In this paper, we study the total displacement statistic of parking functions from the perspective of cooperative game theory. We introduce parking games, which are coalitional cost-sharing games in characteristic function form derived from the total displacement statistic. We show that parking games are supermodular cost-sharing games, indicating that cooperation is difficult (i.e., their core is empty). Next, we study their Shapley value, which formalizes a notion of "fair" cost-sharing and amounts to charging each car for its expected marginal displacement under a random arrival order. Our main contribution is a polynomial-time algorithm to compute the Shapley value of parking games, in contrast with known hardness results on computing the Shapley value of arbitrary games. The algorithm leverages the permutation-invariance of total displacement, combinatorial enumeration, and dynamic programming. We conclude with open questions around alternative solution concepts for supermodular cost-sharing games and connections to other areas in combinatorics.

翻译：本文从合作博弈论的视角研究了停放函数的总位移统计量。我们引入停放博弈——一种基于总位移统计量的特征函数形式的联盟成本分摊博弈。研究表明，停放博弈是超模成本分摊博弈，这意味着合作难以实现（即其核心为空）。接下来，我们分析了其沙普利值，该值形式化了“公平”成本分摊的概念，相当于对每辆车在随机到达顺序下的预期边际位移进行收费。我们的主要贡献在于提出了一种多项式时间算法来计算停放博弈的沙普利值，这与已知一般博弈中沙普利值计算的困难性结果形成对比。该算法利用了总位移的置换不变性、组合枚举和动态规划。最后，我们提出了关于超模成本分摊博弈的替代解概念及其与组合学其他领域联系的开放性问题。