The organiser of the UEFA Champions League, one of the most prestigious football tournaments in the world, faces a non-trivial mechanism design problem each autumn: how to choose a perfect matching in a balanced bipartite graph randomly. For the sake of credibility and transparency, the Round of 16 draw consists of some discrete uniform choices from two urns whose compositions are dynamically updated with computer assistance. Even though the adopted mechanism is unevenly distributed over all valid assignments, it resembles the fairest possible lottery according to a recent result. We challenge this finding by analysing the effect of reversing the order of the urns. The optimal draw procedure is shown to be primarily dependent on the lexicographic order of degree sequences for the two sets of teams. An example is provided where exchanging the urns can reduce unfairness by one-third on average and almost halve the worst bias for all pairs of teams. Nonetheless, the current policy of starting the draw with the runners-up remains the best option if the draw order should be determined before the national associations of the clubs are known.

翻译：欧洲冠军联赛是世界上最具声望的足球赛事之一，其组织者每年秋天都面临一个重要的机制设计问题：如何随机选择一种平衡二分图中的完美匹配。为了确保公信力和透明度，16强抽签采用离散均匀选择的方式，从两个动态更新组成的瓮中抽取。尽管所采用的机制在所有有效分配中并非均匀分布，但根据近期研究，它已接近最公平的抽签方式。我们通过分析瓮的抽取顺序反转效应，对这一结论提出挑战。结果表明，最优抽签程序主要取决于两组球队的度序列的词典顺序。我们提供一个示例，说明交换瓮的顺序可将不公平性平均降低三分之一，并几乎将所有球队对的偏差减少一半。尽管如此，如果抽签顺序需要在知道俱乐部所属国家协会之前确定，当前从亚军开始抽签的政策仍然是最佳选择。