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.

翻译：节日礼物交换游戏是一种具有非平凡策略结构的社会常见活动。本文对该游戏机制进行了形式化处理，定义了状态空间、行动集以及偷窃链的递归结构；我们证明了游戏的终止性，并推导出用于计算不同游戏轨迹数量的算法——其增长速度远超最终分配可能性的空间。在基础机制之外，我们引入了一个修饰模型，该模型融入了不完全信息、社会成本以及基于离散选择理论和挫折-攻击文献的自适应策略。通过对24万场游戏进行全因子模拟，得到三项显著发现：隐性社会成本是攻击行为的主要调节因素，可使偷窃行为减少27%-48%，其影响超过了不确定性和策略复杂性；与预期相反，不完全信息通过不对称不确定性略微增加了偷窃行为；相关性估值放大了所有行为效应，因此，决定竞争激烈程度的是对礼物质量的共识，而非礼物本身特征。首玩家优势在所有情境下均稳健存在。