2026年国际足联世界杯抽签非均匀性研究
On the non-uniformity of the 2026 FIFA World Cup draw
AI总结：
本文研究了2026年国际足联世界杯抽签的非均匀性，提出使用整数规划优化抽签过程，发现官方机制在多个度量标准下最优，但通过调整策略可显著降低非均匀性。
摘要：
体育赛事小组赛阶段通常通过在抽签中引入额外约束来增加吸引力，以促进吸引人且平衡的小组组成。例如，几个世界杯都最小化了区域内部小组比赛的数量。然而，在这种约束下，传统抽签程序可能会变得非均匀，意味着将球队分配到小组的可行分配并非等概率发生。本文量化了2026年国际足联世界杯抽签的非均匀性，包括官方抽签程序以及由四个抽签池和两种小组标签政策的所有排列所暗示的47种合理替代方案。我们展示了使用递归回溯算法进行模拟是不可行的，并提出了一种可行的整数规划实现。官方抽签机制在四个非均匀性度量标准下被发现是最优的。然而，如果组织者希望将第一个抽签池中选出的最佳球队平等对待，非均匀性可以减少超过一半。
英文摘要：
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 championships. 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 two of five non-uniformity measures, including the average distortion of the 771 non-zero assignment probabilities. Nonetheless, non-uniformity can be more than halved if the organiser aims to treat the best teams drawn from the first pot equally.
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