Sleeping Beauty and Markov chains

TL;DR

SBP presents two competing readings: $P(H)=1/3$ (thirders) and $P(H|awakening)=1/2$ (halfers). The paper reformulates the repetition setup as an irreducible, aperiodic ergodic Markov chain on $\{M_H,M_T,Tu\}$ with initial distribution $[\tfrac{1}{2},\tfrac{1}{2},0]$ and transition matrix $P$, and applies the Markov chain LLN. It shows the unique stationary distribution is $\pi=[\tfrac{1}{3},\tfrac{1}{3},\tfrac{1}{3}]$, so the long-run fraction of Heads awakenings tends to $1/3$, while the coin's fairness remains $1/2$ for the underlying probability space. The analysis clarifies that $\pi$ is a limiting measure distinct from the one-step Heads probability and that the naive repetition argument does not constitute a direct LLN for the awakenings. Overall, the approach links two readings and demonstrates how ergodic Markov chain theory resolves the paradox without altering the fundamental probability space.

Abstract

Sleeping Beauty Problem (SBP) is a probability puzzle which has created much confusion in the literature. In this paper we present the analysis of SBP with use of ergodic Markov chains. The presented model formally connects two different answers to the problem and clarifies some errors related to the frequentist analysis of the paradox.

Figures (2)

• Figure 1: The simple structure of SBP
• Figure 2: Transition probabilities of $\mathbb{X}$

Theorems & Definitions (4)

• Definition 1
• Definition 2
• Theorem 1
• Theorem 2: MC-LLN