论文标题
四人赌徒的废墟问题
The 4-player gambler's ruin problem
论文作者
论文摘要
这项工作解释了如何利用P. diaconis，K。Houston-Edwards和第二位作者来估计与4-player赌徒毁灭问题有关的概率。例如，我们表明，一个非常主导的玩家（即，从剩下的玩家中分发的3个芯片开始的所有玩家）首先是$ n^{ - α} $，其中$α$约为$ 5.68 $。在$ 3 $ - 玩家游戏中，此概率是$ n^{ - 3} $。我们注意到，尝试为$α$的价值提供启发式/直观的解释是徒劳的。该值是通过与dirichlet特征值$λ$（零边界条件）相关的明确公式获得的，在单位球体上的球形球形三角形中，球形拉普拉斯人在单位$ \ mathbb s^2 $上，与与一个单位单位相对应的单位符合一个pertex的单位，将其放置在euclidean $ 3 $ 3 $ 3 $ 3 $ 3 $ 3 $ 3 $ 3 $ 3 $ 3 $ 3 $ 3。 $λ$的值是使用Grady Wright开发的有限差异算法估算的。
This work explains how to utilize earlier results by P. Diaconis, K. Houston-Edwards and the second author to estimate probabilities related to the 4-player gambler ruin problem. For instance, we show that the probability that a very dominant player (i.e., a player starting with all but 3 chips distributed among the remaining players) is first to loose is of order $N^{-α}$ where $α$ is approximately $5.68$. In the $3$-player game, this probability is or order $N^{-3}$. We note it is futile to attempt to give heuristic/intuitive explanations for the value of $α$. This value is obtained via an explicit formula relating $α$ to the Dirichlet eigenvalue $λ$ (zero boundary condition) of the spherical Laplacian in the equilateral spherical triangle on the unit sphere $\mathbb S^2$ that corresponds to a unit simplex with one vertex placed at the origin in Euclidean $3$-space. The value of $λ$ is estimated using a finite-difference-type algorithm developed by Grady Wright.
