thus, there is no general tendency,
upon ﬁnding oneself awakened in the experiment, to favor hypotheses implying thatthere are many such awakenings. This means that we are immune from objections ofthe “Presumptuous Philosopher”-type. Our new de-biasing postulate implies that theconditional probability of HEADS given that it is MONDA Y is greater than 50%:
P(HEADS |H1∨T1)=2/3, Constraint P .
Now, if the credence function P
+that Beauty should have upon learning that it is
MONDA Y were obtained by conditionalizing Pon (H1 ∨T1), then we would fall into
the trap illustrated in Beauty the High Roller. To avoid falling into this trap, it seems
as though we require that
P+(HEADS |H1∨T1)=1/2, Constraint P +.
“Presumptuous Philosopher” and “Beauty the High Roller” form a Scylla and
a Charybdis, which we must avoid, and yet it looks like the only way to satisfy theconstraints from these thought experiments involves violating Bayesian
(footnote 15 continued)
of awakenings that would occur so that the calculation still favors her taking the gamble, withoutaffecting the basic point of the thought experiment.
Synthese (2007) 157:59–78 69
conditionalization.16I believe, however, that this sacriﬁce is not necessary. The matter
is somewhat subtle.
Suppose that Beauty will be told after awakening on Monday that it is Monday.
The situation is then different from the one described above. It must instead berepresented as follows:
w1: h1 h1m [The “heads” world ],
w2: t1 t1m t2 [The “tails” world ].
The situation we are confronting involves ﬁvepossible agent-parts, not three. The
added terms, “h1m” and “t1m”, denote the agent-parts of Beauty that know thatit is Monday (in the heads and the tails world, respectively). Let us retain the P
unchanged (i.e. the credence function for Beauty at the times when she is unawarethat it is Monday). Now consider more carefully Constraint P
+,w h i c h seemed like
it was a constraint on the P+(i.e. the credence function that Beauty has when she
knows that it is Monday). Constraint P+contains the expression “H1 ∨T1”. But this
expression does notdescribe what Beauty knows after she has learnt that it is Monday,
for at that point she should set:
P+(H1)=0, P+(T1)=0.
This is because at that point Beauty knows that her current agent-part is either h1m
or t1m. The information she has just obtained is therefore not(H1∨T1), but rather
(H1M ∨T1M), where H1M is the centered proposition expressed by “My current
agent-part is h1m” and T1M is the centered proposition expressed by “My currentagent-part is t1m”. The correct formulation of Constraint P
+is therefore as follows:
P+(HEADS |H1M ∨T1M)=1/2, Constraint P +[corrected ].
This correction eliminates the conﬂict with Constraint Pand allows us to avoid vio-
lating Bayesian conditionalization. The corrected Constraint P+is precisely what we
need to save Beauty from ruin in the High Roller thought experiment.
One may still wonder what conditional credence Beauty should assign, before being
informed about it being Monday, to HEADS given that she is currently an agent-partthat knows that it is Monday:
P(HEADS |H1M ∨T1M)=?
However, there is no need to assign a value to this expression. Note that
P(HEADS |H1M ∨T1M)=P(HEADS &[H1M ∨T1M])/P(H1M ∨T1M).
Since P(H1M ∨T1M) =0, this expression is undeﬁned. And so it should be.
17
Let us take a step back and consider the point more generally. Whenever an agent
receives some evidence E, we could distinguish the earlier agent-part, α−, that lacked
this evidence, and the later agent-part, α+, which has come to possess it. According to
16It has been argued that we should indeed violate conditionalization in the Sleeping Beauty prob-
lem (Kierland and Monton 2005). Kierland and Monton argue for the 1/3 answer on grounds whichthey claim do not lead to the counterintuitive result in the Beauty and Doppelganger. Their positionthus diverges signiﬁcantly from Lewis’s 1/2 view.
17The model used here presupposes that agent-parts know what their evidence is. This simplifying
assumption may be inappropriate in certain cases, but we shall not here discuss how such cases shouldbe modeled.
70 Synthese (2007) 157:59–78
the reasoning just described, we cannot automatically conclude that the conditional
probability P(X|E&“ Ia mc u r r e n t l y α−”), conditionalized on E, yields the correct
posterior credence that α+should assign to X.O n l yi f
P(X|E&“I am currently α−”)=P+(X|E&“I am currently α+”),
can the kinematics be represented in the simpliﬁed form P+(X)=P(X|E). This stan-
dard representation is thus elliptic as it omits some changes in indexical information.
In ordinary cases, such changes in indexical information are irrelevant to the
hypotheses being considered and can hence be safely ignored. The standard ellip-tic representation of Bayesian conditionalization can then be used without danger. Incertain special cases, however, such delicate changes in indexical information can berelevant, and it is then crucial to recognize and make explicit the hidden intermediarystep. Sleeping Beauty, on the model proposed here, turns out to be just such a specialcase.
