“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 the constraints from these thought experiments involves violating Bayesian conditionalization.15 I 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 be represented as follows: w1: h1 h1m [The “heads” world] w2: t1 t1m t2 [The “tails” world] The situation we are confronting involves five possible agent-parts, not three. The added terms, “h1m” and “t1m”, denote the agent-parts of Beauty that know that it 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 unaware that it is Monday). Now consider more carefully Constraint P+, which 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 not describe 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 current agent-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 P and allows us to avoid violating 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 agentpart that knows that it is Monday: P(HEADS | H1M ∨ T1M) = ? However, there is no need to assign a value to this expression. Note that 15 It has been argued that we should indeed violate conditionalization in the Sleeping Beauty problem (Kierland and Monton). Kierland and Monton argue for the 1/3 answer on grounds which they claim do not lead to the counterintuitive result in the Beauty and Doppelganger. Their position thus diverges significantly from Lewis’s 1/2 view. 11 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.16 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 the reasoning just described, we cannot automatically conclude that the conditional probability P(X | E & “I am currently α– “), conditionalized on E, yields the correct posterior credence that α+ should assign to X. Only if 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 standard 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 elliptic representation of Bayesian conditionalization can then be used without danger. In certain special cases, however, such delicate changes in indexical information can be relevant, and it is then crucial to recognize and make explicit the hidden intermediary step. Sleeping Beauty, on the model proposed here, turns out to be just such a special case. To recapitulate, I have argued that a Bayesian can coherently accept both Constraint P and the corrected Constraint P+, even though superficially this seems to violate Bayesian conditionalization. The reason why we not only can but should accept both these constraints was given earlier: to avoid the counterintuitive consequences that follow if either of these constraints is violated, as shown by the “Presumptuous Philosopher” 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 will also 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 2/3 would be tails-awakenings). In the infinite limit, this ratio would, with probability 1, be approached arbitrarily closely. This argument could be buttressed by introducing betting 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 her mouth 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 in each run of the experiment, and if both the bookie and Beauty knew this, then Beauty could 16 The 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 should be modeled. 12 infer from the fact that she was offered a bet that it was Monday. According to the hybrid model, she should then assign credence 1/2 to the proposition that the coin will fall heads in that trial. This will match the long-run frequency of bets that she will win, 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 being offered a bet. In this case, in the long run, Beauty would be expected to lose 2/3 of her bets if she consistently bet on heads. How does this square with the prescription of the hybrid model that Beauty, upon awakening (but before learning which day it is) assigns credence 1/2 to heads? 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 would be a different number of bets placed depending on how the coin fell, one might regard the test as unfair.17 In 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 losses of the collective of all his duplicate agent-parts would be zero. Since there would be a trillion times more duplicates of the agent-part if theory T2 is true then if T1 is true, each agentpart would have to assign a trillion times greater odds to T2 than to T1 in order for the expected value of all the bets made by the collective of agent-parts to be zero. And yet we argued that it seems wrong for an agent-part to assign a trillion times greater credence to T2 than to T1. 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 total proportion of awakenings in tails- and heads-runs of the experiment. Beauty knows that (with probability one) there will actually be two times as many awakenings in tails-trials as in heads-trials. In this case, therefore, betting considerations unambiguously 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 by saying that there would be a different number of bets placed depending on which of the hypotheses under consideration is true, because the total number of bets placed is not (significantly) variable when Beauty is put through a large number of repetitions of the experiment. There is, consequently, strong reason for recommending that Beauty assign credence 1/3 to heads when she knows that the experiment will be repeated very many times. This, however, is not an objection to the hybrid model proposed above. The hybrid model, as we shall now see, implies the very same credence assignment as the betting considerations suggest. Betting considerations, far from being an embarrassment 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 is the simplest case: the world contains no other relevant agent-parts than those existing within a single implementation of the Sleeping Beauty experiment. Let us now apply the hybrid model to the situation that arises if the experiment is repeated many times. But first, as an intermediary step, consider the following case. 17 See also (Arntzenius 2002). 13 Three Thousand Weeks (non-random) Beauty lives for three thousand weeks. On odd-numbered weeks she is awakened once, on Mondays. On even-numbered weeks she is awakened twice, on Mondays and Tuesdays. After each awakening she is given an amnesia drug that causes 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 in an odd-numbered week” (or “ODD” for short). This is because Beauty, when she wakes up, knows that ODD is true for one third of all the agent-parts that are in the same subjective evidential state as her current agent-part. (The credence assignment follows from the very weak indifference principle which Lewis, Elga, and I all accept.) Crucially, these agentparts 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 | MONDAY) = 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 | MONDAY) = 1/2 Note that, for the reasons explained earlier, “MONDAY” denotes a different centered proposition in each of these two conditional credence expressions. (In the first expression, “MONDAY” refers to a proposition that is centered on an agent-part that does not know that it is Monday; in the second expression, specifying Constraint P+, “MONDAY” 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 the same. The hybrid view therefore coincides with the 1/3 view in this example. The key difference between the original Sleeping Beauty problem and ThreeThousand 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 Three Thousand Weeks, the prior credence in ODD, before Beauty learns that it is Monday, is unaffected by the correction we made to eliminate the bias in favor of hypotheses entailing the existence of more duplicates. (Such a bias would result from applying the indifference principle to a class of agent-parts that included merely possible as well as actual agentparts.) In Three-Thousand Weeks, ODD is true for one-third of the agent-parts that are ignorant about whether 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 1/2 for the second type.18 18 We 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 be careful about whether we take this to refer to a specific centered proposition or to a function that yields a centered