Sleeping Kelly is a Thirder

TL;DR

The paper reframes the Sleeping Beauty problem as a decision problem under imperfect recall and multiplicative wealth dynamics, arguing that rational bets should maximize long-run growth via the Kelly criterion rather than traditional expected value. It shows that the optimal pre-sleep and awakening bets are $a=0$ and $b=\tfrac{1}{3}$, with the implied awakening-tail probability $p=\tfrac{2}{3}$, making Sleeping Beauty a thirder in the growth-rate sense. Dutch-book analyses reveal that a thirder who accepts only growth-rate-positive bets is invulnerable to Dutch books, whereas halfers are vulnerable under certain constructions; EV-maximizing strategies are particularly susceptible. The work highlights that a growth-rate, multiplicative approach yields a robust rationality criterion for decision-making under imperfect recall and underlines the nuanced role of probability updating in this context.

Abstract

The Sleeping Beauty problem was presented by Elga and highlights the role of probabilities in situations with imperfect recall. One approach to solving the Sleeping Beauty problem is to allow Sleeping Beauty to make decisions based on her beliefs, and then characterize what it takes for her decisions to be "rational". In particular, she can be allowed to make monetary bets based on her beliefs, with the assumption that she wants to gain wealth rather than lose it. However, this approach is often coupled with the assumption that Sleeping Beauty should maximize the expected value of her bets. Here, I argue instead that it is rational for Sleeping Beauty to maximize the growth rate of her wealth using the Kelly Criterion, which leads us to the "thirder" position. Furthermore, this position is shown to be "rational" by Dutch book arguments. If Sleeping Kelly only accepts bets that have a growth rate greater than 1 as a "thirder" then she is not vulnerable to Dutch books. By contrast, if Sleeping Beauty takes the "halfer" position, she is vulnerable to Dutch books. If the bets offered to Sleeping Beauty were to be structured differently and lead to non-multiplicative wealth dynamics, she may no longer be a "thirder".

Figures (1)

• Figure 1: Wealth multiplier for wager fractions $a$ and $b$.

Theorems & Definitions (4)

• Theorem 1
• Theorem 2
• proof
• Theorem 3