Formal specification and behavioral simulation of the holiday gift exchange game

Abstract: The holiday gift exchange game is a familiar social institution with nontrivial strategic structure. We provide a formal treatment of the game's mechanics, defining the state space, action sets, and the recursive structure of stealing chains; we prove termination and derive an algorithm for counting distinct game trajectories, which grow far faster than the space of possible final allocations. Beyond the base mechanics, we introduce a decorated model incorporating partial information, social costs, and adaptive strategies grounded in discrete choice theory and the frustration-aggression literature. A full factorial simulation of 240,000 games yields three findings of note: implicit social costs are the dominant regulator of aggression, reducing stealing by 27--48\% and outweighing both uncertainty and strategic sophistication; partial information, contrary to expectation, slightly increases stealing through asymmetric uncertainty; correlated valuations amplify every behavioral effect, so that consensus about gift quality, rather than the features themselves, is what intensifies competition. The first-player advantage is robust across all conditions.