Simple Dutch books versus Sleeping Beauty halfers
The Sleeping Beauty problem is a famous philosophical puzzle. Described by Elga (2000):
Some researchers are going to put you to sleep. During the two days that your sleep will last, they will briefly wake you up either once or twice, depending on the toss of a fair coin (Heads: once; Tails: twice). After each waking, they will put you back to sleep with a drug that makes you forget that waking. When you are first awakened, to what degree ought you believe that the outcome of the coin toss is Heads?
Stipulate that Beauty (“you”) wakes only on Monday if Heads, and on Monday and Tuesday if Tails. The standard answers are 1⁄3 and 1⁄2 for the probability of Heads. Among those who answer 1⁄2, we ask a follow-up question: “When you are awakened and then learn it is Monday, to what degree ought you believe that the outcome of the coin is Heads?”. The standard answers are 2⁄3 and 1⁄2. Of those who endorsed 1⁄2 for the original question, those who answer 2⁄3 to this follow-up question are “single halfers”, and those who answer 1⁄2 to this follow-up question are “double halfers”. Generally, people who answer 1⁄3 to the original question answer 1⁄2 to the follow-up question, and are called “thirders”.
In anthropic theory, SSA (self-sampling assumption) generalizes single-halfing, SIA (self-indication assumption) generalizes thirding, and FNC (fully non-indexical conditioning) generalizes double-halfing. (Where relevant, I assume self-sampling over observer moments, in line with Bostrom’s “SSSA”, to handle cases of amnesia.) To summarize these theories:
A SSA agent assigs probabilities as if it is sampled randomly from all observers in the universe who are in its reference class. (Reference class is a parameter SSA needs, but SIA and FNC don’t.)
A SIA agent first multiplies its prior probabilities over universes by the universe’s number of observers, then (as in SSA) conditions on itself being a random sample from observers in a universe sampled according to this modified prior, and that observer having its observation.
A FNC agent conditions its prior over universes on “some observer with my observation exists”, and otherwise assumes it is randomly sampled from observers in the universe with that observation.
One complication: In practice, FNC will agree with thirding in the Sleeping Beauty problem, because Beauty receives random data upon waking up, and the specific random observation sequence is more likely to exist conditional on Tails, because there are more chances for the sequence to happen. FNC endorses double-halfing in a controlled scenario where Beauty is a brain upload and has the exact same information state upon waking up regardless of Heads+Monday, Tails+Monday, or Tails+Tuesday, and regardless of random data that would vary between runs of the scenario. Charitably to double-halfing, I assume this controlled Sleeping Beauty scenario for the purposes of this post.
I explore Dutch book arguments against anthropic theories in the context of Sleeping Beauty. A Dutch book against an agent is a sequence of bets they are offered which leads to a sure loss for them, if they bet according to their probabilities, and linear utility in how much money they end up with. My Dutch books satisfy two strict criteria: (a) the bookie must have no information not had by the agent at the time the bet takes place; (b) the agent’s probabilities must strictly favor the sure-loss sequence of bets if they assign probabilities according to the theory; theories with ambiguity about probabilities can resist Dutch books.
I will focus mainly on CDT, since CDT agents accept or reject bets straightforwardly according to their betting odds. First, I present simple Dutch book arguments against CDT single-halfing and CDT double-halfing. Second, I argue that CDT thirding is resistant to Dutch books. Third, I discuss EDT.
Dutch booking CDT single-halfers
This Dutch book offers one bet on Sunday, and one on Monday. It doesn’t matter if the coin has already been flipped by the time of the bet on Sunday, provided the bookie does not know the result. Beauty only considers the Monday bet after knowing it is Monday.
The Sunday bet is as follows: “-$14 if Heads, +$16 if Tails”. A CDT single-halfer accepts, because Heads and Tails are equally likely; the expected value is 1/2(-$14 + $16) = $1.
The Monday bet is as follows: “+$11 if Heads, -$19 if Tails”. A CDT single-halfer accepts, because they assign 2⁄3 to Heads once they see it is Monday; the expected value is 2⁄3 * $11 + 1⁄3 * -$19 = $1.
If the coin is Heads, then the overall payoff is -$14 + $11 = -$3. If the coin is Tails, then the overall payoff is $16 - $19 = -$3. This is a sure loss.
