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\begin{document}

\title{Acyclicity of Game 2048}

\author{Xingguo Chen,  Xinwen Li, Shangdong Yang, and Wenhao Wang
\thanks{Manuscript received XXXX; revised XXXX; accepted XXXX.
Date of publication XXXX; date of current version XXXX. The work was supported
by the National Natural Science Foundation, China (Nos. 62276142, 62206133, and 62202240).}
\IEEEcompsocitemizethanks{
\IEEEcompsocthanksitem X. Chen,  X. Li and S. Yang are
 with  the Jiangsu Key Laboratory of Big Data Security \&
Intelligent Processing, Nanjing University of Posts and Telecommunications,
 P.R., China \protect (e-mail:
chenxg@njupt.edu.cn).}%
\IEEEcompsocitemizethanks{
\IEEEcompsocthanksitem W. Wang is 
 with the College of Electronic Engineering, National University of Defense Technology, 
 P.R., China \protect (e-mail:
wangwenhao11@nudt.edu.cn).
\emph{(Corresponding author: W. Wang.)}%
}

}

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\markboth{IEEE Transaction on Games,~Vol.~14, No.~8, August~202X}%
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\maketitle

\begin{abstract}
In the reinforcement learning of the 2048 game, 
we observed that existing successful cases 
do not explicitly utilize exploration strategies
 such as $\epsilon-$greedy and softmax. 
 Szubert and Ja{\'s}kowski argued that
  the intrinsic randomness of the 2048 game does 
  not necessitate the use of exploration strategies.
   However, through experiments, 
   we found that incorporating the $\epsilon-$greedy
    exploration strategy into the 2048 game 
    leads to very poor learning outcomes. 
    This suggests that it's not that exploration 
    strategies are unnecessary, but rather that 
    they cannot be used effectively. 
    By combining near-optimal policies with an
     $\epsilon-$greedy exploration strategy
      and comparing the 2048 game with a maze game, 
      we discovered that in the maze game, the $\epsilon-$greedy
       exploration led to an $\epsilon-$greedy policy, 
       whereas this was not the case for the 2048 game. 
    This led us to uncover a crucial property of 2048: its acyclic nature.
    We proved that the 2048 game is acyclic between non-absorbing states.
     This is the fundamental reason why explicit exploration cannot be 
     employed in the 2048 game.
\end{abstract}

\begin{IEEEkeywords}
Acyclicity, 2048 game, ergodicity, backward learning.
\end{IEEEkeywords}


\input{main/introduction}
\input{main/background}
\input{main/acyclic}
\input{main/2048isAcyclic}
\input{main/discussion}


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