Category: mathematics
Associated thinker(s): Adam Elga; Robert Stalnaker; Arnold Zuboff
Original publication or approximate period: Canonical published discussion: Elga, “Self-locating belief and the Sleeping Beauty problem” (2000); earlier precursors
Difficulty: Advanced
Type: thought experiment
Themes: self-location; credence; sampling
Related experiments: monty-hall; st-petersburg-paradox
Before sleeping, Beauty learns the protocol. A fair coin determines her awakenings. Heads produces one awakening, on Monday. Tails produces two, on Monday and Tuesday. After each interview, a treatment removes her memory of that interview. At every interview she knows the protocol but does not know the day.
Upon awakening she is asked for her credence that the coin landed heads. The uncertainty is partly about the world and partly about her location within it. A tails world contains two subjectively indistinguishable interviews, whereas a heads world contains only one.
Should Beauty assign heads probability 1/2 or 1/3 when interviewed?
The question tests how ordinary conditionalization should handle self-locating information. A fair coin defines a distribution over trials. A collection of interviews has a different frequency structure. Translating between them requires a principle about what evidence the agent has and how her present perspective is represented.
Halfers emphasize that Beauty knew she would awaken whichever result occurred and may regard the awakening as no new evidence about the toss. Thirders emphasize the three possible centered situations: heads-Monday, tails-Monday, and tails-Tuesday; Elga argues for equal probabilities and hence 1/3 for heads. Further positions distinguish kinds of credence or updating. Merely pointing out the coin’s physical fairness does not settle the philosophical disagreement.
No one needs to claim that the coin becomes physically biased. Counting interviews does not automatically settle what one should believe about a randomly selected trial. Betting arguments depend on whether stakes are paid per interview, per experiment, or under some other rule; changing that rule can change the decision problem.
Variants change the coin bias, number of awakenings, or whether Beauty is told the day. These changes help isolate whether the disputed update concerns possible worlds or positions within worlds.
Elga’s paper supplies the scenario, the two principal answers, and acknowledgement of precursors. This entry presents the disagreement without advertising the author’s own thirder conclusion as an uncontested theorem.
Verified on 2026-10-30. Source coverage: [1] Original canonical formulation, earlier precursors, half/thirder positions. Reflection questions and illustrative numerical examples are original editorial material.