Goals, Constraints, and Transparently Fair Assignments: A Field Study of Randomization Design in the UEFA Champions League

Abstract

We analyze the design of a randomization procedure in a field setting with high stakes and substantial public interest: matching sports teams in the Union of European Football Association Champions League. While striving for fairness in the chosen lottery—giving teams similar distributions over potential partners—the designers seek to balance two conflicting forces: (i) imposing a series of combinatorially complex constraints on the feasible matches; and (ii) designing an easy-to-understand and credible randomization. We document the tournament’s solution, which focuses on sequences of uniform draws over each element in the final match, assisted by a computer to form the support for each draw. We first show that the constraints’ effects within this procedure are substantial, with shifts in expected prizes of up to a million euro and large distortions in match likelihoods of otherwise comparable team pairs. However, examining all possible counterfactual lotteries over the feasible assignments, we show that the generated inequalities are, for the most part, unavoidable and that the tournament design is close to a constrained-best. In two extensions, we outline how substantially fairer randomizations are possible when the constraints are weakened, and how the developed procedure can be adopted to more-general settings. This paper was accepted by Yan Chen, operations management.