Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

When will (game) wars end?

Manan Bhatia, Byron Chin, Nitya Mani, Elchanan Mossel

Abstract

We study several variants of the classical card game war. As anyone who played this game knows, the game can take some time to terminate, but it usually does. Here, we analyze a number of asymptotic variants of the game, where the number of cards is $n$, and show that all have expected termination time of order $n^2$. This is the same expected termination time as in the game where at each turn a fair coin toss decides which player wins a card, known as Gambler's Ruin and studied by Pascal, Fermat and others in the seventeenth century.

When will (game) wars end?

Abstract

We study several variants of the classical card game war. As anyone who played this game knows, the game can take some time to terminate, but it usually does. Here, we analyze a number of asymptotic variants of the game, where the number of cards is , and show that all have expected termination time of order . This is the same expected termination time as in the game where at each turn a fair coin toss decides which player wins a card, known as Gambler's Ruin and studied by Pascal, Fermat and others in the seventeenth century.
Paper Structure (10 sections, 4 theorems, 14 equations, 1 figure)

This paper contains 10 sections, 4 theorems, 14 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Models and Main Results
  3. Related Work
  4. Random Game
  5. Deterministic gameplay
  6. Martingales
  7. Analysis of martingale war
  8. Simulations and Open questions
  9. Simulations with a standard deck of cards
  10. Open questions

Key Result

Theorem 2.6

Consider a game of $\mathfrak{p}$-war with a deck of $2n$ cards with ranks in $[2n]$ (possibly including repeated ranks) where $\mathfrak{p}$ is a symmetric winning rule. Suppose that $A(0)$ is uniformly distributed among all hands of size $A_0$. Then, $A_t$ is a simple symmetric random walk stopped

Figures (1)

  • Figure 1: Histograms of the number of rounds in three models of war

Theorems & Definitions (19)

  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Theorem 2.6
  • proof
  • Corollary 2.7
  • Definition 3.1: Martingale
  • Theorem 3.2: Optional Stopping Theorem
  • ...and 9 more