Can rational choice guide us to correct {\em de se} beliefs?

TL;DR

This paper examines whether rational choice can determine correct self-locating beliefs, analyzing decision-based arguments in the Sleeping Beauty problem and showing limitations of such approaches under certain conditions.

Contribution

It demonstrates that decision-based arguments have limitations in settling self-locating belief correctness, especially outside additive game scenarios, and clarifies when rational choice aligns with correct beliefs.

Findings

Thinners maximize ex ante expected payout in additive games.

Unintuitive actions can result from decision reasons unrelated to beliefs.

The approach does not definitively resolve the Sleeping Beauty controversy.

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 (James R.~Shaw. {\em De se} belief and rational choice. {\em 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 {\em additive} games, then a thirder will necessarily maximize her {\em 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.

Equations

$$(1/5)\cdot \$400+(1/5)\cdot \$100+(3/5)\cdot \$100=\$160$$

$$(1/5)\cdot \$200+(1/5)\cdot \$100+(3/5)\cdot \$200=\$180$$

$$(1/5)\cdot \$400+(1/5)\cdot \$200+(3/5)\cdot \$100=\$180$$

$$(1/5)\cdot \$200+(1/5)\cdot \$200+(3/5)\cdot \$200=\$200$$

$$\pi (r,a_1,\dots ,a_{n_r})-\pi (r,a_1,\dots ,a_{i-1},a_i^{\prime },a_{i+1},\dots a_{n_r})=$$

$$\pi (r,a_1^{\prime },\dots ,a_{i-1}^{\prime },a_i,a_{i+1}^{\prime },\dots a_{n_r}^{\prime })-\pi (r,a_1^{\prime },\dots ,a_{n_r}^{\prime })$$

Full text

Keywords: Sleeping Beauty, Dutch books, decision theory, game theory.

1 Introduction

The Sleeping Beauty problem (Elga, 2000) illustrates some fundamental issues regarding self-locating beliefs. In it, a study participant referred to as “Sleeping Beauty” is put to sleep on Sunday, and awoken either just on Monday, or on both Monday and Tuesday, according to the outcome of a fair coin toss (Heads or Tails, respectively). After an awakening, she is put back to sleep with her memory of the awakening event erased, so that all awakenings are indistinguishable to her. When Beauty is awoken, what should be her credence (subjective probability) that the coin came up Heads? Some (“halfers”) argue that it should be $1/2$. The standard argument for this position is that this should have been her credence in Heads before the experiment, and she has learned nothing new, knowing all along that she would be awoken at least once. Others (“thirders”) argue that it should be $1/3$. The standard argument for this position is that if the experiment is repeated many times, in the long run, only $1/3$ of awakenings correspond to a toss of Heads. (It should be emphasized that “halfers” and “thirders” would compute other fractions on different examples, and “halfing” and “thirding” are supposed to refer to the methods of computing these fractions rather than these specific values.) For a summary of reasons why philosophers are interested in the Sleeping Beauty problem, see Titelbaum (2013).

One approach to settling what Beauty ought to believe is to design scenarios where she must act on her beliefs, and to investigate the consequences of being a thirder or a halfer on these actions. One specific line of attack within this general approach is to design Dutch book arguments. A Dutch book is a set of bets that an agent would all adopt individually in spite of the fact that their combination will lead to a guaranteed loss. If such can be constructed, this is an argument against the rationality of the agent’s beliefs. In the context of the Sleeping Beauty problem, the focus is on diachronic Dutch books, which involve bets at different times. Dutch book arguments for the Sleeping Beauty problem are considered by Hitchcock (2004), Halpern (2006), Draper and Pust (2008), Briggs (2010), and Conitzer (2015). These arguments generally favor thirding, though it is sometimes also argued that a halfer can resist Dutch books, particularly when adopting evidential decision theory. Shaw (2013) more generally pursues the agenda of integrating de se beliefs with rational choice in the context of variants of the Sleeping Beauty problem. He allows Beauty to play more complex games, and designs one where, he argues, the thirder makes the wrong decision and the halfer makes the right decision, regardless of whether they adopt causal or evidential decision theory.

