Sleeping Beauty and Self-Location: A Hybrid Model (2006) Nick Bostrom Faculty of Philosophy, Oxford University 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 contain indexical information. Opinion on this problem is split between two camps, those who defend the “1/2 view” and those who advocate the “1/3 view”. I argue that both these positions are mistaken. Instead, I propose a new “hybrid” model, which avoids the faults of the standard views while retaining their attractive properties. This model 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 where indexical information is relevant, we discover that the hybrid model is in fact consistent with Bayesian kinematics. If the proposed model is correct, there are important lessons for the study of self-location, observation selection theory, and “anthropic reasoning”. 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 her to 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 on Monday, what should her credence be that the coin will fall heads? 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 theoretic literature.1 It was named by Robert Stalnaker, who had learnt about similar cases from Arnold Zuboff. The problem was brought to the attention of the philosophical community through an exchange between Adam Elga, who argued for the answer 1/3 (hereafter the “1/3 view”), and David Lewis, who defended the 1/2 view. The last few years have seen a burst of publications advocating one or the other of the two competing 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 location. An agent-part that knew exactly which possible world is actual can still be ignorant about its own location in that world. That can happen if the world contains two or more agentparts whose evidential states are subjectively indistinguishable. These agent-parts would then be unable to determine with certainty their own spatiotemporal location. (Even if 1 Robert Stalnaker gave the problem its name, after having of examples of a similar kind in unpublished work by Arnold Zuboff. Closely related problems have also been discussed in the game theory literature; see volume 20 of the journal Games and Economic Behavior (1997). 1 Beauty knew that the outcome of the coin toss would be tails, she could not know whether it was currently Monday or Tuesday.) Another way of expressing this is by saying that the agent-parts would be ignorant about which centered 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”, possess all 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 problem in isolation from this wider context, one risks coming up with answers and principles that do not fit with the other parts of the puzzle. I will first argue that both the 1/3 view 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 is identical to neither. The hybrid view overcomes the problems associated with the purebred answers. It also suggests an explanation for why the 1/3- and the 1/2-view have both managed to appeal to their fan bases: they each capture a part of the truth. 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 Given that the Monday and the Tuesday awakening would be evidentially indistinguishable, 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 2 (Elga 2000). For some other defenses of the 1/3 view, see (Dorr 2002; Monton 2002; Weintraub 2004).