UEFA could have easily improved the fairness of the European Qualifiers to the 2022 FIFA World Cup

Some recent qualifying tournaments, organised by the Union of European Football Associations (UEFA), establish restrictions in the draw of the group stage. In this case, the official mechanism used to assign the teams into groups is unevenly distributed, meaning that not all valid allocations are equally likely. Motivated by a simple illustrative example, the draw of the European Qualifiers to the 2022 FIFA World Cup is proved via simulations to exhibit a systematic bias: certain teams should play more matches on average only due to the design of the draw procedure. However, simple randomisation in the labelling of the groups can reduce that unfairness without any negative effects. Since UEFA has recently introduced an analogous reform, our recommendation has a good chance to be implemented soon.