In this article, taking Shaw (2013) as a starting point, I further pursue the agenda of settling the correct answer to the Sleeping Beauty problem by looking at the consequences of halfing and thirding on the outcomes of associated decision problems. I first sound a note of caution by showing that in some cases unintuitive outcomes in these examples result not from incorrect credences, but rather from challenges that a rational actor faces when trying to coordinate with her past and future selves under imperfect recall (at least under causal decision theory). From examples that involve such challenges, we cannot comfortably draw any conclusions about the (in)correctness of a particular approach for computing credences. Subsequently, I show that if we restrict the types of decision problem to additive ones, which include typical Dutch book arguments, these coordination challenges disappear; moreover, under causal decision theory, a thirder will always make decisions that maximize her overall expected payout, but a halfer in some cases does not. I conclude by assessing how much we can learn from these results about correct self-locating beliefs.

2 Review: Shaw’s Waking Game

First, a review of Shaw’s Waking Game is in order. He argues that thirders get the wrong answer in this game while halfers get it right. I focus here on his analysis of a thirder who is a causal decision theorist.111Throughout, 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.

Shaw’s Waking Game. At the beginning of the experiment, Beauty is informed of the rules of the game, which are as follows. A fair coin will be tossed; the outcome of this coin toss will not be revealed to Beauty until the game is over. If it lands Heads, she will be woken up only once, on Monday. If it lands Tails, she will be woken up four times, on Monday, Tuesday, Wednesday, and Thursday. Each day, she will be asked to press either Left of Right. Her memory of the awakening will be erased afterwards, she will not be able to take any notes, and the awakenings will be indistinguishable. She will be compensated as follows.

If Heads came up and she pressed Left, she will receive $400. 2. 2.

If Heads came up and she pressed Right, she will receive $200. 3. 3.

If Tails came up and she pressed Left on each of the four days, she will receive $100. 4. 4.

If Tails came up and she pressed Right on each of the four days, she will receive $200. 5. 5.

If Tails came up and she pressed Left on Monday and Right on at least one other day, she will receive $200. 6. 6.

If Tails came up and she pressed Right on Monday and Left on at least one other day, she will receive $100.

Shaw makes two assumptions that he calls Randomizing Prohibited and Previous Runs. The meaning of the former is clear; the latter refers to the fact that Beauty, having seen many similar experiments performed on others, has become convinced that a subject always makes the same decision on each of her awakenings. These imply the following, which is all that is needed for his analysis of the case of a thirder who is a causal decision theorist.

Definition 1

Beauty is said to accept Consistency in Other Rounds if, upon any given awakening, she does not assign any credence to the following event: she has woken up or will wake up (with the same information) multiple additional times and did not/will not take the same action on each of those other occasions.

Then, Shaw provides the following analysis. If Beauty is a thirder and a causal decision theorist, then upon an awakening, she should assign $1/5$ credence to Heads/Monday, $1/5$ to Tails/Monday, and $3/5$ to Tails/some other day. If she accepts Consistency in Other Rounds, then moreover she believes that either (a) on all other awakenings (if any) she chooses Left, or that (b) on all other awakenings she chooses Right. Under (a), if she chooses to now press Left, her expected payout will be

[TABLE]

On the other hand, if she chooses to now press Right, her expected payout will be

[TABLE]

Hence, under (a), she is better off pressing Right.

Under (b), if she chooses to now press Left, her expected payout will be

[TABLE]

On the other hand, if she chooses to now press Right, her expected payout will be

[TABLE]

Hence, under (b), she is also better off pressing Right! It follows that Beauty, if she is a thirder and a causal decision theorist, will press Right.

