The restricted group draw problem in sports tournaments: a detailed case study

Many 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, UEFA has adopted a procedure based on a random draw assisted by the computer to indicate the group(s) available for the drawn team. It is shown how this mechanism is connected to generating permutations, and a straightforward modification is proposed to increase the excitement of the draw. The consequences of draw restrictions are investigated through the case study of the European Qualifiers to the 2022 FIFA World Cup. We assess the probability of an obscure situation that threatens the transparency of the draw and quantify the departure of the UEFA mechanism from an evenly distributed random draw among all feasible allocations.