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.
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)".
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 , we imagine that choosing upon gives no evidence about choice in ". As such, optimality for EDT+FNC requires independence assumptions about the agent's distribution over its policy.
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.
The Sleeping Beauty problem is a famous philosophical puzzle. Described by Elga (2000):
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:
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.