Drunk Angel and Hiding Devil

TL;DR

This paper investigates a probabilistic variant of the angel and devil game on infinite grids, demonstrating that the devil can effectively cage a drunk angel in one or two dimensions with high probability, and analyzing related hitting times.

Contribution

It introduces a new probabilistic version of the angel and devil game, providing strategies for the devil and analyzing the game in higher dimensions with numerical simulations.

Findings

Devil can cage a drunk angel with high probability in 1D and 2D.

Strategies depend on the dimension of the grid.

Analysis of hitting times for the angel outside a sphere.

Abstract

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 $\epsilon > 0$, the devil can cage the angel of power $c$ with probability greater than $1 - \epsilon$ 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.