Improving the fairness of constrained assignments: The role of draw order in the UEFA Champions League

The organiser of the UEFA Champions League, one of the most prestigious football tournaments in the world, faces a non-trivial mechanism design problem each autumn: how to choose a perfect matching in a balanced bipartite graph randomly. For the sake of credibility and transparency, the Round of 16 draw consists of some discrete uniform choices from two urns whose compositions are dynamically updated with computer assistance. Even though the adopted mechanism is unevenly distributed over all valid assignments, it resembles the fairest possible lottery according to a recent result. We challenge this finding by analysing the effect of reversing the order of the urns. The optimal draw procedure is shown to be primarily dependent on the lexicographic order of degree sequences for the two sets of teams. An example is provided where exchanging the urns can reduce unfairness by one-third on average and almost halve the worst bias for all pairs of teams. Nonetheless, the current policy of starting the draw with the runners-up remains the best option if the draw order should be determined before the national associations of the clubs are known.