To recapitulate, I have argued that a Bayesian can coherently accept both Con-
straint Pand the corrected Constraint P
+, even though superﬁcially this seems to vio-
late Bayesian conditionalization. The reason why we not only canbut should accept
both these constraints was given earlier: to avoid the counterintuitive consequencesthat follow if either of these constraints is violated, as shown by the “PresumptuousPhilosopher” and the “Beauty the High Roller” thought experiments.
The long-run frequency argument
One other important argument for the 1/3 view needs to be examined as it might be
thought to pose a problem for the hybrid model. The discussion of this argument willalso serve to further elucidate how the proposed model works.
A proponent of the 1/3 view could argue that Sleeping Beauty awakened ought to
have credence 1/3 in HEADS because if the experiment were repeated many times,then approximately 1/3 of all her awakenings would be heads-awakenings (and 2/3would be tails-awakenings). In the inﬁnite limit, this ratio would, with probability 1,be approached arbitrarily closely. This argument could be buttressed by introducingbetting considerations. In the inﬁnite limit, Beauty would have to assign credence
1/3 to HEADS, else she would be guaranteed a loss if she put her money where hermouth is.
For a betting argument to have any bite, the hypothesized bookie must have the
same information as Beauty. If a bet were only offered on the Monday awakening ineach run of the experiment, and if both the bookie and Beauty knew this, then Beautycould infer from the fact that she was offered a bet that it was Monday. According tothe hybrid model, she should then assign credence 1/2 to the proposition that the coinwill fall heads in that trial. This will match the long-run frequency of bets that she willwin, so in this case betting considerations pose no problem.
The betting argument therefore requires that Beauty be offered a bet each time
she is awakened. Then she cannot infer what day it is from the fact that she is beingoffered a bet. In this case, in the long run, Beauty would be expected to lose 2/3 ofher bets if she consistently bet on heads. How does this square with the prescriptionof the hybrid model that Beauty, upon awakening (but before learning which day itis) assigns credence 1/2 to heads?
Synthese (2007) 157:59–78 71
One possible response to this argument is to deny that betting considerations
provide a valid guide to credence assignment in the present case. Since there wouldbe a different number of bets placed depending on how the coin fell, one might regardthe test as unfair.
18In support of this response one may note that the presumptuous
philosopher would also be vindicated if we assumed that an agent-part’s credence
should be determined by the betting-odds at which the expected net gains and lossesof the collective of all his duplicate agent-parts would be zero. Since there would bea trillion times more duplicates of the agent-part if theory T
2is true then if T 1is true,
each agent-part would have to assign a trillion times greater odds to T 2than to T 1in
order for the expected value of all the bets made by the collective of agent-parts tobe zero. And yet we argued that it seems wrong for an agent-part to assign a trilliontimes greater credence to T
2than to T 1.
However, this response does not address the case of the repeated Sleeping Beauty
problem. For in this case, in the inﬁnite limit, there is no uncertainty about the totalproportion of awakenings in tails- and heads-runs of the experiment. Beauty knowsthat (with probability one) there will actually be two times as many awakenings intails-trials as in heads-trials. In this case, therefore, betting considerations unambig-uously suggest that Beauty upon awakening should assign the 2/3 credence to tails.Here one could not justify a divergence of credence assignment from betting odds bysaying that there would be a different number of bets placed depending on which ofthe hypotheses under consideration is true, because the total number of bets placed isnot (signiﬁcantly) variable when Beauty is put through a large number of repetitionsof the experiment.
There is, consequently, strong reason for recommending that Beauty assign cre-
dence 1/3 to heads when she knows that the experiment will be repeated very manytimes. This, however, is not an objection to the hybrid model proposed above. Thehybrid model, as we shall now see, implies the very same credence assignment as thebetting considerations suggest. Betting considerations, far from being an embarrass-ment to the hybrid model, actually agree with its implications and support it.
Up until this section, our discussion has focused on (variations of) the single-shot
Sleeping Beauty problem, where there are no repetitions of the experiment. This isthe simplest case: the world contains no other relevant agent-parts than those existingwithin a single implementation of the Sleeping Beauty experiment. Let us now applythe hybrid model to the situation that arises if the experiment is repeated many times.But ﬁrst, as an intermediary step, consider the following case.
Three Thousand Weeks (non-random)
Beauty lives for three thousand weeks. On odd-numbered weeks she is awak-
ened once, on Mondays. On even-numbered weeks she is awakened twice, onMondays and Tuesdays. After each awakening she is given an amnesia drug thatcauses her to forget her previous awakenings. Beauty knows all this.
The hybrid model that I propose implies that in this case, Beauty should have
credence 1/3 in the centered proposition “My current awakening is taking place inan odd-numbered week” (or “ODD” for short). This is because Beauty, when shewakes up, knows that ODD is true for one third of all the agent-parts that are in thesame subjective evidential state as her current agent-part. (The credence assignment
18(see also Arntzenius 2002).
