The group draw of a sports tournament requires assigning teams to groups of (almost) the same size. The most important criteria for a draw procedure are balance, randomness, and transparency, which could not be satisfied simultaneously if draw constraints exist. Organisers usually use the so-called Skip mechanism, a method based on a random sequential draw of the teams from pots, in order to ensure balance and transparency. However, the Skip mechanism is non-uniformly distributed: the valid assignments are not necessarily equally likely. We quantify this distortion if a group can contain at most two teams from a given set S, which poses a serious challenge for the Skip mechanism. Our study provides exact results for an arbitrary number of teams when there are three pots and two pots contain only one team from the set S, as well as complete enumeration for small problems with three pots and at most five teams per pot. We also analyse three real-world case studies from basketball and football. It turns out that the optimal design considers the pots in decreasing order according to the number of teams in the set S. These results can be used to identify the least distorted transparent draw procedure, and decide whether the extent of non-uniformity calls for further actions.

翻译：体育赛事的小组抽签要求将参赛队伍分配到规模（几乎）相同的组别。抽签程序最重要的标准是均衡性、随机性和透明度，当存在抽签约束时，这些标准无法同时满足。组织者通常采用所谓的跳过机制（Skip mechanism），一种基于从分组池中随机顺序抽取队伍的方法，以确保均衡性和透明度。然而，跳过机制的分布不均一：有效的分配方案并非必然具有相同可能性。当某一给定集合S中的队伍在小组中最多只能有两支时，我们对这种分布失真进行了量化评估，这给跳过机制带来了严峻挑战。本研究针对以下情况提供了精确结果：当存在三个分组池且其中两个分组池仅包含集合S中的一支队伍时，适用于任意数量的参赛队伍；此外，对于三个分组池且每个分组池最多包含五支队伍的小规模问题，我们进行了完整枚举。我们还分析了篮球和足球领域的三个实际案例研究。结果表明，最优设计方案需根据集合S中队伍数量按递减顺序考虑各分组池。这些结果可用于识别失真最小的透明抽签程序，并判定不均一性的程度是否需要采取进一步措施。