On the fairness of the restricted group draw in the 2018 FIFA World Cup

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 usually 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. However, this procedure is known to be unfair: the feasible allocations are not equally likely. Therefore, we quantify the departure of the 2018 FIFA World Cup draw procedure from an evenly distributed random choice among all valid allocations and evaluate its effect on the probability of qualification for the knockout stage for each nation. The official draw order of Pot 1, Pot 2, Pot 3, Pot 4 turns out to be a significantly better option than the 23 other draw orders with respect to the unwanted distortions. Governing bodies in football are encouraged to make similar calculations immediately before the draw of major sporting events in order to avoid using a highly unfair draw order that can be easily improved by a simple relabelling of the pots.