Several sports tournaments contain a round-robin group stage where the teams are assigned to groups subject to some constraints. Since finding an allocation of the teams that satisfies the established criteria is non-trivial, the organisers use a computer-assisted random draw to avoid any dead end, a situation when the teams still to be drawn cannot be assigned to the remaining empty slots. It is shown how this mechanism is connected to systematically generate all permutations of a given sequence, a well-known problem in computer science. We quantify the departure of the 2018 FIFA World Cup draw procedure from an evenly distributed random choice among all feasible allocations and evaluate its effect on the probability of qualification for the knockout stage for each nation. The draw order of Pot 1, Pot 2, Pot 4, Pot 3 turns out to be a better option than the official rule (Pot 1, Pot 2, Pot 3, Pot 4) with respect to these unwanted distortions. Governing bodies in football are encouraged to make similar calculations before the draw of major sporting events in order to find an optimal draw order by appropriate labelling of the pots.
翻译:几个体育锦标赛中包含一个圆形球赛组阶段,将球队分配给受某些限制的团体。由于找到符合既定标准的球队的分配是非三重的,因此组织者使用计算机辅助随机抽抽,以避免任何死胡同,这样,仍待抽选的球队无法被分配到其余的空位上。可以证明这一机制是如何连接到系统地产生一个特定序列的所有变异的,这是计算机科学中众所周知的一个问题。我们量化2018年国际足联世界杯的出发程序,从所有可行拨款中均匀分配随机选择,并评估其对每个国家击球阶段资格可能性的影响。Pot 1、Pot 2、Pot 4、Pot 3的抽奖令比官方规则(Pot 2、Pot 3、Pot 4)中关于这些不想要的扭曲的一条规则(Pot 2、Pot 3、Pot 4)的提奖令比官方规则更好。鼓励足球管理机构在重大体育赛事开始前进行类似的计算,以便通过适当标签找到最佳的提法。