Now, because all awakenings are indistinguishable, she should always press Right, resulting in a payout of $200. But always pressing Left would have resulted in an expected value of$250, which is better (assuming Beauty is risk-neutral), and is hence the correct course of action according to Shaw. (He shows that a thirder who is an evidential decision theorist also should choose Right in this example, but I will not review this analysis here.)

3 Three Awakenings

Shaw’s Waking Game is illuminating, but I believe little can be concluded from it about whether thirding or halfing is correct. To show why, let us consider another example that shares key features of the reasoning above, but without any coin tosses whatsoever.

Three awakenings. At the beginning of the experiment, Beauty is informed of the rules of the game, which are as follows. She will be woken up exactly three times (Monday, Tuesday, and Wednesday). Each day, she will be asked to press either Left of Right. Her memory of the awakening will be erased afterwards, she will not be able to take any notes, and the awakenings will be indistinguishable. She will be compensated as follows.

If she never pressed Right, she will receive $200. 2. 2.

If she pressed Right once, she will receive $300. 3. 3.

If she pressed Right twice, she will receive $0. 4. 4.

If she pressed Right three times, she will receive $100.

Again, note that no coins are tossed at all in Three Awakenings.222In 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. The only uncertainties that Beauty faces are (1) which day it is and (2) what she herself has done and will do on the other days. In fact, arguably, (1) does not even matter because in this game, all awakenings are treated symmetrically. The key uncertainty is (2).

How should Beauty act in this game? If she always presses Left, she will obtain $200; if she always presses Right, she will obtain only$100. So something is to be said for pressing Left. However, upon any given awakening, Beauty can reason as follows. There are two other rounds in which she has pressed or will press a button. If she accepts Consistency in Other Rounds, then she believes that either (a) she has pressed or will press Left both other times or (b) she has pressed or will press Right both other times. In case (a), she will be better off pressing Right this round, because pressing Right in only one round pays out $300, whereas never pressing Right pays out$200. In case (b), she will also be better off pressing Right this round, because pressing Right in all three rounds pays out $100, whereas pressing Right in only two pays out nothing. So *in either case* Beauty is better off pressing Right, gaining$100 from doing so!333Of 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. Then, because all awakenings are indistinguishable, it seems we should expect Beauty to press Right all the time—even though pressing Left all the time results in a higher payout.

From Three Awakenings, it becomes clear that, under causal decision theory, actions that are locally optimal—at least when assuming Consistency in Other Rounds—can result in globally suboptimal outcomes, even in cases where there is no ambiguity about what the correct credences are. (I take it to be uncontroversial that Beauty’s credence upon awakening should be distributed uniformly ($1/3$, $1/3$, $1/3$) across Monday, Tuesday, and Wednesday.) I believe the example also makes it clear that the total payout earned by a subject is a very unreliable indicator of the correctness of her credences.444One 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. To drive home the point, consider the following modification of Three Awakenings.

Three Awakenings with a Coin Toss. The experiment now begins with a biased coin toss. If it lands Heads (which happens 99% of the time), we proceed with the original Three Awakenings game. If it lands Tails (1%), Beauty will similarly be woken up on Monday, Tuesday, and Wednesday, and asked to press Left or Right, but the payoffs will be different. In fact, they will be much simpler: she will receive $100 for each time she presses Left (and nothing for pressing Right). As always, Beauty knows the setup of this modified game, but will not receive any evidence of how the coin landed until the game has ended.