72 Synthese (2007) 157:59–78
follows from the very weak indifference principle which Lewis, Elga, and I all accept.)
Crucially, these agent-parts are all actual agent-parts, as opposed merely possible ones.
We thus have
P(ODD )=1/3.
Further, it is easy to show that
P(ODD |MONDA Y )=1/2.
If we suppose that every Monday, just before being put to sleep, Beauty is told that it
is Monday, we also have
P+(ODD |MONDA Y )=1/2.
Note that, for the reasons explained earlier, “MONDA Y” denotes a different cen-
tered proposition in each of these two conditional credence expressions. (In the ﬁrstexpression, “MONDA Y” refers to a proposition that is centered on an agent-part thatdoes not know that it is Monday; in the second expression, specifying Constraint P
+,
“MONDA Y” refers to a proposition centered on an agent-part that does know that
it is Monday.) In the present case, however, the conditional credences work out thesame. The hybrid view therefore coincides with the 1/3 view in this example.
The key difference between the original Sleeping Beauty problem and 3,000 weeks
and is that in the latter case — but not in the former — there are twice as many actual
awakenings of one type as of the other. This means that in 3,000 weeks, the priorcredence in ODD, before Beauty learns that it is Monday, is unaffected by the cor-rection we made to eliminate the bias in favor of hypotheses entailing the existenceof more duplicates. (Such a bias would result from applying the indifference principleto a class of agent-parts that included merely possible as well as actual agent-parts.)In 3,000 weeks, ODD is true for one-third of the agent-parts that are ignorant aboutwhether it is Monday, and for one half of the agent-parts who know that it is Monday;correspondingly, the credence in ODD is 1/3 for the ﬁrst type of agent-part and 1/2for the second type.
19
Let us now apply this analysis to a more straightforwardly repeated version of the
original Sleeping Beauty problem:
The N-fold Sleeping Beauty Problem
This is like the original Sleeping Beauty problem repeated Ntimes on consec-
utive weeks. Beauty knows that the experiment is repeated Ntimes, but she is
unable to determine which run of the experiment she is currently in.
For N=1, this reduces to the original Sleeping Beauty problem, and Beauty’s
credence in HEADS should be 1/2, both before and after learning that it is Monday. IfNis some large number, such as N=3, 000, then the case approximates 3,000 weeks,
and Beauty’s credence in HEADS should be approximately 1/3 before learning thatit is Monday, and 1/2 after being told that it is Monday. (“HEADS” here stands for“My current awakening is in one of the trials where the coin fell heads”.) The larger N
19We say that ODD “is true for” an agent-part if the centered proposition which that agent-part
would express by saying “ODD” is true. When writing down a symbol like “ODD”, we need to becareful about whether we take this to refer to a speciﬁc centered proposition or to a function thatyields a centered proposition when given an agent-part as an argument. In the text, the context shouldmake it clear what is intended in each case.
Synthese (2007) 157:59–78 73
is, the more exact will the approximation be. The credences in the 3,000-fold Sleeping
Beauty Problem are not exactly equal to those in 3,000 weeks because the total num-
ber of awakenings is not strictly ﬁxed. There is, however, a very high chance that therewill be roughly 3,000 tails-awakenings and 1,500 heads-awakenings in the 3,000-foldSleeping Beauty Problem, so it closely approximates 3,000 weeks.
Illustration: The hybrid model to the N=2c a s e
It may be instructive to calculate the exact credences for the N=2 case. There are
four possible outcomes of the coin tosses: heads-heads, heads–tails, tails–heads, andtails–tails. We can represent these four possibilities along with the possible agent-partsthey would realize as follows:
Week 1 |Week 2
w1: h1 |h2
w2: h3 |t1 t2
w3: t3 t4 |h4
w4: t5 t6 |t7 t8
Each of these four possibilities has an equal chance of occurring ( p=1/4). Since
each of these agent-parts are in the same evidential situation, Beauty’s conditionalcredence, given one of the four possibilities, is divided equally between the agent-parts that that possibility would realize. Hence, her unconditional credence in beingany particular possible agent-part is obtained by multiplying this conditional credencewith her prior credence in the possibility in question (i.e. 1/4). Thus, we get the fol-lowing assignment of credence to the centered propositions that she is currently aparticular agent-part:
Week 1 | Week 2
w1: 1/8 | 1/8
w2: 1/12 |1/12 1/12
w3: 1/12 1/12 | 1/12
w4: 1/16 1/16 |1/16 1/16
We obtain P(HEADS) by summing the credences of the centered propositions that
imply HEADS (indicated with boldface):
P(HEADS )=1/8+1/8+1/12+1/12=5/12.
Since P(HEADS |MONDA Y ) =P(HEADS & MONDA Y)/ P(MONDA Y), we
likewise get
P(HEADS |MONDA Y )=(5/12)/(17/24)=10/17.
