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

L\'aszl\'o Csat\'o

Mar 2023

摘要

The organiser of the UEFA Champions League, one of the most prestigiousfootball tournaments in the world, faces a non-trivial mechanism design problemeach autumn: how to choose a perfect matching in a balanced bipartite graphrandomly. For the sake of credibility and transparency, the Round of 16 drawconsists of some discrete uniform choices from two urns whose compositions aredynamically updated with computer assistance. Even though the adopted mechanismis unevenly distributed over all valid assignments, it resembles the fairestpossible lottery according to a recent result. We challenge this finding byanalysing the effect of reversing the order of the urns. The optimal drawprocedure is shown to be primarily dependent on the lexicographic order ofdegree sequences for the two sets of teams. An example is provided whereexchanging the urns can reduce unfairness by one-third on average and almosthalve the worst bias for all pairs of teams. Nonetheless, the current policy ofstarting the draw with the runners-up remains the best option if the draw ordershould be determined before the national associations of the clubs are known.

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