Random matching in balanced bipartite graphs: The (un)fairness of draw mechanisms used in sports

The draw of some knockout tournaments requires finding a perfect matching in a balanced bipartite graph. The problem becomes challenging with draw constraints: the two field-proven procedures used in sports are known to be non-uniformly distributed (the feasible matchings are not equally likely), which may threaten fairness. We compare the biases of both mechanisms, each of them having two forms, for reasonable subsets of balanced bipartite graphs up to 16 nodes. A mechanism is found to dominate all others in the draw of quarterfinals under reasonable restrictions. The UEFA Champions League Round of 16 draw is verified to apply the best design among the four available options between the 2003/04 and 2023/24 seasons. However, considerable scope remains to improve the performance of these randomisation procedures, especially because they tend to distort the probabilities in the same direction and roughly with the same magnitude.