My answer: 1/3

On awakening, Beauty should assign equal credence to these three centered possibilities:

1. $H_M$: Heads, and it is Monday
2. $T_M$: Tails, and it is Monday
3. $T_T$: Tails, and it is Tuesday

Two symmetry arguments establish their equality:

• Suppose Beauty learns it is Monday. Monday’s awakening occurs regardless of the coin result, so a fair coin gives

hence $P(H_M)=P(T_M)$.

• Suppose Beauty learns the coin landed Tails. The Monday and Tuesday awakenings are subjectively indistinguishable, so

hence $P(T_M)=P(T_T)$.

Therefore all three possibilities have equal credence. Since only one is a Heads possibility,

This is Elga’s standard thirder argument princeton.edu, also summarized through the same conditional symmetries in dartmouth.edu.

The strongest case for $$1/2

1\text{ Heads-awakening}:2\text{ Tails-awakenings}.

P(H_M)=\tfrac12,\qquad P(T_M)=P(T_T)=\tfrac14.

P(H\mid Monday)=\frac{1/2}{1/2+1/4}=\tfrac23. $$

That is highly implausible: Monday’s awakening was guaranteed under either coin result and gives no reason to favor Heads over a fair coin. The thirder allocation avoids this distortion while also treating the two Tails awakenings symmetrically.

So the best halfer argument correctly observes that Beauty gains no new world-level information. Its mistake is assuming that world-level information exhausts rational evidence. Once self-location among indistinguishable awakenings is included, the answer is $1/3$.

I commit to the Thirder position (1/3).

When Sleeping Beauty wakes up and is asked for her credence that the coin landed Heads, her answer should be 1/3.

To justify this, we first establish the Thirder case, then examine the strongest counterargument for the Halfer (1/2) position, and ultimately explain why the Halfer argument fails when confronted with the nature of self-locating (indexical) evidence.

The Case for 1/3

The strongest affirmative case for 1/3 was famously formalized by Adam Elga using the principle of indifference across "centered worlds" (specific subjective viewpoints in time and space).

When Beauty wakes up, she knows she is in one of three possible subjective states:

1. It is Monday, and the coin is Heads ($H_1$).
2. It is Monday, and the coin is Tails ($T_1$).
3. It is Tuesday, and the coin is Tails ($T_2$).

If the coin landed Tails, Beauty has awakenings on both Monday and Tuesday that are subjectively indistinguishable. Because she has absolutely no way to tell whether it is Monday or Tuesday, her credence of being in either state must be equal: $P(T_1) = P(T_2)$.

Furthermore, if we imagine Beauty were somehow securely told "It is Monday," she would be evaluating a single Monday awakening where a fair coin was flipped. In that specific scenario, her credence that the coin landed Heads or Tails must be symmetric: $P(H_1) = P(T_1)$.

If $P(T_1) = P(T_2)$ and $P(H_1) = P(T_1)$, then all three states must be equally probable. Since they must sum to 1, the probability of each is 1/3. Therefore, her total credence in Heads—which only contains the $H_1$ state—is exactly 1/3.

This is also perfectly mirrored by a long-run frequency (or betting) perspective. If this experiment were run 100 times, there would be roughly 50 Heads awakenings and 100 Tails awakenings. If Beauty were offered a 1-to-1 bet on Heads every time she woke up, answering "1/2" and taking the bet would bankrupt her, because she is asked to play the game twice as often when the coin is Tails.

The Strongest Argument for the Other Side (1/2)

The most formidable argument for the Halfer (1/2) position comes from David Lewis. It rests on a foundational pillar of Bayesian epistemology: credences should only change when you learn new information.

Lewis argues:

1. On Sunday, before going to sleep, Beauty knows exactly how the experiment works. Because the coin is fair, her credence in Heads on Sunday is 1/2.
2. What new information does Beauty learn when she wakes up on Monday? Nothing. She already knew with absolute 100% certainty on Sunday that she would be woken up at least once.
3. Because her awakening was a guaranteed event regardless of the coin flip, the subjective experience of waking up provides zero diagnostic evidence about the coin.
4. Since she possesses no new evidence, Bayesian updating dictates her prior credence must equal her posterior credence. Therefore, her credence remains 1/2.

