Combinatorial Aspects of the Card Game War
Tanya Khovanova, Atharva Pathak

TL;DR
This paper explores the combinatorial structure of a simplified single-suit War card game, analyzing game states, sequences, and trees to understand winning strategies and enumeration of game configurations.
Contribution
It introduces new combinatorial objects and relationships specific to War, providing a detailed enumeration framework for game states and outcomes.
Findings
Defined the concept of passthroughs in the game
Established relationships among game graphs, sequences, and trees
Enumerated game states based on rounds and passthroughs
Abstract
This paper studies a single-suit version of the card game War on a finite deck of cards. There are varying methods of how players put the cards that they win back into their hands, but we primarily consider randomly putting the cards back and deterministically always putting the winning card before the losing card. The concept of a \emph{passthrough} is defined, which refers to a player playing through all cards in their hand from a particular point in the game. We consider games in which the second player wins during their first passthrough. We introduce several combinatorial objects related to the game: game graphs, win-loss sequences, win-loss binary trees, and game posets. We show how these objects relate to each other. We enumerate states depending on the number of rounds and the number of passthroughs.
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Taxonomy
TopicsArtificial Intelligence in Games · Gambling Behavior and Treatments · Digital Games and Media