(For prior work on Dutch books for CDT single-halfers, see Hitchcock 2004 and Draper and Pust 2008; see Briggs 2010 on CDT/EDT divergence.)
Dutch booking CDT double-halfers
This Dutch book offers two bets. Bet A is offered when Beauty has woken up, and otherwise has no posterior information; we imagine the bookie is woken up and administered amnesia drugs just like Beauty. Upon Tails, bet A is offered twice; we label the Monday A-bet as AM, and the Tuesday A-bet as AT. Bet B is offered on Monday after Beauty has accepted or rejected bet AM, and has been told the day (we imagine the bookie learns the day at the same time as Beauty, and only offers bet B if it is Monday). Importantly, no bets are offered prior to Monday.
Bet A is as follows: “+$10 if Heads; -$8 if Tails”. The CDT double-halfer accepts, because they assign 50% Heads; the expected value is 1⁄2 * ($10 - $8) = $1.
Bet B is as follows: “-$11 if Heads; +$15 if Tails”. The CDT double-halfer accepts, because they assign 1⁄2 to Heads and 1⁄2 to Tails upon learning it is Monday; the expected value is 1⁄2 * (-$11 + $15) = $2.
If the coin is Heads, the CDT double-halfer accepts bets AM and B; the overall payoff is $10 - $11 = -$1. If the coin is Tails, then CDT double-halfer accepts bets AM, B, and AT; the overall payoff is -$8 + $15 - $8 = -$1. This is a sure loss.
(Hitchcock’s 2004 Dutch book also applies to double-halfers, but requires a bet to happen before any awakening, unlike this section’s Dutch book. “Anthropics: Full Non-indexical Conditioning (FNC) is inconsistent” (Armstrong 2019) notes diachronic inconsistency of FNC, which this Dutch book exploits.)
CDT thirders resist Dutch books
To make a positive case for thirding, we examine ex ante optimality arguments for CDT+SIA (which generalizes betting according to thirding odds). A policy is ex ante optimal when it is expected to attain maximal expected utility if adopted ahead of time. “Can rational choice guide us to correct de se beliefs?” (Conitzer 2015) supports CDT+SIA ex ante optimality in additive games. Additive games do not require Beauty to “worry about coordinating her actions with her selves from other awakenings”, and include ordinary imperfect-recall betting scenarios. CDT+SIA is ex ante optimal for additive games: “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)”.
In the other direction, all ex ante optimal policies are compatible with CDT+SIA in a wide class of imperfect-recall scenarios, a superset of additive games. See my “In memoryless Cartesian environments, every UDT policy is a CDT+SIA policy” (2016) and prior work (“On the Interpretation of Decision Problems with Imperfect Recall”, Piccione and Rubinstein, 1997, Proposition 3: “If a behavioral strategy is optimal then it is modified multiself consistent”).
Together, these considerations show that CDT+SIA policies are exactly ex ante optimal policies in additive games. Therefore, there are no Dutch books against CDT+SIA in additive games, and even in non-additive games (of the class studied in Piccione 1997), there are ex ante optimal (therefore non-Dutch-bookable) policies compatible with CDT+SIA.
“A Dutch Book for CDT thirders” (Korzukhin 2020) presents a non-additive game where CDT+SIA can lose; the scenario does not meet my strict condition (b). Quoting the paper: “if we allow Sleeping Beauty to deliberate dynamically, she will end up in one of two possible stable states: she may converge on rejecting, or she may converge on accepting.” Indeed, not every CDT+SIA policy is ex ante optimal in non-additive games. Still, this failure mode pales in comparison to those of CDT+SSA and CDT+FNC, which compel ex ante suboptimal policies even in additive games (such as the situations of the above Dutch books).
EDT and Dutch books
The Dutch book against CDT single-halfers also works against EDT single-halfers, because the Monday bet is only considered once Beauty is certain that it is Monday. Similar considerations have been noted in the literature; “the reasoning utilized by the evidential decision theorist to evade Hitchcock’s [diachronic Dutch book] cannot be employed here” (Draper and Pust 2008).
