Finiteness in the Card Game of War

TL;DR

This paper investigates the card game of War, demonstrating that allowing players to occasionally change the order of returning cards results in a finite expected game length, supported by graph-theoretic analysis.

Contribution

It proves that with occasional rule modifications, the game of War has a finite expected duration, extending understanding of its chaotic dynamics.

Findings

Expected game length is finite under certain rule variations.

The game can be modeled as a graph with finite paths to termination.

Changing return order influences the game's long-term behavior.

Abstract

The game of war is one of the most popular international children's card games. In the beginning of the game, the pack is split into two parts, then on each move the players reveal their top cards. The player having the highest card collects both and returns them to the bottom of his hand. The player left with no cards loses. Those who played this game in their childhood did not always have enough patience to wait until the end of the game. A player who has collected almost all the cards can lose all but a few cards in the next 3 minutes. That way the children essentially conduct mathematical experiments observing chaotic dynamics. However, it is not quite so, as the rules of the game do not prescribe the order in which the winning player will put his take to the bottom of his hand: own card, then rival's or vice versa: rival's card, then own. We provide an example of a cycling game with fixed rules. Assume now that each player can seldom but regularly change the returning order. We have managed to prove that in this case the mathematical expectation of the length of the game is finite. In principle it is equivalent to the graph of the game, which has got edges corresponding to all acceptable transitions, having got the following property: from each initial configuration there is at least one path to the end of the game.