2021/09/28 by Csató, László · 1 citation
#62-08 #90-10 #90B90 #91B14 #Applications (stat.AP) #FOS: Computer and information sciences #FOS: Physical sciences #Physics and Society (physics.soc-ph)
paper · doi:10.48550/arxiv.2109.13785
The group draw of a sports tournament requires assigning teams to groups of (almost) the same size. The most important criteria for a draw procedure are balance, randomness, and transparency, which could not be satisfied simultaneously if draw constraints exist. Organisers usually use the so-called Skip mechanism, a method based on a random sequential draw of the teams from pots. While this method is balanced and transparent, it is non-uniformly distributed, the valid assignments are not equally likely. We quantify the distortion of the Skip mechanism with different orders of the pots if a group can contain at most two teams from a given set, which is verified to pose a serious challenge for the draw procedure. Our study provides exact results for an arbitrary number of teams, complete enumeration for small problems, 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. The findings can be used to identify the least distorted transparent draw procedure, and decide whether the extent of non-uniformity calls for further actions.