A draft paper (“Can de se choice be ex ante reasonable in games of imperfect recall? A complete analysis”, Oesterheld and Conitzer, 2024) claims CDT+SSA, EDT+SSA, EDT+SIA, CDT+FNC are all Dutch bookable, but that EDT+FNC is ex ante optimal (and accordingly resistant to Dutch books). This post supports Dutch books for CDT+SSA, CDT+FNC, and EDT+SSA, though I have not checked all claims in the draft paper. I will briefly examine EDT+FNC on this post’s Dutch book for double-halfers.
An EDT+FNC agent will, when offered bet A, use FNC to assign probability 50% to Heads, and 50% to Tails, with “Tails and Monday” equally probable with “Tails and Tuesday” (both 25%). To simplify, we first suppose bet B is not offered. The agent believes its general policy of whether to accept bet A is uncorrelated with Heads/Tails. Using EDT, the agent estimates that, conditional on accepting bet A as a general policy, the expected payoff is 1⁄2 * $10 + 1⁄2 * (-$8 + -$8) = -$3, whereas the expected payoff is $0 conditional on rejecting bet A as a general policy. And the agent’s action is good evidence about its general policy for what it does in this very information state. So the agent rejects bet A, and adding bet B into the mix does not change this.
The main upshot is that EDT’s betting behavior differs from naively betting according to posterior anthropic probabilities, unlike CDT, which matches naive betting behavior. As such, traditional Dutch book analysis must be modified when considering EDT agents. EDT+FNC avoids this post’s Dutch book against double-halfers, but only by agreeing with CDT+SIA about which bets to accept.
Intuitively, an EDT+FNC agent acts as if it considers the consequences of its information state’s action, rather than its body’s action. Upon being offered bet A, an EDT+FNC agent can reason that its information state would exist either way the coin went (so its existence tells it nothing about the coin), but that its information state has one embodied occurrence if Heads and two embodied occurrences if Tails; the reason to reject bet A is that the bet’s consequences are doubled in the Tails case due to two embodied occurrences each accepting an episode of bet A.
“A Dutch book against sleeping beauties who are evidential decision theorists” (Conitzer 2015) argues that EDT is subject to Dutch books in Sleeping Beauty variants due to correlations between actions taken in different information states. Oesterheld and Conitzer (2024) address this by specifying that in their version of EDT, “For all other
Conclusion
Decision-theoretic considerations support betting according to CDT+SIA, and against betting according to CDT+SSA and CDT+FNC. How much does this support thirding in Sleeping Beauty, at a philosophical level?
First, philosophical problems with SSA (and therefore single-halfing) are well-known; see Bostrom’s Paradoxes of the Self-Sampling Assumption. CDT+SSA and EDT+SSA are easily Dutch booked. An important general problem for combining SSA with decision theory is that adding “dummies” (copies of the agent, who receive an observation informing them they are a dummy, and whose actions don’t matter) to a situation changes SSA’s posterior probabilities (given non-dummy observations), and by default this change causes decision-theoretic problems, since dummy counts do not affect ex ante strategy optimality.
Second, while CDT+FNC is easily Dutch booked, EDT+FNC is a serious contender, by the analysis of Oesterheld and Conitzer (2024), though this analysis relies on independence assumptions. Of course, FNC ‘thirds’ in realistic Sleeping Beauty scenarios (due to random observations), but double-halves in controlled brain upload versions. FNC is non-Bayesian, which is why this post’s Dutch book against CDT double-halfers succeeds, even without needing any pre-experimental bets: the update between bet AM and bet B (from learning it is Monday) is non-Bayesian. Standard Dutch book arguments for Bayesian updating would apply under CDT, but EDT agents bet at odds not matching their subjective probabilities.
I prefer CDT+SIA to EDT+FNC since it is more theoretically elegant, has more concordance with standard theories of probability such as Bayesianism, and has ex ante optimality results (especially for additive games) that do not depend on strong independence assumptions. Still, EDT+FNC appears to be the strongest alternative anthropic decision theory at the moment, and I don’t have a decisive argument against it.
I have found that assuming “one’s total realityfluid is a non-increasing quantity over time”, and that multiple wakings / amnesia drugs divide and combine your realityfluid in the natural way makes these kinds of problems easier to think about.
With that assumption, you can then simply calculate EV weighted by realityfluid and choose the realityfluid-weighted EV-maximizing bets (or use a proper scoring rule to recover probabilities one should give) without needing to think about the specifics of decision theory too much.