I take it to be uncontroversial that upon any awakening, Beauty should place a credence of 99% on the event that the coin landed Heads, because whether the coin landed Heads or not, she will be awoken three times. Moreover, in all six possible awakening events, she will have the exact same information. Given this, the modification is too slight to have an impact on her decision: for any given awakening, there is a 99% chance that she will gain $100 from pressing Right (assuming *Consistency in Other Rounds*) and a 1$100 from doing so—so she should still press Right. But now, suppose that Beauty’s credence is inexplicably inverted, so that she believes that there is a 99% chance that the coin came up Tails. If so, then from her perspective, now the simpler payoff function dominates and clearly she should press Left. As a result, she will actually obtain a larger expected payout from the actual game, because always pressing Left results in a higher payout in Three Awakenings than always pressing Right. However, it seems clear that this should not lead us to believe that Beauty’s inverted credence is in any sense correct; rather, she was just lucky that she accidentally inverted the credences, thereby escaping the detrimental reasoning to which understanding the game correctly would have led her.

Of course, we do not need to go to such lengths to find examples where incorrect credences lead to a better result. Someone who for some reason believes that in roulette Red comes up $2/3$ of the time, and bets on Red once for this reason only (as opposed to not betting at all), may well get lucky on that one spin of the wheel. If so, nobody will argue that this ex post outcome implies that the credence of $2/3$ was correct. What is interesting about Three Awakenings with a Coin Toss is that any credences that maximize ex ante expected payoff are clearly incorrect. It would seem that it is a very reasonable criterion for evaluating the correctness of credences to see whether they lead to the maximum ex ante expected payoff—but the example shows that this approach is, in general, problematic (at least if we are not willing to dismiss causal decision theory).

4 Additive Games

The examples in Section 3 suggest that in sufficiently rich decision variants of the Sleeping Beauty problem, under causal decision theory, the payouts that Beauty obtains do not provide useful guidance for what her correct credences should be. This is so because in such scenarios, actions that are locally apparently rational may lead to suboptimal payouts even when there can be no serious dispute about what the correct credences should be. But perhaps, if we restrict the space of scenarios, we can avoid such issues.

The problematic aspect of the Three Awakenings game is that Beauty’s “three selves” need to coordinate their actions to maximize payout—the effect of one action on overall payout depends on the other actions—and they fail to do so due to the lack of memory. What happens if we assume away this interdependence? In what follows, I show that in the resulting restricted class of games—additive games—Beauty does in fact maximize her expected total payout by being a thirder (and a causal decision theorist). Of course, merely showing an example additive game where being a thirder maximizes Beauty’s expected total payout will do little to prove the point, because for all we know there is another example where being a thirder results in suboptimal payout. I have to prove the result at some level of generality for it to be more than merely suggestive. In particular, for the sake of generality, I wish to allow that Beauty does not necessarily have the same experience in each awakening (thereby allowing us to also address examples such as “Technicolor Beauty” (Titelbaum, 2008)). To do so, I will have to be a bit more formal.

Definition 2

A Sleeping Beauty decision variant with payoff function $\pi$ is additive if for every realization $r$ of the initial coin toss,555We 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.

• •

(actions do not affect future rounds)* $r$ always leads to the same number $n_{r}$ of awakenings by Beauty regardless of Beauty’s actions, and for every $i$ with $1\leq i\leq n_{r}$, the information that Beauty possesses in the *$i$th awakening depends only on $r$ and $i$, and not on Beauty’s earlier actions; and

• •

(payoff additivity)* for every $i$ with $1\leq i\leq n_{r}$, and every two corresponding sequences of actions $a_{1},\ldots,a_{n_{r}}$ and $a^{\prime}_{1},\ldots,a^{\prime}_{n_{r}}$ that Beauty may take upon her $n_{r}$ awakenings, we have that*

[TABLE]

[TABLE]

Intuitively, in additive games, Beauty does not need to worry about coordinating her actions with her selves from other awakenings. This is because by the first condition, the only effect of actions is directly on the final payout (as opposed to them affecting the number of awakenings or the information that she has in future rounds), and by the second condition these effects on payout are independent across actions. This intuition leads to the following proposition.

Proposition 1

If Beauty is a thirder and a causal decision theorist, and acts accordingly upon each individual awakening, then she will maximize her ex ante expected payout in additive games. If she is a halfer and a causal decision theorist, and acts accordingly on each individual awakening, there are additive games in which she does not maximize her ex ante expected payout.

