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

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 $m$.