Fairness versus transparency: The optimal design of a randomised mechanism for constrained assignment

The problem of assigning items to balanced sets of (almost) the same size emerges in various settings. Organisers of sports tournaments usually solve it with the so-called Skip mechanism, a procedure based on a random sequential draw of the teams from pots. Its main advantage is transparency since the computer-assisted calculations that ensure the satisfaction of draw constraints can be easily verified. However, the mechanism is not uniformly distributed, the valid assignments are not equally likely, which might threaten fairness. We quantify this distortion of the Skip mechanism with different orders of the pots if a group can contain at most two teams from a given set. Our study provides exact results for an arbitrary number of teams, complete enumeration if the problem size is small, as well as two real-world case studies addressed by Monte Carlo simulations. The results deliver a key insight: the best design is a decreasing order of the pots according to the number of teams affected by the constraint examined here. Sports administrators can use these findings to reduce the distortions of a draw at most a marginal price in credibility.