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 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 permutation generation algorithms. 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 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.