From the perspective of Sunday Beauty, 1⁄2 of her realityfluid will be in the heads world and 1⁄2 is in the tails world. The tails-world could be further subdivided into 1⁄4 in Monday-tails, 1⁄4 in Tuesday-tails, but administering amnesia drugs and putting Beauty back to sleep recombines them[1], so for most bet calculations you should use the total weight of the tails-world.
e.g. in your first example, you know that you’re in either H-M or T-M branches, with 1⁄2 or 1⁄4 of your total realityfluid respectively. But in the T-M world, you’ll be put back to sleep and combined with the T-T world which has the other 1⁄4 of your realityfluid and which will be the benefactor of any winning bets. Using those weights, the weighted-EV of the second bet is 1⁄2 * 11 + 1⁄2 * −19 < 0.
(Not sure whether this is closer to SIA or EDT+FNC, but I think they all give the same answer on your examples?)
And then the philosophical discomfort people have with the actual credences varying according to the observer and specifics of the setup and generally not matching intuition in many cases is resolved by accepting that one’s reported credences are just a report of one’s own mind’s scaffolding; they’re facts about a (mind, world), not a world alone. (https://glowfic.com/replies/1746953#reply-1746953)
Alternatively, one can think of the short-lived Beauty as just a kind of counterfactual that never gets any realityfluid in the first place; I think this is an equivalent formulation to everyone except the short-lived T-M Beauty.
Not sure how you’re using reality fluid to calculate expected values here. If H-M has 1⁄2 of your reality fluid and T-M has 1⁄4, then after seeing it’s Monday, it looks like you should assign 2⁄3 to H-M in common with single halfing. After that you’re arguing that because of who is a benefactor you should bet as if at 50⁄50 odds? I don’t think I understand or agree.
We could assume the benefactor in either case is Wednesday Beauty, who has more or less money in her bank account. In that case I don’t see a good reason why Monday Beauty should reason, “if it was Tails then my reality fluid gets combined and who I combine with is a benefactor so I should double importance and bet at 50⁄50 odds even though I have twice as much reality fluid in Heads/Monday as Tails/Monday”.
I don’t see how the “short-lived Beauty is a counterfactual that never gets any realityfluid” helps either; naively one would expect that under this assumption, upon learning it’s Monday, you should assign 100% to Heads, which seems clearly wrong (and that’s clearly the wrong betting odds). Also it seems adopting this view would mean dis-believing in one’s own mortality quite generally, as long as there is a sliver of a chance of immortality.
I agree “probabilities over centered worlds” is the right idea
From the perspective of Sunday (or Wednesday) Beauty, who both have 100% realityfluid.
Why shouldn’t Monday Beauty reason exactly that way for the benefit of Wednesday Beauty, knowing her own existence is short-lived and nothing else she does has any effect on her own experience or existence? You could imagine a different setup, where, after asking Monday Beauty to bet, you could separately ask her to guess how the coin actually came up, and if she answers incorrectly give her a painful electric shock before being put to sleep, which Wednesday Beauty won’t remember either way. And then Monday Beauty might answer that question differently from how she bets.
Mm, maybe the counterfactual formulation obscures things or is ill-posed. In general, I think the point of a counterfactual is that, with high probability, you never expect to actually find yourself inside of one, but you can still use them to reason about decision theory and probability theory usefully. “Never expect to find yourself as tails-Monday Beauty” is counter to the stated setup, but it’s counterfactual in the sense that no long-lived Beauty anywhere remembers being any kind of Monday Beauty, ever.
My question to you is how you counteract the Heads+Monday > Tails+Monday reality fluid ratio. If there is no good reason for this, then it seems Monday Beauty should accept the Monday bet, because she has 2⁄3 of her posterior reality fluid on Heads+Monday. It seemed you were earlier offering a reason why she should not take the Monday bet even though she has twice as much reality fluid on Heads+Monday as Tails+Monday, and that’s what I’m not convinced by.
We can distinguish “altruistic utility function” (like caring about Wednesday Beauty as the benefactor, or shrimp welfare, etc) from “hedonistic utility function”. As I said, I don’t see why reality fluid based decision theory would reject the Monday bet in the altruistic case. Let’s analyze the hedonistic case.