This, however, is not the credence that Beauty should assign to HEADS if she
were told that it is Monday. For the same reasons as noted above in the discussionof the original (one fold) Sleeping Beauty problem, the relevant quantity is insteadP
+(HEADS |MONDA Y). To determine this quantity, we again represent four possi-
bilities, but these now include agent-parts that know that it is Monday (these are theagent-parts in the middle columns, whose names end with the letter ‘m’):
74 Synthese (2007) 157:59–78
Week 1 | Week 2
w1: h1 h1m |h2 h2m
w2: h3 h3m |t1 t1m t2
w3: t3 t2m t4 |h4 h4m
w4: t5 t3m t6 |t7 t4m t8
Since the number of agent-moments that know that it is Monday is the same in all
four possibilities (i.e., two in each case), each of these agent-parts (who are in thesame evidential situation) should assign the same credence to being a particular oneof these agent-parts, namely (1/4)(1/2)=1/8, and they should assign zero credence
to being some other agent-part. Thus:
Week 1 | Week 2
w1: 0 1/8 |0 1/8
w2: 0 1/8 |01 / 80
w3: 0 1/8 0 |0 1/8
w4: 0 1/8 0 |01 / 80
To obtain P+(HEADS |MONDA Y), we sum the credences of the centered proposi-
tions that imply both HEADS and MONDA Y (indicated in boldface), and divide thisby the sum of the credences that imply MONDA Y:
P
+(HEADS |MONDA Y )=(1/8+1/8+1/8+1/8)/1=1/2.
The hybrid model thus implies that when Beauty learns that it is Monday, she
should have credence 1/2 in HEADS. This is so both in the original one-shot versionof the Sleeping Beauty problem and in the repeated (“ N-fold”) versions where N≥1.
Discussion
We have argued that the standard arguments for the standard positions on the Sleep-
ing Beauty problem, the 1/2 view and the 1/3 view, are, if not directly question-beg-ging then at least inconclusive in that they rely on eminently deniable premises. Toevaluate the standard positions, therefore, we need to seek for further constraints.We presented two such constraints in the form of two thought experiments. ThePresumptuous Philosopher thought experiment, in a version adapted for applicationto the Sleeping Beauty case, strongly suggests that the 1/3 view is wrong. The Beautythe High Roller thought experiment strongly suggests that the 1/2 view is wrong.On these grounds, we concluded that both the standard models for reasoning about
self-location are unacceptable.
In the second, constructive part of the paper we proposed a new model. This model
seeks to combine the most attractive features of the 1/3- and the 1/2-view, so wetermed it the hybrid model. It implies that Beauty should not take the fact that she iscurrently awake as evidence that there are large numbers of awakenings. But it alsoimplies that when Beauty discovers that it is currently Monday, she should not takethis as evidence against the hypothesis that there will be many more awakenings inthe future.
If the hybrid model is correct, it might explain the fact that both the 1/3- and the
1/2-views have some intuitive appeal. According to the hybrid model, both these views
Synthese (2007) 157:59–78 75
get something right. The 1/3 view is right that Beauty’s posterior credence in HEADS
after being informed that it is Monday should be one-half. The 1/2 view is right thatBeauty’s prior credence in HEADS, after awakening but before learning that it is
Monday, should be one-half.
The 1/3 view is also right that in the version of the Sleeping Beauty where the
experiment is repeated a large number of times, Beauty should (in the inﬁnite limit),upon awakening, assign a prior credence of 1/3 to the centered proposition that thecoin fell heads in that particular trial. The hybrid view distinguishes between actualand merely possible agent-parts. In the N-fold Sleeping Beauty problem, for N/greatermuch1, it
is (almost certainly) the case that approximately one-third of all actual agent-parts ofBeauty are in trials in which the coin fell heads, and the total number of awakeningsis (with high probability) approximately determined in advance. By contrast, in theone fold version, it is notthe case that one-third of all actual agent-parts of Beauty are
in a heads-trial. There, either all are, or none. Moreover, in the one fold version, thetotal number of awakenings is strongly correlated with which hypothesis, HEADS orTAILS, is true. The hybrid model corrects for the bias in favor of many awakeningsthat is inherent in the 1/3 view. (In cases where Nis small but larger than 1, the hybrid
model gives a prior credence that is intermediate between the that of the 1/3 view andthe 1/2 view, thus avoiding any sharp discontinuity. In general, for N≥1, we have
1/3≤P(HEADS )≤1/2.)
The main concern about the hybrid model is that it appears to violate Bayesian
conditionalization. I argued, however, that this violation is merely apparent. If we payclose attention to the changing indexical information available to different agent-seg-ments, we ﬁnd that the model does not violate Bayesian conditionalization. A lessonhere is that while indexical evidence is irrelevant and can be ignored in most ordinarycases of Bayesian updating, there are special cases — Sleeping Beauty included — wheresuch evidence is relevant. In these special cases, certain implicit assumptions in thecommon way of applying Bayesian conditionalization are false.
In closing, I will address one challenge that could be directed at the hybrid model.