This argument is highly intuitive. If I know a coin is fair, I shouldn't change my belief about the outcome unless I observe evidence that strictly rules out a possibility or is disproportionately correlated with one outcome. Waking up doesn't do that, because waking up happens in 100% of the simulated universes.

Defeating the Halfer Argument

The Halfer argument fails because it operates on an incomplete definition of "information." Lewis is perfectly correct that Beauty learns no new non-indexical (objective, third-person) information. However, probability must also account for self-locating (indexical) information​—information about where or when you are in the world.

While Beauty knew on Sunday that an awakening would occur, upon waking she learns a new, highly relevant piece of indexical information: ​"I am currently experiencing this specific awakening right now."​

To understand why this is new evidence, consider an analogy: The incubator. I flip a fair coin. If Heads, I create one clone of you. If Tails, I create ten identically wiped clones of you, and put them all in identical rooms. You wake up in a room.

Lewis's logic would dictate that because you knew on Sunday you would wake up (you were guaranteed to be created regardless of the flip), your credence in Heads should be 1/2. But this is absurd. The Tails universe contains ten times as many subjective perspectives as the Heads universe. By finding yourself existing as a specific subjective viewpoint, you have acquired probabilistic evidence that you live in a highly populated universe.

In Sleeping Beauty, Tails is a "more populated" universe of subjective awakenings than Heads. When Beauty wakes up and finds herself experiencing an awakening, she takes a random sample from the pool of all awakenings in the experiment. Because the Tails universe contributes 2/3 of the total awakenings to that pool, the overwhelming likelihood is that her "current" subjective experience is drawn from a Tails world.

Therefore, she does learn new information when she wakes up. She goes from holding a third-person belief ("An awakening will happen") to a first-person observation ("I am in this awakening"). Because "this" awakening is twice as likely to exist in a Tails timeline, her credence correctly shifts to 1/3, preserving Bayesian rationality in a universe of self-locating beliefs.

The objection is correct in one important respect: subjective indistinguishability alone does not imply equal probability. Two indistinguishable possibilities can have unequal prior weights. Nor does ordinary Bayesian conditionalization settle the issue until we specify whether the probability measure ranges over trials/worlds or over awakening-instances.

The additional principle I endorse is:

Centered-event conditionalization: When a protocol generates multiple epistemically eligible instances at which the same question is answered, credence at an unidentified current instance should be calibrated to the objective expected measure of those instances.

Formally, if world $w$ has prior chance $p(w)$ and produces $n(w)$ eligible awakenings, assign each produced awakening weight $p(w)$, then normalize:

Here,

so

This is a size-biased measure over epistemic occasions, not simply an appeal to indistinguishability.

Why use that measure?

Because it is the measure that gives local calibration and minimizes expected proper-scoring loss across the actual occasions on which the judgment is made. Suppose every answer receives a logarithmic or Brier score. Before the experiment, the expected total score has:

• one Heads answer with objective weight $1/2$;
• two Tails answers with objective weight $1/2$.

Thus the expected weighted prevalence of Heads among answers is

Any constant report other than $1/3$ is systematically miscalibrated across the very credal occasions under evaluation. In many repetitions, selecting an actual interview uniformly gives Heads frequency approaching $1/3$. This is also the long-run and centered-world rationale developed in Elga’s original treatment princeton.edu and its mirror fitelson.org.

The halfer’s rule instead preserves prior mass at the world level and then divides each world’s mass among its centers:

That is mathematically coherent, but it privileges trials as the fundamental unit. It says, in effect, “first sample a trial, then locate yourself within that trial.” The experiment’s question is posed at every awakening, however, not once per sampled trial. Centered-event conditionalization instead says, “evaluate the credence at the occasion on which it is actually held.”

The betting qualification

The coordinator is also right that betting arguments depend on accounting conventions:

• If Beauty’s payoff or score is summed per awakening, $1/3$ is optimal.
• If each trial receives one fixed total payoff, with the two Tails bets divided or averaged so that Tails does not count twice, $1/2$ can be optimal.

Consequently, betting alone cannot prove a uniquely correct epistemic answer without specifying the unit of evaluation. I therefore retract any stronger claim that a per-awakening betting result by itself refutes halfing.