Our Dutch books now involve meal quality rather than money. On Sunday, Beauty may accept a meal of quality 0, or the following lottery: “-14 meal quality if Heads, +16 meal quality if Tails”. Assume her utility is linear in meal quality. It seems she should accept this, as the expected change to meal quality is positive.
Now when Beauty learns it is Monday, she may accept a meal of quality 0, or the following lottery: “+11 meal quality if Heads, −19 meal quality if Tails”. Here’s how I’d try to answer this using the “reality fluid” method. She believes there is twice as much reality fluid on Monday+Heads as Monday+Tails. As such the meal quality values should be scaled by reality fluid. And 2*11 > 19. So she should accept the meal lottery on Monday.
But combined, the Sunday and the Monday meal lotteries lead to −3 overall meal quality relative to taking the 0-quality meal on each day.
So in both cases (altruistic or hedonistic goal) it seems like the reality fluid method is going to get Dutch booked? (If you could explain the general method, maybe it would seem less ad hoc to me, and easier to evaluate.)
Yup, my position is that Beauty should accept the meal lottery but not the dollar lottery (and in general I claim the thing I am proposing agrees with EDT+FNC in the original examples and doesn’t get dutch booked).
But I don’t think this is about altruistic vs. hedonistic—the setup in either case is that Wednesday Beauty remembers nothing of the Monday or Tuesday experiences. In the original setup with dollar lottery, there’s nothing positive or negative that happens to Monday Beauty either way, so she might as well choose for Wednesday Beauty. In the meal lottery, there’s nothing that Wednesday Beauty remembers either way, so Monday Beauty might as well choose for Monday Beauty.
I think you’re saying you’re not convinced by this, but I’m saying that you shouldn’t need a particularly strong or good reason to be convinced—realityfluid doesn’t directly translate to a posterior probability that has to be “overcome”.
Maybe there’s a mixed dollar / meal case where the philosophical question about to what degree, if any, Monday Beauty considers herself to have altruisitc preferences about Wednesday Beauty or considers herself to be Wednesday Beauty matters more. But in the original dollar lottery, unless Monday Beauty specifically wants to spite Wednesday Beauty, she only has to weakly believe that she’s at least a little bit Wednesday Beauty to reject it.
More specifically, I’m not convinced computing the reality fluid distribution is helpful for making decisions. If “realityfluid doesn’t directly translate to a posterior probability that has to be ‘overcome’” then why don’t I skip the step of computing realityfluid, and do CDT+SIA or EDT+FNC, or just UDT (compute ex ante optimal policy, consider anthropic probabilities extraneous)?
I’m not sure how relevant Wednesday Beauty’s memories are; we could either imagine she remembers Heads+Monday and Tails+Tuesday, or not. If she doesn’t remember it seems somewhat like shrimp welfare, where if Monday Beauty contributes to a shrimp welfare fund, the shrimp don’t remember being her.
So in the shrimp welfare case (or Wednesday beauty remembers nothing case) it seems you’re agreeing with EDT+FNC which means the reality fluid distribution (SSA / single-halfing) is irrelevant. So we can just skip the reality fluid computation here.
In the meal case, it seems you are using reality fluid as an input to the utility function? You said earlier that reality fluid doesn’t translate to posterior probabilities, in which case, it could only be relevant for the utility function. In which case, I’m not convinced that these are preferences that meet the reasonable person standard.
Let’s accept “Monday Beauty might as well choose for Monday Beauty”. So then it seems her utility function is “meal quality for Monday Beauty”. Since reality fluid is irrelevant for posterior probabilities in your view, this raises the question of why she doesn’t use EDT+FNC to maximize expected meal quality for Monday Beauty. That computation would be as follows: (1) compute FNC probabilities: “50% on Heads+Monday, 50% on Tails+Monday”, (2) apply EDT: “Expected Monday Beauty meal quality given I take the bet < 0, so I reject”. In which case she would reject the bet.
Now, there is a possible utility function which considers meal quality for Heads+Monday Beauty twice as important as meal quality for Tails+Monday Beauty. But it is (a) not reasonable to summarize that as “Monday Beauty might as well choose for Monday Beauty”, (b) I don’t think it meets the reasonable person standard (assuming both meals are being forgotten anyway).
