论文标题
醉天使和藏着魔鬼
Drunk Angel and Hiding Devil
论文作者
论文摘要
天使游戏以$ 2 $二维的无限网格的播放，$ 2 $ 2 $，天使和魔鬼。在每个转弯中，权力的天使$ c \ in \ mathbb {n} $从她的当前点$（x，y）$移动到point $（x'，y'）$，$（x'，y'）$，$ \ max \ {| x -x -x -x'|，| y -y -y -y -y'| \ y'| \ y'| \ y'| \ y'| \ y'| \ leq c $，而魔鬼选择了毁灭他的点销毁的点。然后，天使再也无法降落在这些被摧毁的点上。如果天使能够永远逃离魔鬼，而魔鬼能够以有限的转弯击败天使，那么他将获胜。在2007年证明，权力的天使至少$ 2 $总是获胜。在本文中，当天使喝醉时，我们会引起问题。她在每个回合中随机移动到力量范围内的任何点。在我们的游戏版本中，魔鬼必须以特定的有限圈笼罩天使，否则天使赢了。我们提出了一种魔鬼的策略：如果魔鬼使用这种策略，那么对于\ mathbb {n} $和$ε> 0 $，给定的$ c \，魔鬼可以在$ 1-ε$的情况下笼罩着Power $ c $的天使，并且仅在$ n $ n $ n $ n $ n $ dimimential -d $ dimemional Infinite $ n $ n时，当时$ 1 -ε$。一旦天使第一次在给定半径的$ n $维度领域以外，我们还建立了与打击时间有关的结果。在最后一部分中也显示了数值仿真结果。
The angel game is played on $2$-dimensional infinite grid by $2$ players, the angel and the devil. In each turn, the angel of power $c \in \mathbb{N}$ moves from her current point $(x, y)$ to a point $(x', y')$ which $\max\{|x - x'|, |y - y'|\} \leq c$ while the devil chooses a point to destroy in his turn. Then, the angel can no longer land on these destroyed points. The angel wins if she has a strategy to escape from the devil forever and the devil wins if he can cage the angel in his destroyed points by a finite number of turns. It was proved in 2007 that the angel of power at least $2$ always wins. In this paper, we rise the problem when the angel is drunk. She randomly moves to any point in the range of her power in each turn. In our game version, the devil must cage the angel by a given finite number of turns, otherwise, the angel wins. We present a strategy for the devil that: if the devil plays with this strategy, then for given $c \in \mathbb{N}$ and $ε> 0$, the devil can cage the angel of power $c$ with probability greater than $1 - ε$ if and only if the game is played on an $n$-dimensional infinite grid when $n \leq 2$. We also establish the results related to the hitting time once the angel is first time outside an $n$-dimensional sphere of a given radius. The numerical simulation results are also presented in the last section.