20
If Beauty follows this model and agrees to betting odds matching her credence func-tion, she can be Dutch-booked.
The Beauty and the Bookie
This is like the original one-shot version but with an added Bookie, who is put to
sleep at the same time as Beauty and given the same amnesia drug. (We put theBookie through this procedure to make sure that he does not have any relevantinformation that Beauty lacks.) Upon awakening, on both Monday and Tuesday,before either knows what day it is, the Bookie offers Beauty the following bet:
Beauty gets $10 if HEADS and MONDA Y.
Beauty pays $20 if TAILS and MONDA Y.(If TUESDA Y, then no money changes hands.)
20I’m grateful here to one anonymous referee. A similar Dutch-book argument has recently been
advanced in Hitchcock (2004).
76 Synthese (2007) 157:59–78
On Monday, after both the Bookie and Beauty have been informed that it is
Monday, the Bookie offers Beauty a further bet:
Beauty gets $15 if TAILS.
Beauty pays $15 if HEADS.
If Beauty accepts these bets, she will emerge $5 poorer.
Since Beauty is able to anticipate the result of accepting all the bets, it is clear that
she should not do so.
Following the hybrid model, Beauty should have no objection to accepting the
second Monday bet. The hybrid model implies that P+(HEADS |MONDA Y) =
P+(TAILS |MONDA Y) =1/2. Being offered a single straightforward bet on HEADS
at even odds, knowing that it is Monday, she has no reason to refuse it.
It is the other set of bets that she should reject. The hybrid model implies that
Beauty, before learning that it is Monday, assigns P(HEADS |MONDA Y) =2/3. This
appears to justify her accepting the bookie’s ﬁrst offer. But here the situation is morecomplicated. Since neither party knows whether it is Monday, the Bookie cannot offerthis bet only on Monday. He must offer it on both awakenings. This means that thetotal number of bets will vary depending on how the coin falls: if heads, the ﬁrst typeof bet is offered only once; but if tails, it is offered twice. Moreover, we may assumethat Beauty will either accept it on both occasions or reject it on both occasions, asshe has no effective way of telling which occasion she is currently encountering.
21So
Beauty knows that she would be accepting two bets if TAILS and one bet if HEADS.
Now, we already know from other examples that when the number of bets depends
on whether the proposition betted on is true, then the fair betting odds can divergefrom the correct credence assignment. For instance, suppose you assign credence 9/10to the proposition that the trillionth digit in the decimal expansion of πis some num-
ber other than 7. A man from the city wants to bet against you: he says he has a gutfeeling that the digit is number 7, and he offers you even odds—a dollar for a dollar.Seems ﬁne, but there is a catch: if the digit is number 7, then you will have to repeatexactly the same bet with him one hundred times; otherwise there will just be onebet. If this proviso is speciﬁed in the contract, the realbet that is being offered you is
one where you get $1 if the digit is not 7 and you lose $100 if it is 7. That you shouldreject thisbet is quite unproblematic and does not in any way undermine your original
assessment that the probability of the trillionth digit being 7 is 1/10.
A similar situation can arise in a more subtle way. We can construct a scenario
where, even though no “catch” is explicitly part of the contract, you neverthelessknow that you will be put in a position where you will end up betting a hundred timesif you are wrong but only one time if you are right. This could happen, e.g. if there isa machine that will determine the correct answer and then, on the basis of what thisanswer is, will decide whether to repeatedly administer an amnesia drug to you thatmakes you forget whether you have already betted. The machine could do this in sucha way that you end up making a larger number of bets if you are wrong. If you believethat you are facing a situation of this kind, you should take corrective action to limitthe distortive effects of the memory erasure on your decision-making. In particular,you may decide to reject bets that seem fair to you and that may have been perfectlyacceptable in the absence of the forced irrationality constraint.
21If Beauty could opt for a mixed strategy, she could decide to accept the bet at a given occasion
with a certain probability. This would complicate the argument but would not affect the conclusion.
Synthese (2007) 157:59–78 77
Let us return to the case of Beauty and the Bookie. Beauty knows that she faces the
risk of having her memory erased and thus of becoming irrational. (Memory erasureentails a form of irrationality.) For reasons such as those described above, Beauty maytherefore reject the bookie’s ﬁrst set of bets as a form of damage control to minimizethe impact of the failures of rationality from which she knows she is at risk. If thedeviation of her optimal betting odds from her credence assignment can be justiﬁedon these grounds, then she can use the hybrid model and still avoid being Dutchbooked.
It is interesting that in Beauty and the Bookie, Beauty’s betting odds should deviate
from her credence assignment even though the bet that might be placed on Tuesdaywould not result in any money changing hands. In a sense, the bet that Beauty and thebookie would agree to on Tuesday is void. Nevertheless, it is essential that this bet isincluded in the example. The bookie is unable to pursue the policy of only offering betson Monday since he does not know which day it is when he wakes up. If we changedthe example so that the bookie knew that is was Monday immediately upon awaken-ing, then Beauty and the bookie would no longer have the same relevant information,and the Dutch book argument would fail. If instead we changed the example so thatBeauty as well as the bookie knew that it was Monday immediately upon awakening,then Beauty’s credence in HEADS & MONDA Y would be 1/2 throughout Monday,so again she would avoid a Dutch book.
