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Multiplayer Games of War

Axel Adjei, Neil Krishnan, Elchanan Mossel

TL;DR

This work extends stochastic War models to any number of players and shows that the evolving deck configuration is equivalent to a sticky random walk on the $m$-simplex, with game termination matching absorption. Under symmetry and suitable size conditions ($n/m$ large and bounded initial hands), the absorption time—and thus the War termination time—scales as $\Theta(n^2)$. The analysis combines martingale arguments with bounds on the number of positive coordinates and extends to $f$-war variants; simulations corroborate the $n^2$ scaling, and an approximate formula $f^*(\mathbf{x})$ is proposed to estimate expected times, matching known $m=3$ results. The paper also outlines directions for tightening constants and analyzing post-loss card distributions.

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 $Θ(n^2)$. 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 $Θ(n^2)$ for any number of players.

Multiplayer Games of War

TL;DR

This work extends stochastic War models to any number of players and shows that the evolving deck configuration is equivalent to a sticky random walk on the -simplex, with game termination matching absorption. Under symmetry and suitable size conditions ( large and bounded initial hands), the absorption time—and thus the War termination time—scales as . The analysis combines martingale arguments with bounds on the number of positive coordinates and extends to -war variants; simulations corroborate the scaling, and an approximate formula is proposed to estimate expected times, matching known results. The paper also outlines directions for tightening constants and analyzing post-loss card distributions.

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 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 . In this paper, we generalize these models to {\em any number of players} . We prove that the game with players is equivalent to a sticky random walk on an -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 for any number of players.
Paper Structure (12 sections, 15 theorems, 82 equations, 1 figure, 1 table)

This paper contains 12 sections, 15 theorems, 82 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. The Sticky Random Walk
  3. Simple Sticky Random Walk
  4. Sticky Random Walk
  5. Bounds on Absorption Time
  6. Expected Amount of Time with $r$ Positive Martingales
  7. Expected absorption time
  8. Simulations
  9. Approximate Formula
  10. Motivation
  11. Formula
  12. Further Directions

Key Result

Proposition 2.2

Consider a game of $\mathfrak{p}$-war with a deck of $n$ cards where $\mathfrak{p}$ is a symmetric winning rule. Then, $(A_{i, t})_{i\in [m]}$ is a simple sticky random walk.

Figures (1)

  • Figure 1: Simulations of a real game of War with $2,$$4,$ and $13$ players in the game.

Theorems & Definitions (35)

  • Definition 1.1
  • Remark 1.3
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Corollary 2.3
  • Theorem 2.4
  • proof
  • Definition 2.5
  • Theorem 2.6
  • ...and 25 more