How to improve the fairness of group draw in sports tournaments

Several sports tournaments establish restrictions on the draw of the group stage. In this case, the usual mechanism to assign the teams into groups is unevenly distributed: the valid allocations are not equally likely. We show by two illustrative examples how the lack of randomisation in the size of the groups and the inflexible treatment of a sophisticated draw constraint can increase unfairness. Based on these ideas, the draw of the European Qualifiers for the 2022 FIFA World Cup could have been closer to uniform distribution without any negative effects. Our recommendation for a careful relabelling of the pots from which the teams are drawn has a decent chance to be implemented in practice.