22
In conclusion, the hybrid model combines the comely aspects of the 1/2 view and
the 1/3 view while avoiding their faults. The main concern with the hybrid model isthat it may appear to violate Bayesian conditionalization. I have presented (tentative)arguments suggesting that the violation is merely apparent. At any rate, one mighthope that having a third contender for how Beauty should reason will help stimulatenew ideas in the study of self-location.
23
References
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22If Beauty would know on Monday that it is Monday, then she would also be able to infer on
Tuesday — from the fact that she does not know then that it is Monday — that it is Tuesday. So she wouldalways know what day it is. (We assume that Beauty always know the general setup of the experimentshe is in.)
23For comments and discussions, I am grateful to Adam Elga, Bradley Monton, Brian Kierland,
Simon Saunders, and anonymous referees.
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EDITOR LETTER
Introduction: Open Questions in Roboethics
John P . Sullins
Published online: 12 August 2011
#Springer-V erlag 2011
1 The Rise of the Robotethicist
Roboethics is the recent offshoot of computer ethics that pays special attention to the
alterations that need to be made to computer ethics when we give the computer mobility
and a means to interact directly in the human environment. The closely related field of
machine morality explores how ethical systems and behaviors may be programmed intosocial robotics applications. As robots move from the factory floor into our homes andwork lives, they stand to change key aspects of the way our lives are lived. In order to besuccessful, these machines must be programmed with the ability to navigate the humanlife world without committing ethical faux pas or moral outrage. Thus, the roboethicist istasked not only with critiquing the attempts of robots engineers to achieve theintegration of these machines into our life world, but also, and more importantly , with
suggesting means of achieving better results than what is presently on offer.
The undeniable roots of roboethics begin in the world of science fiction. The very
coining of the word “robot ”in Karel Čapek ’s 1936 play, RUR, is loaded with ethical
import. The Czech word “Robota ”refers to labor or servitude, which gives us the
uncomfortable inference that roboethics refers to a kind of slave ethics. I reject thisconnotation and it is just an unfortunate byproduct from the literary trope of therobot rebellion that Čapek began with his play and Fritz Lang masterfully solidified
in the human psyche with his film Metropolis , something which Hollywood has been
reiterating ever since. There is no need to reenact this unfortunate future in reality.
As the great science fiction writer Philip K. Dick once observed, the duty of sciencefiction is to imagine dystopian futures so that we don ’t actually have to live them.
With this in mind, we can then see that the job of the roboethicist is not simplyscience fiction, it is instead to help avoid the imagined robo-apocalypse and helpbuild an alternative future where robots are not resentful slaves or out of controlkilling machines, but instead more like pets and perhaps someday even friends orpossibly, in the very far future, even colleagues. In the near future, the job of thePhilos. T echnol. (2011) 24:233 –238
DOI 10.1007/s13347-011-0043-6
J. P . Sullins ( \*)
Sonoma State University , 1801 East Cotati A veune, Rohnert Park, CA 94928, USA
e-mail: John.sullins@sonoma.edu
roboethicist is to ensure that we do not harm each other too deeply with these
machines as they grow in complexity and capability.
Gianmarco V eruggio seems to have coined the term “roboethics ”in 2002 at the first
roboethics workshop organized around a nIEEE robotics conference.1At that time, it
was decided to separate the field into two allied subfields. One is machine ethics ormachine morality , which is concerned with describing how machines could behave
ethically towards humans. The other is roboethics itself, which is concerned with how
humans relate to these machines in both the design and use phase of their operation. Inthe last 9 years though, these terms have drifted a bit and you will hear expressionssuch as “machine ethics, ”“machine morality , ”“roboethics, ”“robot ethics, ”and
“moral machines ”all used somewhat synonymously to refer to the ethical concerns
raised by robotics technologies.
Since that time there have been numerous articles printed, workshops and
conference tracks organized, special issues of journals, blogs and Facebook groups
formed, as well as a few important book projects. But there is much left to do and it
is my purpose here to try to interest more people to join this growing area ofresearch. Robotics technology ’s move into the home is roughly where the personal
computer was in the 1970 ’s. If such trend continues, then we can expect personal
robotics and military robotics to move quickly into the home, workplace andbattlefield. It is therefore our duty to stay ahead of that curve in order to anticipateand help alleviate the ethical impacts of these technologies.
One further conceptual complexity needs to be mentioned here as well. Robots come
in two broad categories, autonomous and non-autonomous. Roughly speaking,
“autonomy ”typically refers to the level of human control and oversight over the
robot ’s actions and decisions. When one speaks of “autonomous robots, ”one is
generally not making any strong claim regarding the philosophical free will of themachine. It is simply the acknowledgement that autonomous robots make the majorityof their decisions using computational systems, whereas non-autonomous or telerobotshave at least some human oversight and input into the decisions they make.