We could also think of a variation of meal quality where these are meals for shrimp, in which case it looks more like the altruistic case. Then the question is: “If you accept the Monday bet for yourself, why do you reject it for shrimp?”. And the possible answer seems to be “Inherently, I value Heads+Monday version of myself having a good meal more than Tails+Monday version of myself having a good meal, whereas I have no such discrimination for meals fed to shrimp given Heads or Tails.”. Which… again, I don’t see the point of, as a preference.
Hmm, we are maybe not on the same page then, because I think the memories vs. not is fundamental and what makes the original setup interesting as an anthropics problem in the first place? If there’s no amnesia drugs at all, then there aren’t really distinct Mon-Tues-Weds Beauties, just a single Beauty being offered a negative EV lottery. If you’re postulating something else where Beauty regains memories of valenced experiences that she had during the wakings after the fact, that’s different / more complicated.
Either way, I don’t see what the comparison to shrimp welfare is doing; I think the interesting setups are well-captured by (a) something happens which has a lasting impact on Wednesday Beauty (bank account balance) vs. (b) something happens which affects only the temporary / memory-erased Beauties (meal quality), or some mixture of the two.
I’m not saying “no amnesia drugs at all”, I’m trying to disambiguate whether, if Heads, then she is administered amnesia drugs during Monday night. Similarly, if Tails, is she administered drugs during Tuesday night?
The most straightforward answer to those is “no to both” because that reduces complications of “maybe it is Wednesday?” upon waking up. Which is fine for bank balance purposes. However when reasoning about meals it’s harder because then Monday Beauty might take into account that, since she might not be administered amnesia drugs (in the Heads case), then she is going to remember this meal on Wednesday, and that might affect things.
That’s part of why I brought up shrimp welfare: it means we can assume no amnesia drugs on Heads+Monday night or Tails+Tuesday night.
In case (a) it seems you agree with EDT+FNC and that reality fluid is irrelevant to computing anthropic probabilities, so I don’t see any purpose in computing reality fluid at all. In case (b) it seems, if you are (consistently with (a)) saying reality fluid is irrelevant to computing anthropic probabilities, then reality fluid could only be relevant to the utility function (if it is relevant at all), and then my puzzlement re: this looking like an arbitrary preference (for Heads+Monday meal quality being more important than Tails+Monday meal quality) applies.
Ah OK, I was assuming the opposite, because the original setup and most variations of Sleeping Beauty I am aware of stipulate that amnesia drugs follow every waking, not all but the last. I would have to reconsider your variant. I don’t think “maybe it’s Wednesday” is very complicated either way though, other than maybe momentarily. On Wednesday the experiment is over and Beauty leaves the research facility...
Ah. Yeah, reading the original again, I do see the amnesia drugs are administered on every awakening, and that can affect probabilities! (Since one can reason: “either Heads+Monday, Heads+Tuesday, Tails+Monday, Tails+Tuesday, Tails+Wednesday”, so ‘thirder’/SIA rules will say 2⁄5 heads… until they observe it’s not Wednesday, and then they go back to 1⁄3. )
In that scenario I think my comments still apply. With (a) you’re saying that reality fluid is irrelevant to anthropic probabilities, and that your decision rule matches EDT+FNC. With (b) it seems, to be consistent, you should also say reality fluid is irrelevant to anthropic probabilities, so the question is, why take the Monday bet? It looks like an arbitrary preference that the Tails+Monday meal is more important than the Heads+Monday meal.
Like the Monty Hall problem, the answer may depend on the exact way it’s phrased and seemingly minor changes in the phrasing can drastically change the answer. “When you are first awakened?” Does that mean I know it’s the first time I’m awakened, or are you asking what I’ll do when I’m first awakened based on a decision procedure that doesn’t take into account that that is my first awakening? And what does it mean to believe the outcome to a particular degree—am I asked to choose a value that expresses the proportion of heads on a per-awakening basis?
You don’t know if it’s the first awakening when you are asked the main question, the one distinguishing thirders from halfrrs. The follow up question which distinguished halfers from double halfers involes knowing it is Monday (therefore the first awakening). This part is not ambiguous.
Regarding “belief to a particular degree”, the question is what subjective probability to assign. (“There’s no good answer / it’s ambiguous” is a reasonably common answer here.) It relates to questions of how to generalize Bayesian probability theory to imperfect recall situations.