When Will (Game) Wars End?

Abstract

We study several variants of the classical card game war. As anyone who has 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 an expected termination time of order 𝑛2. This is the same expected termination time as the game where at each turn a fair coin toss decides which player wins a card, known as Gambler’s Ruin and was studied by Blaise Pascal, Pierre de Fermat and others in the seventeenth century.