Sleeping Beauty as a Mind Killer

Sleeping Beauty (SB) is a very popular logical puzzle, and there is an enormous volume of writing on the topic. No one can read it all.

Here I suggest that the SB problem was naturally selected to become maximally philosophically inflammatory. As a result, it loses much of its explanatory potential. If a correct answer exists, it is buried in tons of literature and depends on a number of assumptions.

The science-fictional setup does not help either. There are no practical situations in which powerful amnesia is used without damaging reasoning abilities. There is an analogue of SB involving twin brothers, but it has important differences: no sequentially appearing tests, such as Tuesday–Tails following Monday–Tails, are possible.

There are 161 posts about SB on LessWrong alone, compared with 3,200 about superintelligence.

The best minds, many of whom also work on AI safety, are spending their time on a puzzle that future generations, if any appear, may see as analogous to counting angels on the head of a pin. That problem also has depth: it requires calculating the smallest invisible thing, a task that could not be solved without a theory of light at the time.

SB simultaneously tests several ideas:

1. Probability vs. credence concerning a given toss.

2. Path-dependent identity across Monday and Tuesday under Tails vs. state-dependent identity across the two Mondays.

3. Different ways of aggregating bets.

4. Actual copies vs. possible copies: days vs. coin outcomes.

5. Whether possible copies should count as real in some sense.

6. Changes in the reference class after a new question, as in Bostrom's hybrid model.

7. Prior policy vs. individual action.

8. A one-shot game vs. repeated games.

9. First-person vs. third-person perspectives.

10. Amnesia vs. copying.

11. Many-worlds interpretation (MWI) vs. classical models.

12. Different decision theories.

13. Bayesian vs. frequentist probability.

14. The nature of self-locating beliefs: SSA vs. SIA.

The perceived simplicity of SB conceals substantial complexity. It tricks the mind into proposing solutions that address only one of the distinctions above.

Below are several hidden caveats, or complexity bombs, within SB.

1. Probability realism is problematic

SB assumes "probability realism": probabilities are real things, and we can correctly infer them.

Probability can typically be tested through frequencies or betting. However, SB is constructed so that both frequency-based and betting-based approaches can distort the result.

The frequentist approach does not work straightforwardly because, if the SB experiment runs only once, its outcomes are mutually exclusive with respect to the coin toss, though not with respect to the day. Under Tails, there are no instances of Heads. This supports the halfer position. If we run SB many times, however, the observations are no longer mutually exclusive and converge toward the thirder position. If the experiment runs only twice, the result is more complicated; see [Bostrom's hybrid approach ].

Beauty's optimal betting behavior can also be calculated, but only if we assume that the experiment occurs many times. We also need assumptions about counterfactuals and about who receives which benefits.

Double-Extreme Sleeping Beauty

It is possible to imagine a one-shot SB experiment in which the stakes are extremely high:

God creates, just once, an unfair coin with a 0.999 probability of Heads.

If Heads, He creates one copy of me.

If Tails, He creates one million copies of me.

I have one attempt to guess the result of the coin toss. If I am wrong, I will be tortured and killed. MWI is false, and no one will ever repeat the experiment.

Tails or Heads?

When the stakes are this high, it becomes much harder to feel certain about what to do. In the ordinary problem, the difference between 1/3 and 1/2 may seem negligible. In Double-Extreme Sleeping Beauty, an error means almost certain pain and death.

SIA and the problem of exact copies

SIA and expected-utility reasoning favor Tails. The difficulty in applying SIA here is that the copies under Tails must be different, so that we can count them as distinct objects drawn from a pool of possible objects.

This can be illustrated by simplifying SIA into a box-of-souls problem. Imagine a box containing possible souls, each numbered from 1 to 100. I toss a coin. If Heads, I take one soul from the box; if Tails, I take ten souls. In this setup, soul 17 is ten times more likely to be selected under Tails, so any selected soul can treat the fact that it was chosen as evidence for Tails.

However, if the souls have no numbers, or all have the same number, 17, then soul 17 gains no information merely from having been chosen.

Thus, SIA is not simply an assumption but a provable theory that works only for different souls. In SB, however, Monday–Tails and Tuesday–Tails are internally indistinguishable. Perhaps we could simulate SIA in SB by assigning both states different random names before awakening. There is also a view that subjectively indistinguishable souls should still count as different: haecceitism.

In the "normal" SB problem, the awakenings are exact copies from the inside, and we do not know whether the universe is finite. Therefore, SIA may not be applicable to SB. Moreover, although SIA may be provably valid under certain assumptions, it can become self-defeating: it immediately favors an infinite universe in which all possible observers exist, after which we must return to SSA. See [ SIA Becomes SSA in the Multiverse ].

2. A coin's intrinsic probability vs. credence about a particular toss

There is a subtle difference between the probability that a coin will land Heads, which is a property of the coin, and our credence that a particular past toss resulted in Heads.

We know that the coin has a 1/2 chance of landing Heads. Before observing the result of a single toss, we assign P(H)=1/2. We may then collect additional information about that particular toss, updating our credence about its result. SB is a form of such measurement, in which the intrinsic global probability and the credence assigned to a particular case can differ.

For example, if a Heads toss produced a louder sound, that could provide additional information about the result and raise our credence in Heads to, say, 0.6.

A useful term here is observation-selection effects, which may be clearer than "anthropics." Some results are more likely to be observed. SB is a classical example: Tails is more likely to be observed.

The coin-in-a-crowd thought experiment

Suppose I toss a coin. If it lands Heads, I tell one person in a crowded room; if it lands Tails, I tell two people. A person who is told about the toss should then assign a 2/3 probability to Tails for that particular toss.

This is not an exact analogue of SB because the other members of the crowd continue to exist. Those who were not told can still assign probability 1/2 to Tails.

3. The no-MWI assumption

SB assumes that MWI is false, or at least that there are no other indistinguishable copies of Beauty. Thus, the alternative Heads branch does not exist when the result is Tails.

This assumption is necessary if SB is to support SIA through the claim that the Tails universe contains more observers.

However, if SIA implies that we live in the largest possible universe, and therefore in an MWI universe, then SB cannot serve as a test of SIA. This negative circularity weakens attempts to prove SIA through SB.

4. The sequential nature of events under Tails

Under Tails, Monday–Tails and Tuesday–Tails are not independent events, as explained in [ ape-in-the-coat's solution ].

Beauty can predict her own actions on the other betting day because an exact counterpart of her exists there. This is the basis of the unusual two-thirder approach, discussed below, in which Beauty counts not only her own bets but also those of her exact counterparts.

5. There is no factual or test…