From abfd66aede831ae1a070c0797944dc634210fcc5 Mon Sep 17 00:00:00 2001 From: Lenovo Date: Sun, 26 May 2024 08:34:49 +0800 Subject: [PATCH] 架构已经搭好，需要李昕闻补充 --- document.tex | 1 + main/2048isNonergodic.tex | 30 ++++++++++++++++++++++++++++++ main/background.tex | 15 ++++++++------- main/nonergodic.tex | 69 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++-- 4 files changed, 106 insertions(+), 9 deletions(-) create mode 100644 main/2048isNonergodic.tex diff --git a/document.tex b/document.tex index 739e89b..ebf07ad 100644 --- a/document.tex +++ b/document.tex @@ -75,6 +75,7 @@ wangwenhao11@nudt.edu.cn). \input{main/introduction} \input{main/background} \input{main/nonergodic} +\input{main/2048isNonergodic} %\input{main/nonergodicity} %\input{main/paradox} diff --git a/main/2048isNonergodic.tex b/main/2048isNonergodic.tex new file mode 100644 index 0000000..5017391 --- /dev/null +++ b/main/2048isNonergodic.tex @@ -0,0 +1,30 @@ +\section{Non-ergodicity of 2048} + + +\begin{theorem} +2048 game is non-ergodic between non-absorbing states. +\end{theorem} +\begin{IEEEproof} + +\end{IEEEproof} + + + + + + + + + + + + + + + + + + + + + diff --git a/main/background.tex b/main/background.tex index 68dc1c5..680badb 100644 --- a/main/background.tex +++ b/main/background.tex @@ -54,9 +54,9 @@ reaching the leftmost or rightmost node where it terminates. The terminal states are usually called absorbing states. The transition probobility matrix of random walk with absorbing states -$P_{\text{absorbing}}$ is defined as follows: +$P_{\text{ab}}$ is defined as follows: \[ -P_{\text{absorbing}}\dot{=}\begin{array}{c|ccccccc} +P_{\text{ab}}\dot{=}\begin{array}{c|ccccccc} &\text{T} & \text{A} & \text{B} & \text{C} & \text{D} & \text{E} \\\hline \text{T} & 1 & 0 & 0 & 0 & 0 & 0 \\ \text{A} & \frac{1}{2} & 0 & \frac{1}{2} & 0 & 0 & 0 \\ @@ -122,6 +122,7 @@ where $Q$ is the matrix of transition probabilities between is \begin{equation} N\dot{=} \sum_{i=0}^{\infty}Q^i=(I_{n-1}-Q)^{-1}, + \label{definitionN} \end{equation} where $I_{n-1}$ is the $(n-1)\times(n-1)$ identity matrix. It is now easy to define whether the non-absorbing states @@ -139,15 +140,15 @@ Assume that $N$ exists for any policy $\pi$ For random walk with absorbing states, \[ -P_{\text{absorbing}} = +P_{\text{ab}} = \begin{bmatrix} -Q_{\text{absorbing}} & R_{\text{absorbing}} \\ -0 & I_{\text{absorbing}} +Q_{\text{ab}} & R_{\text{ab}} \\ +0 & I_{\text{ab}} \end{bmatrix}, \] where \[ -Q_{\text{absorbing}}\dot{=}\begin{array}{c|ccccc} +Q_{\text{ab}}\dot{=}\begin{array}{c|ccccc} & \text{A} & \text{B} & \text{C} & \text{D} & \text{E} \\\hline \text{A} & 0 & \frac{1}{2} & 0 & 0 & 0 \\ \text{B} & \frac{1}{2} & 0 & \frac{1}{2} & 0 & 0 \\ @@ -175,7 +176,7 @@ Q_{\text{absorbing}}\dot{=}\begin{array}{c|ccccc} Then, \[ -N_{\text{absorbing}}=(I_5-Q_{\text{absorbing}})^{-1}=\begin{array}{c|ccccc} +N_{\text{ab}}=(I_5-Q_{\text{ab}})^{-1}=\begin{array}{c|ccccc} & \text{A} & \text{B} & \text{C} & \text{D} & \text{E} \\\hline \text{A} & \frac{5}{3} & \frac{4}{3} & 1 & \frac{2}{3} & \frac{1}{3} \\ \text{B} & \frac{4}{3} & \frac{8}{3} & 2 & \frac{4}{3} & \frac{2}{3} \\ diff --git a/main/nonergodic.tex b/main/nonergodic.tex index 2a04d68..0b6deac 100644 --- a/main/nonergodic.tex +++ b/main/nonergodic.tex @@ -62,7 +62,7 @@ Figure \ref{TruncatedPetersburg} is a truncated version of the St. Petersburg paradox. The transition probabilities between non-absorbing states are as follows: \[ -Q_{\text{truncated}}\dot{=}\begin{array}{c|ccccc} +Q_{\text{st}}\dot{=}\begin{array}{c|ccccc} & \text{S}_1 & \text{S}_2 & \text{S}_3 & \text{S}_4 & \text{S}_5 \\\hline \text{S}_1 & 0 & \frac{1}{2} & 0 & 0 & 0 \\ \text{S}_2 & 0 & 0 & \frac{1}{2} & 0 & 0 \\ @@ -73,7 +73,7 @@ Q_{\text{truncated}}\dot{=}\begin{array}{c|ccccc} \] Then, \[ -N_{\text{truncated}}=(I_5-Q_{\text{truncated}})^{-1}=\begin{array}{c|ccccc} +N_{\text{st}}=(I_5-Q_{\text{st}})^{-1}=\begin{array}{c|ccccc} & \text{S}_1 & \text{S}_2 & \text{S}_3 & \text{S}_4 & \text{S}_5 \\\hline \text{S}_1 & 1 & \frac{1}{2} & \frac{1}{4} & \frac{1}{8} & \frac{1}{16} \\ \text{S}_2 & 0 & 1 & \frac{1}{2} & \frac{1}{4} & \frac{1}{8} \\ @@ -86,6 +86,71 @@ Bases on Definition \ref{definition3}, the truncated St. Petersburg paradox is non-ergodic between non-absorbing states. +\subsection{A sufficient condition for non-ergodicity between non-absorbing states} +Based on the truncated St. Petersburg paradox, +it is easy to provide a sufficient condition for non-ergodicity between non-absorbing states. +\begin{theorem}[A sufficient condition for non-ergodicity between non-absorbing states] +Given a Markov chain with absorbing states, +suppose the size of the non-absorbing states $|S\setminus\{\text{T}\}|\geq 2$. +If the transition probabilities $Q$ between non-absorbing states satifies, +$\forall i,j \in S\setminus\{\text{T}\}$, +\begin{equation} +Q_{i,j}=\begin{cases} +0, & \text{if } i\leq j; \\ +\geq 0, & \text{otherwise.} +\end{cases} +\label{condition} +\end{equation} +Then, the Markov chain is non-ergodic between non-absorbing states. +\end{theorem} +\begin{IEEEproof} +Based on the assumption, the $Q$ matrix is an upper triangular matrix. +The product of two upper triangular matrices is still an upper triangular matrix. +Furthermore, the sum of two upper triangular matrices +is still an upper triangular matrix. +Based on Definition \ref{definitionN}, +the $N$ matrix is product and sum of upper triangular matrices. +Then, the $N$ matrix is an upper triangular matrix. +The claim now follows based on Definition \ref{definition3}. +\end{IEEEproof} + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + -- libgit2 0.26.0