Abstract
Significant controversy remains about what constitute correct self-locating beliefs in scenarios such as the Sleeping Beauty problem, with proponents on both the “halfer” and “thirder” sides. To attempt to settle the issue, one natural approach consists in creating decision variants of the problem, determining what actions the various candidate beliefs prescribe, and assessing whether these actions are reasonable when we step back. Dutch book arguments are a special case of this approach, but other Sleeping Beauty games have also been constructed to make similar points. Building on a recent article (Shaw, Synthese 190(3):491–508, 2013), I show that in general we should be wary of such arguments, because unintuitive actions may result for reasons that are unrelated to the beliefs. On the other hand, I show that, when we restrict our attention to additive games, then a thirder will necessarily maximize her ex ante expected payout, but a halfer in some cases will not (assuming causal decision theory). I conclude that this does not necessarily settle the issue and speculate about what might.
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Notes
Throughout, unless otherwise noted, I will focus on causal decision theory. Therefore, some of the conclusions I reach can be avoided by dismissing causal decision theory. If the reader feels compelled to do so by the examples provided here, then that might be an even more significant impact for them to have—but I myself am not willing to go that far.
In this sense, it is closer to the example of O’Leary awakening twice in his trunk (Stalnaker 1981), except that I need three rather than two awakenings. Nevertheless, I will stick with the Beauty terminology for expository purposes, and will reintroduce coin tosses soon.
Of course, to reason this way, Beauty must be a causal decision theorist; if she were an evidential decision theorist, then she would prefer to press Left and therefore believe that she presses Left in the other rounds as well. The example may thus provide some ammunition for evidential decision theorists, but again, I will attempt to steer clear of that debate here as much as possible.
One might, of course, argue that this is so only because we are using causal decision theory and causal decision theory is flawed. Still, given the prominence of causal decision theory, I believe the example should leave us generally cautious about the strategy of using rational choice to determine what the correct credences are.
We may assume without loss of generality that a single coin toss at the beginning provides all the randomness needed for the duration of the game, since we can keep as much of this randomness hidden from Beauty as we must, for as long as we must. Indeed, it is commonly agreed that moving the coin toss between Sunday night and Monday night in the standard version of the Sleeping Beauty problem makes no difference.
To be specific, we can choose, for every \(v\), a default action \(d_v\). Let \(r(v)\) denote the coin toss realization that leads to \(v\). Then, for any action \(a_v\) that can be taken at \(v\), we let \(\pi _v(a_v) = \pi (r(v), d_{v(r,1)}, \ldots , d_{v(r,i-1)}, a_v, d_{v(r,i+1)}, \ldots , d_{v(r,n_r)}) - \pi (r(v), d_{v(r,1)}, \ldots , d_{v(r,i-1)}, d_{v}, d_{v(r,i+1)}, \ldots , d_{v(r,n_r)})\), where \(v=v(r,i)\). By payoff additivity it then follows that \(\pi (r, a_1, \ldots , a_{n_r}) = \pi _{v(r,1)}(a_1) + \pi (r, d_{v(r,1)}, a_2, \ldots , a_{n_r}) = \pi _{v(r,1)}(a_1) + \pi _{v(r,2)}(a_2) + \pi (r, d_{v(r,1)}, d_{v(r,2)}, a_3, \ldots , a_{n_r}) =\) \( \ldots = (\sum _{i \in \{1,\ldots ,n_r\}} \pi _{v(r,i)}(a_i)) + \pi (r(v), d_{v(r,1)}, \ldots , d_{v(r,n_r)})\), so we can set \(c(r)=\pi (r(v), d_{v(r,1)}, \ldots , d_{v(r,n_r)})\). (It is easy to see that conversely the existence of such \(\pi _v(\cdot )\) implies payoff additivity.)
Note that one awakening event corresponds to many nodes in the standard extensive-form representation of the game—one for each sequence of actions that Beauty has taken so far. However, because of the “actions do not affect future rounds” condition, all these nodes must lie in the same information set.
Note that an agent cannot have different sets of actions available to her in two awakening events that are in the same information set, because then she would be able to rule out some of the awakening events in the information set based on the actions available to her. Some Dutch book arguments are flawed because they violate this criterion.
To see this, note that the first summation sums over all \(v\) by first summing over all \(r\) and then over all \(v\) corresponding to that \(r\). The second summation also sums over all \(v\), but instead by first summing over all information sets and then over all \(v\) in that information set. In both cases, the summand for \(v\) is \(P(r(v))\pi _v(a_{I(v)})\).
At least, it would appear natural to split the credence equally across these \(\nu (I,r(v))\) awakening events—but note that the counterexample does not actually rely on this.
