On the expected absorption times of sticky random walks and multiple players war games

Axel Adjei, Elchanan Mossel
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Abstract

A recent paper by Bhatia, Chin, Mani, and Mossel (2024) defined stochastic processes which aim to model the game of war for two players for $n$ cards. They showed that these models are equivalent to gambler's ruin and therefore have expected termination time of $\Theta(n^2)$. In this paper, we generalize these model to any number of players $m$. We prove for the game with $m$ players is equivalent to a sticky random walk on an $m$-simplex. We show that this implies that the expected termination time is $O(n^2)$. We further provide a lower bound of $\Omega\left(\frac{n^2}{m^2}\right)$. We conjecture that when $m$ divides $n$, and $n > m$ the termination time or the war game and the absorption times of the sticky random walk are in fact $\Theta(n^2)$ uniformly in $m$.
论粘性随机漫步和多人战争博弈的预期吸收时间
Bhatia、Chin、Mani 和 Mossel(2024 年)最近发表的一篇论文定义了随机过程,旨在对 $n$ 纸牌的双人战争游戏进行建模。在本文中,我们将这些模型推广到任意数目的玩家 $m$。我们证明有 $m$ 玩家的博弈等同于 $m$ 复数上的粘性随机行走。我们证明,这意味着预期终止时间为 $O(n^2)$。我们进一步提供了一个下限:$\Omega\left(\frac{n^2}{m^2}\right)$。我们猜想,当$m$除以$n$,且$n > m$时,战争博弈的终止时间和粘性随机游走的吸收时间实际上在$m$内均匀为$\theta(n^2)$。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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