Sleeping Beauty Problem Self-Locating Beliefs

Scenario

Sleeping Beauty is put to sleep on Sunday. A fair coin is flipped.

• If the coin lands Heads: Beauty is awakened on Monday morning, interviewed, and put back to sleep until the experiment ends.
• If the coin lands Tails: Beauty is awakened on Monday morning, interviewed, administered an amnesia drug, and awakened again on Tuesday morning for an identical interview. Whenever Beauty is awakened, she has no memory of any previous awakenings and cannot tell what day it is. Upon each awakening, she is asked: 'What is your subjective credence $\mathbb{P}(\text{Heads})$?'

Your Task

1. Formalize the sample space of centered worlds (self-locating awakenings): $H_1 = (H, \text{Mon})$, $T_1 = (T, \text{Mon})$, $T_2 = (T, \text{Tue})$.
2. Derive the Thirder credence $\mathbb{P}(\text{Heads}) = 1/3$ using epistemic symmetry and conditional probability.
3. Explain why long-run frequency and Dutch book / betting arguments confirm the Thirder solution over the Halfer position ($1/2$).

Constraints

• Fair coin with $P(H) = P(T) = 0.5$.

Examples Over 1000 repetitions of the experiment, there are $\approx 500$ Heads awakenings and $\approx 1000$ Tails awakenings, giving a frequency of $500 / 1500 = 1/3$.