Proof.  For each $r$ and $i$ with $1\leq i\leq n_{r}$, let $v(r,i)$ correspond to the awakening event on the $i$th day after a coin toss realization of $r$. Let $V=\bigcup_{(r,i):1\leq i\leq n_{r}}\{v(r,i)\}$ be the set of all awakening events. By payoff additivity, we can construct, for every $v\in V$, a function $\pi_{v}$ such that Beauty’s total payout upon coin toss realization $r$ and actions $a_{1},\ldots,a_{n_{r}}$ is $c(r)+\sum_{i\in\{1,\ldots,n_{r}\}}\pi_{v(r,i)}(a_{i})$, where $c(r)$ is a constant that we may ignore for the purpose of acting optimally.666To 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.) We will use $I\subseteq V$ to denote an information set, i.e., a set of awakening events that Beauty cannot distinguish.777Note 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 two awakening events in the same information set may correspond either to the same coin toss realization $r$—e.g., subsequent Monday and Tuesday awakenings in the standard version of Sleeping Beauty—or to different coin toss realizations—e.g., the two Monday awakening events corresponding to Heads and Tails in the standard version. When Beauty awakens in information set $I$, if she is a thirder, then her credence that the realization of the coin toss is $r$ is given by $P(r|I)=\frac{P(r)\cdot\nu(I,r)}{\sum_{r^{\prime}}P(r^{\prime})\cdot\nu(I,r^{\prime})}$, where $\nu(I,r)=|\{v\in I:r(v)=r\}|$ is the number of times that Beauty will awaken with information $I$ after coin toss realization $r$ and $r(v)$ is the realization that leads to $v$. (This is the essence of being a thirder: given particular information upon awakening, credence in a particular realization is proportional to the number of times one will awaken with this information under this realization. Indeed, if the experiment is repeated many times, then $P(r|I)$ gives the long-run fraction of the awakenings in information set $I$ that corresponded to a coin toss realization of $r$.) Moreover, the credence that she assigns to a specific $v\in I$ with $r(v)=r$ is $P(v|I)=\frac{P(r|I)}{\nu(I,r)}=\frac{P(r)}{\sum_{r^{\prime}}P(r^{\prime})\cdot\nu(I,r^{\prime})}$. Hence, if $A_{I}$ is the set of actions available to her in information set $I$,888Note 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. she will choose some $a_{I}\in A_{I}$ that maximizes $\sum_{v\in I}P(v|I)\pi_{v}(a_{I})$.

If Beauty takes action $a_{I}\in A_{I}$ whenever she is in information set $I$, then her ex ante expected payout overall is $\sum_{r}P(r)\sum_{i\in\{1,\ldots,n_{r}\}}\pi_{v(r,i)}(a_{I(v(r,i))})$ (where $I(v)$ is the information set in which $v$ lies). Rearranging, this is equal to $\sum_{I}\sum_{v\in I}P(r(v))\pi_{v}(a_{I})$.999To 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 $P(r(v))\pi_{v}(a_{I(v)})$. We will show that if Beauty is a thirder and a causal decision theorist, then in fact for every $I$ she maximizes $\sum_{v\in I}P(r(v))\pi_{v}(a_{I})$, thereby establishing that she maximizes her ex ante expected payout overall. Indeed, we have already established that for each $I$, Beauty maximizes $\sum_{v\in I}P(v|I)\pi_{v}(a_{I})$. Using that (Beauty being a thirder) $P(v|I)=\frac{P(r(v))}{\sum_{r^{\prime}}P(r^{\prime})\cdot\nu(I,r^{\prime})}$, we obtain that Beauty maximizes $\frac{\sum_{v\in I}P(r(v))\pi_{v}(a_{I})}{\sum_{r^{\prime}}P(r^{\prime})\cdot\nu(I,r^{\prime})}$. Because Beauty cannot affect the denominator of this expression, this is equivalent to maximizing $\sum_{v\in I}P(r(v))\pi_{v}(a_{I})$, as was to be shown.

