# The answer to the sleeping beauty problem is 1/2

### Too many perpetually-almost-done drafts. I’m entering my ‘just post stuff even if it kinda sucks’ arc.

**Note**: skip down to the “The halfers are right” section if you’re familiar with the problem and usual positions. Also see the subtitle.

# Problem

According to Wikipedia, the *Sleeping Beauty problem* is a “puzzle in decision theory” that goes like this:

Sleeping Beauty [an “ideally rational epistemic agent”] volunteers to undergo the following experiment and is told all of the following details: On Sunday she will be put to sleep. Once or twice, during the experiment, Sleeping Beauty will be awakened, interviewed, and put back to sleep with an amnesia-inducing drug that makes her forget that awakening. A fair coin will be tossed to determine which experimental procedure to undertake:

If the coin comes up heads, Sleeping Beauty will be awakened and interviewed on Monday only.

If the coin comes up tails, she will be awakened and interviewed on Monday and Tuesday.In either case, she will be awakened on Wednesday without interview and the experiment ends.

Any time Sleeping Beauty is awakened and interviewed she will not be able to tell which day it is or whether she has been awakened before. During the interview Sleeping Beauty is asked:

"What is your credence now for the proposition that the coin landed heads?"

# (Purported) solutions

There are basically two competing claims about Sleeping Beauty’s subjective probability (recall, as an ‘ideally rational agent’) in the given scenario. Once again, I’ll plagiarize some pseudonymous Wikipedia editors:

## Thirder position

The thirder position argues that the probability of heads is 1/3. Adam Elga argued for this position originally as follows:

Suppose Sleeping Beauty is told and she comes to fully believe that the coin landed tails. By even a highly restricted principle of indifference, given that the coin lands tails, her credence that it is Monday should equal her credence that it is Tuesday, since being in one situation would be subjectively indistinguishable from the other. In other words, P(Monday | Tails) = P(Tuesday | Tails), and thus

\(P(\text{Tails and Tues}) = P(\text{Tails and Mon})\)Suppose now that Sleeping Beauty is told upon awakening and comes to fully believe that it is Monday. Guided by the objective chance of heads landing being equal to the chance of tails landing, it should hold that P(Tails | Monday) = P(Heads | Monday), and thus

\(P(\text{Tails and Tues}) = P(\text{Tails and Mon})=P(\text{Heads and Mon}) \)Since these three outcomes are exhaustive and exclusive for one trial (and thus their probabilities must add to 1), the probability of each is then 1/3 by the previous two steps in the argument:

\(P(\text{Tails and Tues}) = P(\text{Tails and Mon})=P(\text{Hears and Mon}) =1/3\)## Halfer position

David Lewis responded to Elga's paper with the position that Sleeping Beauty's credence that the coin landed heads should be 1/2…

“Sleeping Beauty receives no new non-self-locating information throughout the experiment because she is told the details of the experiment. Since her credence before the experiment is

P(Heads) = 1/2, she ought to continue to have a credence ofP(Heads) = 1/2since she gains no new relevant evidence when she wakes up during the experiment. This directly contradicts one of the thirder's premises, since it meansP(Tails | Monday) = 1/3andP(Heads | Monday) = 2/3.

## Maybe we should care?

Apparently this might somehow matter for philosophical anthropics, which in turn might somehow matter for the fate of the humanity.

Again, Wikipedia: “credence about what precedes awakenings is a core question in connection with the anthropic principle.”

# The halfers are right

Argument:

By assumption, P(Heads) = P(Tails) = 1/2, so Sleeping Beauty would answer “1/2” before being put to sleep for the first time.

Sleeping Beauty gains no new information upon waking.

Therefore, she should not change her answer upon waking.

Point 2

is where the dispute lies and so is worth defending more fully.## Claim: waking is not evidence that the coin landed tails

Let us examine the probabilities at each stage of events in this setup. Initially, the coin's probability *p *is 1/2 before it is flipped. At this point, Sleeping Beauty would also say p equals one-half. For this to change, some new information must be introduced.

Before she goes to sleep for the first time, has anything occurred between when the coin landed (out of her sight) and her falling asleep? The answer is no; everything remains symmetrical with a probability of one-half. Since all subsequent events are deterministic, Sleeping Beauty can devise an algorithmic plan based on what happens next.

However, for the probability to shift from one-half to any other value like one-third or even zero (as thirders claim), Sleeping Beauty would have to acknowledge that upon waking up – despite knowing she cannot differentiate between worlds – she will instantly believe that heads now have a lower likelihood than previously thought.

#### An intuition pump

To further illustrate where thirders err in their reasoning, consider an alternative scenario: instead of being woken up twice if tails, imagine it happening a billion times or even countably infinite times. In such cases, p should equal zero or an infinitesimally small number according to the same reasoning that implies p=1/3.

Now place yourself in Sleeping Beauty's shoes as you awaken; do you really feel 99.999% sure that the coin landed tails? I certainly don’t, in a sense that seems intuitively much clearer than introspecting on the 1/2 vs 1/3 case.

Indeed, it seems implausible given there remains a 50% chance that heads appeared and only required waking on just one occasion - which, of course, might have occurred just a moment ago.

#### So where does the thirder intuition come from?

It’s hard to say, but I think a case in which p=1/3 really *does* hold up gestures towards the answer:

Suppose you’re one of the experimenters, and divide the days of Monday and Tuesday into half-hour chunks. You then randomly select one, and find out that during this block Sleeping Beauty happened to be woken by your colleague.

Since this time you *could* have observed otherwise (unlike Sleeping Beauty in the thought experiment discussed), you’d be correct to conclude that that *P(Tails)=2*P(Heads)*, which implies in this case P(Heads)=1/3.

Really, this subtitle is just a more forthright reformulation of “Sleeping Beauty gains no new information upon waking.”

## The answer to the sleeping beauty problem is 1/2

That intuition pump doesn’t really work for me. The extreme cases seems to make it more apparent that thirding is correct.

There are ethical implications to accepting different assumptions about observation selection effects. For example, SSA has the possible assumption of a near term doomsday. One implication of the SIA, in my view, is that multiple existences is possible.

If you wake up and I offer you a bet on what day it is, what odds do you accept?