When will (game) wars end?

TL;DR

This paper analyzes various versions of the card game War, demonstrating that all variants have an expected termination time proportional to the square of the number of cards, similar to classical probabilistic models like Gambler's Ruin.

Contribution

It provides a rigorous asymptotic analysis of the expected duration of several War game variants, connecting them to well-known stochastic processes.

Findings

Expected termination time is of order n^2 for all variants.

The analysis links War variants to Gambler's Ruin, a classical stochastic process.

Results generalize previous understandings of game duration.

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.