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The Sleeping Beauty Problem: Sleeping Kelly is a Thirder

This paper argues that Sleeping Beauty should adopt the "thirder" position because maximizing the expected growth rate of her wealth using the Kelly Criterion under multiplicative dynamics renders her impervious to Dutch books, whereas the "halfer" position remains vulnerable.

Original authors: Ben Abramowitz

Published 2026-01-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ben Abramowitz

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are Sleeping Beauty (let's call her "Sleeping Kelly" for this story). You are about to go on a strange adventure involving a fair coin, a drug that makes you forget everything, and a very specific set of rules about money.

Here is the simple breakdown of what Ben Abramowitz's paper argues, using everyday analogies.

The Setup: The Amnesia Hotel

You are put to sleep on Sunday. A researcher flips a fair coin.

If it lands Heads: You are woken up once (Monday), given a task, and then put back to sleep until Wednesday.
If it lands Tails: You are woken up twice (Monday and Tuesday). Each time, you are given the same task, but the drug wipes your memory so you don't know if it's your first or second time waking up.

When you wake up, you have to answer one question: "What are the odds the coin landed Heads?"

For decades, smart people have argued about this:

The "Halfers" say: "It's 50/50. The coin is fair. I knew I would wake up, so nothing changed. It's still 50/50."
The "Thirders" say: "Imagine this happens 100 times. You'll wake up 100 times for Heads, but 200 times for Tails. So, out of every 3 times you wake up, 2 are Tails and 1 is Heads. It's 1/3 Heads, 2/3 Tails."

The Old Way of Thinking: The "Expected Value" Trap

Most people tried to solve this by asking: "If you bet money, how do you maximize your average winnings?"

The paper says this approach is flawed. It uses a famous analogy called the PGM Coin Flip:

Imagine a game where you bet $100.
Heads: You win 50% (you have $150).
Tails: You lose 40% (you have $60).
The Math: The average result is $105. So, "Expected Value" says: "Yes, play this game! You'll make money!"

But here's the catch: If you play this game over and over again, you will go broke.
Why? Because losing 40% hurts more than winning 50% helps. If you lose twice in a row, you have \ $36. If you win twice, you have \$ 225. But if you win once and lose once, you end up with $90.
Over time, the "average" number is misleading. What matters is your growth rate—how fast your wealth actually multiplies over time.

The New Way: The "Kelly Criterion" (The Gardener's Approach)

The paper argues that Sleeping Kelly shouldn't try to maximize her average winnings. She should try to maximize her wealth growth rate over the long run. This is called the Kelly Criterion.

Think of it like gardening:

If you bet too much on a risky plant, one bad storm (a loss) wipes out your whole garden.
If you bet too little, your garden grows too slowly.
The Kelly Criterion tells you the perfect amount to bet to make your garden grow as fast as possible without dying.

The Paper's Big Discovery

When Sleeping Kelly uses this "Gardener's Logic" (maximizing growth rate) to decide how much of her money to bet each time she wakes up, the math forces her to act exactly like a Thirder.

Here is the logic flow:

The Goal: Maximize the long-term growth of her money.
The Strategy: She calculates the perfect bet size (using the Kelly formula).
The Result: To make the math work, she must act as if the chance of Tails is 2/3 and Heads is 1/3.
The Conclusion: If she wants to be a rational money-growth maximizer, she must be a Thirder.

The paper also checks if this strategy is "safe." In gambling, a Dutch Book is a trap where a clever bookie offers you a series of bets that you think are fair individually, but when you take them all, you are guaranteed to lose money.

Halfers are vulnerable: If you believe the odds are 50/50, a clever bookie can trick you into a series of bets that will drain your bank account.
Thirders are safe: If you believe the odds are 1/3 Heads and 2/3 Tails (and bet accordingly), no one can trick you into a losing streak. You are immune to these traps.

The Final Verdict

The paper concludes that the "Halfer" view is mathematically dangerous. If you want to make smart decisions about money in a world where you might wake up multiple times and forget, you have to think like a Thirder.

In short:

Old View: "I'll bet on the average." (Result: You go broke or get tricked).
New View: "I'll bet to grow my wealth steadily." (Result: You act like a Thirder, you grow your wealth, and you can't be tricked).

Sleeping Kelly is a Thirder because that is the only way to keep her wallet full and her garden growing.

Technical Summary: "The Sleeping Beauty Problem: Sleeping Beauty is a Thirder"

Problem Statement

The paper addresses the Sleeping Beauty problem, a classic problem of imperfect recall introduced by Elga [2000]. In this scenario, an agent (Sleeping Beauty) is put to sleep on Sunday. A fair coin is flipped: if Heads, she is awakened once (Monday); if Tails, she is awakened twice (Monday and Tuesday). After each awakening, she is given a drug that erases her memory of the waking. Upon awakening, she must assign a credence (probability) to the coin having landed Heads.

