Abstract
In the context of the Sleeping Beauty problem, it has been argued that so-called “halfers” can avoid Dutch book arguments by adopting evidential decision theory. I introduce a Dutch book for a variant of the Sleeping Beauty problem and argue that evidential decision theorists fall prey to it, whether they are halfers or thirders. The argument crucially requires that an action can provide evidence for what the agent would do not only at other decision points where she has exactly the same information, but also at decision points where she has different but “symmetric” information.
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Notes
Titelbaum (2013) gives a useful overview of these connections.
To minimize clutter, I will not specify currency units such as dollars or euros.
Draper and Pust do propose a modified Dutch book that involves telling Beauty that it is Monday and then offering a bet; this Dutch book works against some halfers, whether they are causal or evidential decision theorists, but not against so-called “double-halfers” who hold that the correct credence remains \(1/2\) even after Beauty is told it is Monday.
The idea that thirding goes hand in hand with CDT, and halfing with EDT, also finds support elsewhere, for example in the context of the absentminded driver problem (Schwarz 2014b).
A standard interpretation of how the halfer assigns credences in general [e.g., Halpern (2006), Meacham (2008), Briggs (2010)] would indeed, in the Technicolor Beauty variant, assign credences of \(1/3\), \(1/3\), \(1/3\) in the possible worlds HR, TR, TB upon seeing a red piece of paper (where the first letter indicates the outcome of the original coin toss and the second letter the color seen on Monday), as these are the possible worlds that are not ruled out by the evidence. This, surprisingly, results in a credence of \(1/3\) in Heads for the halfer. However, Pittard (2015) has objected to this conclusion and suggested that another interpretation of halfing should be found that keeps the credence in Heads at \(1/2\) in Technicolor Beauty. One interpretation of halfing that would achieve this is to treat Beauty’s awakening as selected uniformly at random from her awakenings in the experiment in the actual world. Under this interpretation, we would have \(P(\text{ see } \text{ red } | \text{ HR }) = 1\) (because HR has only one awakening) but \(P(\text{ see } \text{ red } | \text{ TR }) = P(\text{ see } \text{ red } | \text{ TB }) = 1/2\) (because in each of TR and TB, only one of two awakenings results in seeing red). Hence, by Bayes’ rule,
$$\begin{aligned} P(\text{ HR } | \text{ see } \text{ red })&= \frac{P(\text{ see } \text{ red } | \text{ HR }) P(\text{ HR })}{ P(\text{ see } \text{ red } | \text{ HR }) P(\text{ HR }) + P(\text{ see } \text{ red } | \text{ TR }) P(\text{ TR }) + P(\text{ see } \text{ red } | \text{ TB }) P(\text{ TB })}\\&= \frac{1 \cdot (1/4)}{1 \cdot (1/4) + (1/2) \cdot (1/4) + (1/2) \cdot (1/4)} = 1/2 \end{aligned}$$as desired by Pittard. In contrast, if we apply this interpretation of halfing to the WBG variant, we still obtain
$$\begin{aligned} P(\text{ WG } | \text{ see } \text{ white })&= \frac{P(\text{ see } \text{ white }| \text{ WG })P(\text{ WG })}{P(\text{ see } \text{ white }| \text{ WG })P(\text{ WG }) + P(\text{ see } \text{ white }| \text{ WO })P(\text{ WO }) + P(\text{ see } \text{ white }| \text{ BO })P(\text{ BO })}\\&= \frac{(1/2) \cdot (1/4)}{(1/2) \cdot (1/4) + (1/2) \cdot (1/4) + (1/2) \cdot (1/4)} = 1/3 \end{aligned}$$as desired for the Dutch book presented in this paper. The key difference from Technicolor Beauty is that \(P(\text{ see } \text{ white }| \text{ WG }) = 1/2\) because there is also an awakening in the grey room. Of course, the standard interpretation of halfing also results in credences of \(1/3\), \(1/3\), \(1/3\) in the WBG variant. The point of discussing this other interpretation of halfing here is not to argue for it, but rather merely to show that while interpretations of halfing may disagree about the correct credences in Technicolor Beauty, it is hard to imagine that they would disagree in the WBG variant. One possible approach to finding an interpretation that disagrees is to take Bostrom’s approach of classifying awakenings into “reference classes” (Bostrom 2002, 2007) and argue that the awakenings in the white and black rooms belong to the same reference class, but not those in the grey room, thereby eliminating the grey room from the picture in the calculation above. However, it seems hard to justify this classification without reference to the particular details of the bets offered, and it seems difficult to swallow that credences should depend on these details.
This assumes that she will be offered the same bet upon each awakening, but this is a reasonable requirement: see Sect. 4.3.
Again, note that she should always be offered the same bet whenever she has information \(I\); otherwise, the bet offered would in fact give her additional information. See Sect. 4.3 for further discussion.
One might also suppose that knowledge of her decision in a white room makes Beauty only (say) 99 % confident in what her decision would be in a black room, where the remaining 1 % is intended to capture a small probability that the room color is somehow relevant to the decision. It is easy to see that the Dutch book still goes through under these conditions.
As already pointed out by Hitchcock, to be precise, what information the bookie has is not exactly what is at issue. If the bookie does not know what day it is, but someone else prevents the bookie from offering the second bet on Tuesday to make the Dutch book work, this is just as problematic. The point is that the process as a whole by which bets are offered to Beauty cannot use information that is unavailable to Beauty.
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I thank the anonymous reviewers for many useful comments that have helped to significantly improve the paper.
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Conitzer, V. A Dutch book against sleeping beauties who are evidential decision theorists. Synthese 192, 2887–2899 (2015). https://doi.org/10.1007/s11229-015-0691-7
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DOI: https://doi.org/10.1007/s11229-015-0691-7