One may wonder whether, along the lines of Briggs (2010), the halfer could correct for this by adopting evidential decision theory instead. The idea would be that her decision provides evidence for what she does in all the \(\nu (I,r(v))\) awakenings, thereby undoing the problematic division by \(\nu (I,r(v))\) above. Unfortunately, if she adopts evidential decision theory, then in general her decision will also provide evidence about what she does in other information sets (especially, very similar ones) and this prevents the proof from going through. To illustrate, consider the following example (an additive game). We toss a three-sided coin (Heads, Tails, and Edge with probability 1/3 each). On Heads, Beauty will be awakened once in information set \(I_1\); on Tails, once in information set \(I_2\); on Edge, once in \(I_1\) and once in \(I_2\). On every awakening, Beauty must choose Left or Right. If the world is Heads or Tails, Left pays out 3 and Right 0; if it’s Edge, Left pays out 0 and Right 2. Note that \(I_1\) and \(I_2\) are completely symmetric. The optimal thing to do from the perspective of ex ante expected payout is to always play Left (and get \((2/3) \cdot 3\) rather than \((1/3) \cdot 2 \cdot 2\) from Right). What will the EDT halfer do? Upon awakening in (say) \(I_1\), she will assign credence \(1/2\) to each of Heads and Edge (and \(0\) to Tails). (In fact, some variants of halfing will result in different credences; to address such a variant, we can modify the example by adding another awakening in both Heads and Tails—but not Edge—worlds, in an information set \(I_3\) where no action is taken. All variants of halfing—and, for that matter, thirding—of which I am aware will result in the desired credences of \(1/2\) Heads, \(1/2\) Edge in this modified example.) Now, the key point is that if she plays Right (Left) now, this is very strong evidence that she would play Right (Left) in \(I_2\) as well—after all the situation is entirely symmetric. Thus, conditional on playing Left, she will expect to get \(3\) in the Heads world and \(0\) in the Edge world; conditional on playing Right, she will expect to get \(0\) in the Heads world and \(2 \cdot 2=4\) in the Edge world. Hence she will choose Right (and by symmetry she will also choose Right in \(I_2\)), which does not maximize ex ante expected payout. Conitzer (2015) provides a more elaborate example along these lines in the form of a Dutch book to which evidential decision theorists fall prey, along with further discussion. (Incidentally, an evidential decision theorist who is a thirder fails to maximize ex ante expected payoff on a much simpler example: in the counterexample at the end of the proof of Proposition 1, just change the payoff for choosing Left on Heads to \(5\). Now Left maximizes ex ante expected payoff, but an evidential decision theorist who is a thirder will calculate \((1/3)\cdot 5 = 5/3 < 8/3 = (2/3)\cdot 2 \cdot 2\) and choose Right. What goes wrong is that \(\nu (I,\text {Tails})=2\) now occurs twice on the right-hand side, once due to thirding (\(2/3\)) and once due to evidential decision theory (the second \(2\); the third \(2\) is the payoff for choosing Right on Tails). I thank an anonymous reviewer for providing this counterexample. It should also be noted that Briggs (2010) already gives a Dutch book for an evidential decision theorist who is a thirder.)
It should be noted that doing so appears nontrivial. For example, suppose we continue to insist that the number of awakenings depends only on the outcome of the coin toss, but we attempt to relax the requirement that actions do not affect the information that Beauty has in future awakenings. Then, an action’s value may come less from the payoff resulting directly from it and more from allowing Beauty to obtain increased payoffs in later rounds by improving her information. It is possible that these latter, indirect effects on payoffs are not additive even when the direct payoffs are additive (so that payoff additivity is technically satisfied), and that this would still allow us to embed problematic examples such as the Three Awakenings game.
Perhaps such examples are more palatable when we consider variants of the Sleeping Beauty problem that involve clones—see, e.g., Elga (2004) and Schwarz (2014). The example where she hopes that today is Tuesday then is analogous to the “After the Train Crash” case in Hare (2007), where a victim of a train crash who has forgotten his name, upon learning that the victim named “A” will have to undergo painful surgery, hopes that he is victim “B”. [See also Hare (2009, 2010).]
Not all of these bits of evidence would concern decision variants, especially as surprising connections from the Sleeping Beauty problem to other problems continue to be drawn. For example, Pittard (2015) makes an interesting connection to epistemic implications of disagreement that provides a challenge to halfers (and argues that this challenge can be met). Of course, there are also many direct probabilistic arguments. Many of these were already made early on in the debate about Sleeping Beauty (Elga 2000; Lewis 2001; Arntzenius 2002; Dorr 2002, etc.), but new ones continue to be made (Titelbaum 2012; Conitzer 2014, e.g.).
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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. Can rational choice guide us to correct de se beliefs?. Synthese 192, 4107–4119 (2015). https://doi.org/10.1007/s11229-015-0737-x
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DOI: https://doi.org/10.1007/s11229-015-0737-x