Finiteness in the Card Game of War

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.

💡 Research Summary

The paper investigates the classic children’s card game “War” from a rigorous mathematical perspective, focusing on the long‑standing question of whether the game can continue indefinitely. In the standard deterministic version, after each battle the winner places the two revealed cards at the bottom of his hand in a fixed order—either his own card first followed by the opponent’s, or the reverse. It is well known that with a fixed ordering certain initial deck splits generate cycles: the same configurations repeat forever and the game never reaches a terminal state where one player has no cards. The authors first illustrate such a cycling example, confirming that the state‑transition graph of the deterministic game contains closed directed cycles.

To break these cycles they introduce a minimal amount of randomness: on each turn each player may, with a small but positive probability p (the same for both players in the analysis), swap the order in which the two cards are returned to the bottom of his hand. This “seldom but regular” change creates a set of admissible transitions that is larger than the deterministic set, turning the game into a finite Markov chain on the state space S consisting of all possible ordered splits of the deck. The size of S is finite ((2n)!/(n!·n!) for a deck of 2n cards), guaranteeing that the chain has a well‑defined absorbing state – the configuration where one player’s hand is empty.

The core of the proof proceeds in two steps. First, the authors show that for every state s∈S there exists at least one directed path to the absorbing state. They exploit a monotonicity property: whenever the player with the larger hand wins a battle, his hand size increases by at least one. By judiciously choosing the return order (which is always possible because the players can flip the order with probability p), the player who currently holds the most cards can be forced to capture an extra card from the opponent, thereby strictly increasing the gap between the two hands. Repeating this argument finitely many times inevitably yields a state where one player possesses all cards, which is directly adjacent to the absorbing state. Consequently, the transition graph, enriched by the random order‑swap edges, is “absorbing‑connected”: from any configuration there is a finite sequence of moves leading to termination.

Second, the authors establish that the expected time to absorption is finite. Since every admissible edge is taken with probability at least p_min>0 (where p_min is the smallest probability among all possible order‑swap choices), any particular finite path of length L is traversed with probability at least p_min^L. Let d_max denote the maximal length of a shortest path from any state to the absorbing state; d_max is bounded by the size of the state space, which is exponential in n but still finite. Standard results for absorbing Markov chains then give an upper bound on the expected hitting time of order O(d_max / p_min). In practical terms, even a tiny p (e.g., 0.01) reduces the expected number of rounds to a few thousand, far below the astronomically large numbers that can arise in the deterministic game.

The paper supplements the theoretical analysis with Monte‑Carlo simulations. The experiments confirm two key observations: (1) as p approaches zero the mean game length grows sharply, reflecting the underlying deterministic cycles; however, for any p>0 the mean length remains bounded and grows only modestly with deck size; (2) configurations that are cyclic under a fixed ordering are quickly “escaped” when the occasional order swap is allowed, because the random edge provides a “breakout” from the cycle.

In conclusion, the authors prove that introducing a small, positive probability of reversing the card‑return order eliminates infinite loops in War and guarantees a finite expected game duration. The result demonstrates how a minimal injection of randomness can convert a non‑terminating deterministic system into an absorbing stochastic process. This insight not only resolves a classic puzzle about a popular children’s game but also illustrates a broader principle: in finite combinatorial games, occasional nondeterministic moves can ensure convergence to terminal states, a concept that may be applicable to other card games, board games, and dynamical systems exhibiting cyclic behavior.