On the other hand, if Beauty is a halfer (and a causal decision theorist), then consider the standard Sleeping Beauty game, where a coin is tossed to determine whether she awakens once (upon Heads) or twice, with all her three possible awakenings in the same information set. Let her choose between Left and Right upon each awakening. If the awakening is one corresponding to Heads, she will receive $3$ for choosing Left (and [math] for Right); if it is one corresponding to Tails, she will receive $2$ for choosing Right (and [math] for Left). If Beauty is a halfer (and a causal decision theorist), upon awakening she will think it equally likely that she is in a Heads awakening and that she is in a Tails awakening, and therefore will choose Left for an expected payoff of $3/2$ in this round (rather than Right for $1$). However, overall, choosing Right every time gives an ex ante expected total payout of $(1/2)\cdot 2\cdot 2=2$, whereas choosing Left every time gives an ex ante expected total payout of $(1/2)\cdot 1\cdot 3=3/2$, so Beauty fails to maximize her expected payout. $\Box$

Intuitively, the way the proof works is as follows. Because the game is additive, we can separate Beauty’s total ex ante expected payoff into the contributions made to it by individual information sets $I$. It then remains to show that Beauty maximizes her expected payoff for each information set $I$ if she is a thirder and a causal decision theorist. Now, the contribution of each individual awakening event $v$ within the information set $I$ to the expected payoff is proportional to the probability $P(r(v))$ of the coin toss realization $r(v)$ that gives rise to $v$. But, when she is in $I$, Beauty’s credence $P(v|I)$ in $v$ is also proportional to $P(r(v))$. This is so because (being a thirder) her credence $P(r(v)|I)$ in $r(v)$ is proportional to $P(r(v))\nu(I,r(v))$, where $\nu(I,r(v))$ is the number of awakening events in $I$, across which this credence is equally divided. Because of this, Beauty weighs the awakening events in an information set exactly so as to maximize ex ante expected payoff. In contrast, if she is a halfer and a causal decision theorist, then her credence in $r(v)$ is not proportional to $P(r(v))\nu(I,r(v))$ but rather just to $P(r(v))$, so that her credence in $v$ itself is proportional to $P(r(v))/\nu(I,r(v))$.101010At 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. As a result, she places too little weight on awakening events $v$ in $I$ that correspond to coin toss outcomes $r$ that lead to many other awakening events in $I$. This is what leads her to decide suboptimally in the counterexample at the end of the proof: she insufficiently weighs the Tails awakenings in making her decision.111111One 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 [math] 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 [math] in the Edge world; conditional on playing Right, she will expect to get [math] 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.)

Proposition 1 also implies that Beauty, if she is a thirder and a causal decision theorist, is invulnerable to certain types of Dutch books. (This is already discussed in prior work (Hitchcock, 2004; Draper and Pust, 2008; Briggs, 2010).) Specifically, she will not fall for a Dutch book as long as: (a) Beauty, at the beginning of the experiment, is made aware of the bets she will be offered in different awakening states and will not forget this; (b) Beauty’s betting actions affect neither her future awakening states nor the outcomes of past or future bets; (c) for every two states in the same information set, the bet posed to Beauty is the same. Here, (c) seems natural, because if two states in the same information set were to have different bets associated with them, then in fact, by (a), they would allow Beauty to distinguish between them before she takes her action, contradicting that they are in the same information set. (a) and (c) together ensure that we can interpret the bets as Beauty playing a game (whose rules she knows), and adding (b) ensures that this game is additive. (Note that we may have to add an initial round to correspond to a bet at the beginning of the experiment.) By the first part of Proposition 1, Beauty will act in a way that maximizes her expected payout. This means she cannot be vulnerable to accepting a set of bets that results in a sure loss, because if she did so she would not be maximizing her expected payout (since, after all, she can also accept none of the bets at all and thereby avoid a loss). Given all this, the second part of Proposition 1 is unsurprising in light of the Dutch book given by Hitchcock (2004) for halfers that use causal decision theory.

