Multiplayer Games of War

TL;DR

This paper generalizes stochastic models of the game War from two players to any number of players, showing that the game’s expected termination time remains quadratic in the number of cards, regardless of the number of players.

Contribution

It extends existing models to multiple players and links the game to a new class of sticky random walks, analyzing their absorption times.

Findings

Expected termination time is Θ(n^2) for any number of players.

Models are equivalent to a sticky random walk on an m-simplex.

The absorption time of the walk determines the game's termination time.

Abstract

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.