Several sports tournaments establish restrictions on the draw of the group stage. In this case, the usual mechanism to assign the teams into groups is unevenly distributed: the valid allocations are not equally likely. We show by two illustrative examples how the lack of randomisation in the size of the groups and the inflexible treatment of a sophisticated draw constraint can increase unfairness. Based on these ideas, the draw of the European Qualifiers for the 2022 FIFA World Cup could have been closer to uniform distribution without any negative effects. Our recommendation for a careful relabelling of the pots from which the teams are drawn has a decent chance to be implemented in practice.

翻译：多项体育赛事对小组赛阶段的抽签设置了限制条件。在此情况下，将球队分配至各小组的常规机制存在分布不均的问题：有效分配方案的生成概率并不相等。我们通过两个示例说明，小组规模随机化不足以及复杂抽签约束的僵化处理如何加剧不公平性。基于这些思路，2022年国际足联世界杯欧洲区预选赛的抽签本可在不产生任何负面影响的前提下更接近均匀分布。我们提出的"对球队抽签所使用的种子队进行审慎重标号"这一建议，在实践中具有相当的可实施性。