2
While the media has us all used to the idea of autonomous robots, as it turns out
they are very difficult to make and so the robots we see in use today are all largely
telerobots. Therefore roboethicists should focus a bit more on how telerobots alterthe ethical thinking of their users since machines making autonomous ethicaldecisions are still only a theoretical possibility.
2 Open Questions in Roboethics
As roboethics is a young field of study, there are many interesting open questions and
subfields of study . My list here is not meant to be exhaustive but it is what I believe to bethe most interesting at this time.
1It is possible that the term was used prior to this date but the 2002 workshop is certainly where the term
became more widely used.
2There is more complexity in the case of machines that house the majority of their artificial intelligence
capabilities in a remote machine and then teleoperate a drone or drones of some type. I will gloss over this
here and just treat these machine architectures as autonomous robots for now.234 J.P . Sullins
2.1 Military Applications
This is by far the most important of the subfields of roboethics. It would have been
preferable had we worked through all the problems of programming a robot to thinkand act ethically before we had them make life and death decisions, but it looks likethat is not to be. While teleoperated weapons systems have been used experimentally
since the Second W orld W ar, there are now thousands of robotic weapons systems
deployed all over the world in every advanced military organization and in an ad hocway by rebel forces in the Middle East (Singer 2009 ). Some of the primary ethical
issues to be address here revolve around the application of just war theory. Can theseweapons be used ethically by programing rules of warfare, the law of war and justwar theory into the machine itself? Perhaps machines so programmed would makethe battlefield a much more ethically constrained space? How should they be builtand programmed to help war fighters make sound and ethical decisions on the
battlefield? Do they lower the bar to entry into conflict too low? Will politicians see
them as easy ways to wage covert wars on a nearly continuous level? In an effort tokeep the soldier away from harm, will we in fact bring the war to our own front dooras soldiers telecommute to the battlefield? What happens as these systems becomemore autonomous? Is it reasonable to claim that humans will always be “in”or“on
the loop ”as a robot decides to use lethal force?
2.2 Privacy
Robots need data to operate. In the course of collecting, data they will collect some
that people may not want shared but which the machine needs nonetheless tooperate. There will be many tricky conundrums that have to be solved as more andmore home robotics applications evolve. For instance, if we imagine a general-purpose household robot of the reasonably near future, how much data of thefamily ’s day-to-day life should it store? Who owns that data? Might that data be used
in divorce or custody settlements? Will the robot be another entry for directed
marketing to enter the home?
2.3 Robotic Ethical A warenessHow does a machine determine if it is in an ethically charged situation? And
assuming it can deal with that problem, which ethical system should it use to helpmake its decision? Philosophers such as John Dewey and later Mario Bunge haveargued that a technology of ethics is possible and in some ways preferable (Sullins
2009 ). I am certain they were not thinking of robots when they made these
arguments, but their view that ethics is transactional and instrumental allow us toextend their ideas to the claim that ethics is computational. Thus, it is not out of thequestion that machine ethics is possible. Y et we are sorely lacking on the specificsneeded to make any of these claims anything more than theoretical. Engineers arewonderfully opportunistic and do not tend to have emotional commitments to this orthat school of thought in ethics. Therefore, what we see occurring today is that theytend to make a pastiche of the ethical theories that are on offer in philosophy and
pick and choose the aspects of each theory that seem to work and deliver real results.Introduction: Open Questions in Roboethics 235
2.4 Affective Robotics
Personal robots need to be able to act in a fr iendly and inviting way. This field is often
called social robotics, soci able robotics, or affective computing, and was largely the
brainchild of Cynthia Breazeal, from the Mas sachusetts Institute of T echnology (MIT)
robotics lab ( 2002). The interesting ethical question here is: if your robot acts like your
friend, is it really your friend? Perhaps t hat distinction does not even matter? With
sociable robotics, the m achine looks for subtle clues gat hered from facial expression,
body language, perhaps heat signatures or other biometrics and uses this data toascertain the user ’s emotional state. The machine then alters its behavior to suit the
emotional situation and hopefully make the use r feel more comfortable with the machine.
If we come to accept this simulacrum of frien dship, will this degrade our ability to form
friendship with other humans? W e might begin to prefer the company of machines.
