Multiplayer Games of War

A recent paper by Bhatia, Chin, Mani, and Mossel (2026) defined stochastic processes aimed at modeling the game of War for {\em two players} with $n$ cards. That paper showed that these models, assuming uniform random decks, are equivalent to the Gambler's Ruin problem and therefore have an expected termination time of $\Theta(n^2)$. In this paper, we generalize these models to {\em any number of players} $m$. We prove that the game with $m$ players is equivalent to a sticky random walk on an $m$-simplex; therefore, the termination time is the same as the absorption time of the sticky random walk. Interestingly, it seems that this absorption time has not been analyzed before. We show that the absorption time of the walk and the termination time of the game are both $\Theta(n^2)$ for any number of players.