平衡二分图中的随机匹配：体育赛事抽签机制的（不）公平性
Random matching in balanced bipartite graphs: The (un)fairness of draw mechanisms used in sports
AI总结：
针对淘汰赛抽签中平衡二分图完美匹配的公平性问题，比较两种体育抽签机制（各两种形式）的偏差，发现欧冠16强抽签扭曲度最低且四分之一决赛存在最优机制，但现有程序仍需改进。
摘要：
部分淘汰赛抽签需在平衡二分图中寻找完美匹配。抽签约束使问题更具挑战性：体育赛事中使用的两种抽签程序已知为非均匀分布（可行匹配并非等可能），可能威胁公平性。我们比较了两种机制（各有两种形式）在节点数最多为16的合理平衡二分图子集上的偏差。在合理限制下，四分之一决赛抽签存在最优机制。2003/04至2023/24赛季期间，欧冠16强抽签被验证为四种可用选项中扭曲程度最低的设计，但这些随机化程序仍有较大改进空间。
英文摘要：
The draw of some knockout tournaments requires finding a perfect matching in a balanced bipartite graph. The problem becomes challenging with draw constraints: the two draw procedures used in sports are known to be non-uniformly distributed (the feasible matchings are not equally likely), which may threaten fairness. We compare the biases of both mechanisms, each of them having two forms, for reasonable subsets of balanced bipartite graphs up to 16 nodes. An optimal mechanism exist in the draw of quarterfinals under reasonable restrictions. The UEFA Champions League Round of 16 draw is verified to apply the least distorted design among the four available options between the 2003/04 and 2023/24 seasons. However, a considerable scope remains to improve these randomisation procedures.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
