The group stage of a sports tournament is often made more appealing by introducing additional constraints in the group draw that promote an attractive and balanced group composition. For example, the number of intra-regional group matches is minimised in several World Cups. However, under such constraints, the traditional draw procedure may become non-uniform, meaning that the feasible allocations of the teams into groups are not equally likely to occur. Our paper quantifies this non-uniformity of the 2026 FIFA World Cup draw for the official draw procedure, as well as for 47 reasonable alternatives implied by all permutations of the four pots and two group labelling policies. We show why simulating with a recursive backtracking algorithm is intractable, and propose a workable implementation using integer programming. The official draw mechanism is found to be optimal based on four measures of non-uniformity. Nonetheless, non-uniformity can be more than halved if the organiser aims to treat the best teams drawn from the first pot equally.

翻译：体育赛事的小组赛阶段常通过引入额外抽签约束以提升吸引力并促进小组构成的平衡性与观赏性。例如，多届世界杯均致力于最小化同区域球队在小组内的比赛数量。然而，在此类约束下，传统抽签程序可能变得非均匀，即球队分配到各小组的可行方案并非等概率出现。本文针对2026年国际足联世界杯的官方抽签程序，以及由四个抽签池与两种小组标签策略的所有排列组合所隐含的47种合理替代方案，量化了其非均匀性。我们阐释了为何采用递归回溯算法进行模拟具有计算不可行性，并提出一种基于整数规划的可行实施方案。基于四种非均匀性度量指标，官方抽签机制被证明是最优的。尽管如此，若赛事组织者旨在平等对待从第一抽签池中抽出的最优球队，非均匀性可降低超过一半。