The debate centers on two primary positions:

The Halfer position: Assigns a credence of $1/2$ to Heads, arguing that no new information is gained upon waking.
The Thirder position: Assigns a credence of $1/3$ to Heads, arguing that in a long run of experiments, $1/3$ of all awakenings occur in Heads-worlds.

Existing literature often attempts to resolve this by modeling the agent's decision-making through betting games, typically assuming the agent seeks to maximize the expected value (EV) of her payoffs. The paper argues that this assumption is flawed when applied to sequential betting scenarios involving multiplicative wealth dynamics.

Methodology

The author proposes a decision-theoretic framework that shifts the objective from maximizing expected payoff to maximizing the expected growth rate of wealth (the wealth multiplier) under multiplicative dynamics.

1. Sequential Betting Setup

The problem is re-framed as a sequential betting game:

Sleeping Beauty starts with an initial wealth $x$ .
She can wager a fraction of her current wealth on the coin outcome at 1:1 odds.
She makes a wager before sleeping (Sunday) and a wager each time she is awakened (Monday, and potentially Tuesday).
Crucially, she has no memory of previous wagers or her current wealth level; she must decide on a fraction of wealth to wager.
The experiment may be repeated sequentially over time, where the final wealth of one run becomes the initial wealth of the next.

2. Critique of Expected Value Maximization

The paper utilizes the Peters and Gell-Mann (PGM) coin flip [2016] to demonstrate the failure of EV maximization in sequential contexts.

In a single-stage game, a bet with a positive expected value (e.g., +50% gain, -40% loss) appears rational.
However, in a sequential setting where wealth compounds, the geometric mean of the outcomes determines long-term survival. A sequence of such bets leads to exponential decay of wealth (bankruptcy) because the product of gains and losses is less than 1 ( $1.5 \times 0.6 = 0.9$ ).
The paper argues that existing Dutch Book arguments against the Thirder position rely on the incorrect assumption that agents should maximize the sum of expected payoffs rather than the growth rate of wealth.

3. The Kelly Criterion

The author applies the Kelly Criterion, which maximizes the long-term expected geometric growth rate of wealth. For a bet with 1:1 odds, the optimal fraction $b$ to wager is given by $b = p - (1-p)$ , where $p$ is the probability of the winning outcome.

Key Contributions and Results

1. Optimal Strategy Derivation

By modeling the Sleeping Beauty problem as a sequence of bets where the goal is to maximize the wealth multiplier, the paper derives the optimal betting fractions:

Before sleeping (Sunday): The optimal fraction wagered is $a = 0$ .
Upon waking: The optimal fraction wagered on Tails is $b = 1/3$ .

2. Inference of Credence

Using the derived optimal betting fraction ( $b = 1/3$ ) and the Kelly Criterion formula ( $b = p_{tails} - p_{heads}$ ), the paper infers the implicit probability assignment:

$1/3 = p_{tails} - (1 - p_{tails}) \implies 2p_{tails} = 4/3 \implies p_{tails} = 2/3$ .
Consequently, $p_{heads} = 1/3$ .
Result: An agent maximizing wealth growth in this sequential setting implicitly assigns a credence of $1/3$ to Heads, aligning with the Thirder position.

3. Dutch Book Analysis

The paper investigates whether the Thirder or Halfer positions are vulnerable to diachronic Dutch books (sets of bets that an agent accepts individually but which guarantee a net loss).

Thirder (Sleeping Kelly): Theorem 2 proves that if Sleeping Beauty acts as a Thirder and only accepts bets with a wealth multiplier greater than 1, she cannot be subjected to a Dutch book. The conditions required to construct a Dutch book (guaranteed loss in both Heads and Tails scenarios) mathematically contradict the conditions required for her to accept the bets (wealth multiplier > 1).
Halfer: Theorem 3 demonstrates that a Halfer (assigning $p=1/2$ upon waking) is vulnerable to a Dutch book. Even if the Halfer only accepts bets with a positive wealth multiplier, a specific set of offers can be constructed that guarantees a loss in both Heads and Tails scenarios.

Significance and Claims

The paper claims to resolve the Sleeping Beauty problem by grounding the agent's credence in rational wealth management under multiplicative dynamics rather than abstract probability updating.

Rejection of EV Maximization: The paper asserts that the standard assumption of maximizing expected payoff is incorrect for problems involving imperfect recall and sequential betting.
Validation of the Thirder Position: It provides a decision-theoretic justification for the Thirder position, showing that maximizing the growth rate of wealth naturally leads to a $1/3$ credence for Heads.
Robustness: The Thirder position is shown to be "impervious to Dutch books" under the proposed wealth-maximization framework, whereas the Halfer position remains vulnerable.

The author concludes that "Sleeping Kelly" (Sleeping Beauty acting as a rational wealth maximizer) is a Thirder, and that this position is the only one consistent with avoiding ruin in sequential betting environments.