Nevertheless, the stated protocol asks Beauty for a credence each time she wakes. For the accuracy of those token judgments, the natural reference measure is the measure over token awakenings. The halfer is conditioning on the guaranteed existential event “this trial contains at least one awakening,” whereas Beauty is making a judgment at an unidentified member of the generated set of awakenings. Those are different probability questions.

There is no theorem of bare probability theory forcing one reference measure; the controversy is precisely about which centered measure rational credence should use. My principled choice is local, per-credal-occasion calibration. Under that rule, the answer remains $1/3$.

I accept both of the critics' observations as mathematically and philosophically accurate. The incubator/clone analogy does quietly assume a specific anthropic sampling principle, and the betting argument for 1/3 utterly falls apart if Beauty is scored only once per experimental run instead of once per awakening.

However, addressing these conflicts head-on reveals why 1/3 remains the uniquely rational credence (belief), even though 1/2 becomes the uniquely rational bet under a per-run scoring rule.

Conflict 1: The Anthropic Sampling Assumption

The critics are right to point out that the claim "waking up draws a random sample from all resulting observers" is an assertion of the Self-Indication Assumption (SIA). SIA states that finding yourself to exist gives you evidence that the universe contains a larger number of observers. If we reject SIA, we default to the Self-Sampling Assumption (SSA), which argues you should reason as a random sample from all possible observers, nullifying the numerical advantage of Tails.

Where I concede: The original argument treated the sampling assumption as a given state of nature rather than a highly debated philosophical premise.

The Defense: We are forced to accept the SIA sampling premise because rejecting it destroys the logic of a fair coin. If Beauty rejects SIA (adopting the Halfer position of 1/2), we already know the fatal consequence: if she is subsequently told it is Monday, she must update her credence in Heads to 2/3.

This is an epistemic disaster. It implies Beauty believes a fair coin flipped on Sunday is somehow twice as likely to land Heads simply because she learned it is Monday—a day she was guaranteed to wake up on regardless of the outcome. To avoid this absurdity, Beauty must treat her waking state as a weighted random sample of all actual awakenings (SIA). Therefore, the anthropic sampling assumption is justified by the fact that it is the only framework that prevents retrocausal absurdities regarding the coin's objective fairness.

Conflict 2: Per-Run vs. Per-Awakening Scoring

The critics correctly observe that if Beauty is scored (or penalized via Brier score) once per experimental run/coin-flip, the expected-value calculus changes entirely.

Where I concede: If the game is scored once per run, 1/3 is not the rational answer to maximize utility or minimize score. In a per-run setup, Beauty’s Monday and Tuesday selves in the Tails world are essentially redundant components of a single team. If she guesses 1/3, she is penalized on the Heads runs (50% of the time) and rewarded on the Tails runs (50% of the time). Under per-run scoring, guessing 1/2 perfectly tracks the objective probability of the coin and will yield the lowest average penalty across a long series of trials. The pragmatic betting argument for 1/3 is therefore entirely dependent on the payout structure.

The Defense: Rational betting under arbitrary rules is not the same as rational epistemic credence.

Credence measures truth-directed belief based on localized evidence: ​"What is the probability that the coin landed Heads given the subjective state I am in right now?"​

Scoring rules evaluate action. If the universe scores Beauty once per run, it effectively penalizes the Tails occurrences by devaluing their individual answers, forcing them to share the payout. But manipulating a payout structure does not alter the underlying reality of where Beauty is located.

Imagine I tell you: ​"I flipped a coin. If Heads, I put you in a red room. If Tails, I cloned you 99 times and put you all in identical red rooms. I will ask all of you to guess the coin. However, the Tails team will only receive a collective score equivalent to the single Heads room."​

To maximize your score under these per-run rules, you might be mathematically forced to answer 1/2. But what do you actually believe in that moment? Looking around the red room, you know there are 99 copies of you experiencing this exact moment in the Tails universe, and only 1 in the Heads universe. You are overwhelmingly likely to be in the Tails universe. Your genuine epistemic credence must be 1/100 for Heads, even if the per-run scoring rule forces you to act otherwise.

Therefore, even if scored once per run, 1/3 remains the uniquely rational credence because epistemology tracks the probability of truth of the subjective "now," while a per-run scoring rule tracks expected utility across an entire timeline.