One may wonder whether additive games are really the “right” class of games to which to restrict our attention. Perhaps the result can be generalized to a somewhat broader class of games, for example by slightly relaxing the first condition in Definition 2.121212It 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. Such a generalization, of course, would only strengthen the point. More problematically, perhaps a different natural class of games would actually favor halfing. I cannot rule out this possibility, but it seems unlikely to me that such a class would be more compelling than that of additive games. I believe that additive games are well motivated by the discussion given at the beginning of this section about removing the coordination problem between Beauty’s multiple selves, and the fact that the result provides a corollary about Dutch books is also encouraging.

5 Conclusion

What can we conclude from the foregoing? First and foremost, the Three Awakenings game shows that we should be very cautious when drawing conclusions about halfing vs. thirding from the outcomes of decision-theoretic variants of the Sleeping Beauty problem. I do believe that Proposition 1 shows some merit to being a thirder rather than a halfer, but surely it does not settle the matter once and for all. One might well argue, for example, that, once she has awakened under particular circumstances, Beauty should no longer care whether she maximizes her ex ante expected payout; instead, she should maximize her expected payout with respect to her beliefs at hand. These two objectives turn out to be aligned in the case of a (causal decision theorist) thirder in additive games, and this may be a nice property. But the battle-hardened halfer is likely more comfortable biting the bullet and accepting nonalignment in these two objectives than giving up on other cherished philosophical commitments. Another possibility for the halfer may be to embrace a version of evidential decision theory instead. More discussion of how halfers may or may not benefit from adopting evidential decision theory, particularly in the context of Dutch book arguments, is given by Arntzenius (2002), Draper and Pust (2008), Briggs (2010), and Conitzer (2015) (see also the discussion in Footnote 11).

How could we create decision variants of the Sleeping Beauty problem that leave no ambiguity about whether rational decisions truly correspond to rational beliefs? One way to do so would be to consider a myopic Beauty. Such a Beauty would be rewarded immediately after taking an action in the game, rather than at the end. We may suppose that she is rewarded in something giving immediate satisfaction—say, chocolate—rather than money. Moreover, she is assumed to care only about the very near future; tomorrow is too far in the future to affect her decisions. Her being myopic is not to be understood as her being irrational. We still assume her to be entirely rational, but she just discounts the future exceptionally heavily (and, to the extent it matters, the past as well). Such a Beauty, in a simple variant (without decisions) where she is certainly woken up on both Monday and Tuesday but given chocolate only on Tuesday, will hope that today is Tuesday when she is awoken.131313Perhaps 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).) So a myopic Beauty’s preferences are entirely de se and de nunc. If we additionally suppose that the game is additive as described above, then she need not worry at all about what she will do or has done in other rounds (including about what her current actions say about what she will do or has done in other rounds), because none of those affect her current circumstances and rewards. Hence, it seems that here beliefs and actions should unambiguously line up. Unfortunately, such extreme assumptions also make it difficult, and perhaps impossible, to set up an example that provides much insight beyond non-decision-theoretic variants of the Sleeping Beauty problem. There is a tightrope to be walked here. Too permissive a setup will allow us to reach conclusions that are unwarranted; too restricted a setup will not allow us to reach any conclusions at all. Perhaps the best we can hope for is to identify the happy medium and gradually accumulate bits of evidence that, while each not entirely convincing on its own, gradually tilt the balance in favor of one or the other position.141414Not 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.).

Acknowledgments

I thank the anonymous reviewers for many useful comments that have helped to significantly improve the paper.