2.5 Sex T obots
It seems strange but it is true that there are already semiresponsive sex dolls that do count
as a minor type of robot. These machines are such a tantalizing dream for someroboticists that there is little doubt that this industry will continue to grow . This categoryof robotics supercharges the worries raised by affective robotics and adds a few more.Sociable robots examine the user biometrics so the robot can elicit friendly relations, buthere the robot examines biometrics to elicit sexual relations. A sex robot is manipulating
very strong emotions and if we thought video games were addictive, then imagine what
kind of behavior might be produced by a game consul with which one could have sex.These machines are likely to remain on the fringe of society for some time, but theroboticist David Levy has argued that since this technology can fulfill so many of ourdreams and desires, it is inevitable that it will make deep market penetration andeventually will be widespread in our society (Levy 2007 ). This will result in many
situations that will run the spectrum from tragic, to sad, to humorous. The key pointhere is: whether the machines can really be filled with love and grace or whether we
are just fooling ourselves with incredibly expensive and expressive love dolls. I can
easily grant that engineers can build a machine with which many would like to havesex, but can they build a machine that delivers the erotic in a philosophical sense? Canthey build a machine that can make us a better person for having made love to it?
2.6 CarebotsSomewhat related to the above are carebots. These machines are meant to provide
primary or secondary care to children, the elderly and medical patients. There are
already a number of these machines, such as the Paro robot, in service around the world.On one end of the scale, one has something like Paro, a robot that is meant to provideartificial pet therapy for its users. T owards the middle of the scale, one would havemachines built to assist medical caregivers in lifting and moving patients or helping tomonitor their medications or just to check in with patients during their stay . At the far endof the scale, one would have autonomous or semi-autonomous machines that wouldhave nearly full responsibility in looking after children or the elderly in a home setting.
Here again, we have some of the same issues raised by social robotics and the236 J.P . Sullins
concomitant privacy issues. But in addition to those you have the troubling problem of
why other humans are not taking care of their own children and elderly . What kind ofsociety are we creating where we wish to outsource these important human relations to amachine allowing younger generations to simply ignore the elderly?
2.7 Medibots
These are related to carebots but I am specifically thinking here of robots that assist
in surgery and other life and death medical practices such as administeringmedication. Often, the surgeons using these machines are close by the operatingtheater, but this technology is also used to allow a surgeon to work on a patient manythousands of miles away. This technology can be useful when dealing with awounded soldier on a distant battlefield or a patient with serious conditions who isliving in remote or economically depressed places of the world. This technology puts
a new wrinkle on many of the standard medical ethics issues and we need more
medical ethicists to study this phenomenon in depth.
2.8 Autonomous V ehiclesOur roadways could change in a very radical way. Autos and large transportation
vehicles of the near future may have no human driver. Already some luxury vehicleswill take over in emergency breaking situations or when the driver falls asleep at the
wheel. A number of autos will park themselves completely autonomously. The vast
majority of the ethical issues involved here will be legal in nature, but there will alsobe issues of trust involved. For instance, can one trust a vehicle to make the rightdecisions when those decisions mean the lives of you, your family and all thosearound you? There have already been deaths caused by faulty automatic navigationservices because people robotically follow the directions of the GPS machine nomatter what it says, even if it is giving incorrect directions that lead one intodangerous situations.
2.9 Attribution of Moral Blame
This is one of the biggest conundrums in roboethics. Nearly all moral systems have
some way of assessing which moral agent involved in a system is to blame whenthings go wrong. Most humans respond to blame and punishment and might modifytheir behavior to avoid it when possible. But how does one blame a machine? Willpeople use robots as proxies for the bad behavior in order to remove themselves
from blame? When a military robot kills an innocent civilian, who is to blame? If
you are asleep in your robotic car and it runs down a pedestrian, did you commitmanslaughter or are you just an innocent bystander?
2.10 Environmental RoboticsThere are two ways to look at the environmental ethics impacts of robotics. One is to
look at the impact of the manufacture, use and disposal of robots. Currently, there is
no green robotics movement and we should push for this to be developed. A secondIntroduction: Open Questions in Roboethics 237
interesting idea is that robotics could provide an invaluable tool for gathering data
about environmental change. The very same robots that are used to monitor enemytroops and scour the ocean floor for enemy activity can be easily re-tasked tomonitor forests, ocean ecosystems, protect whale and dolphins or any number ofenvironmental tasks that unaided humans find difficult.
3 Robotics, War and Peace
This special issue is an attempt to advance our understanding of the many issues
raised above. The articles collected here represent some of the very best thought onthese subjects. I would like to thank the many referees who worked on this project;their unsung efforts ensured that this issue is a valuable contribution to the growingscholarship on the ethical impacts of robotics technology. Robotics and warfare
dominates the conversation at this time but it is my sincere hope that the many
conflicts that plague our world can diminish and we can move to the study of morepeaceful applications of this fascinating technology.
References
Breazeal, C. (2002). Designing sociable robots . Cambridge, MA: The MIT Press.
L e v y ,D .( 2 0 0 7 ) . Love and sex with robots, the evolution of human –robot relationships . Harper/HarperCollins:
New Y ork, USA.
Singer, P . W . (2009). Wired for war . New Y ork: Penguin Press HC.
Sullins, J. (2009). Artificial moral agency in technoethics. In R. Luppicini & R. Adell (Eds.), Handbook of
research on technoethics . New Y ork: IGI Global.238 J.P . Sullins