Random matching in balanced bipartite graphs: The (un)fairness of draw mechanisms used in sports

TL;DR

This paper examines the fairness of draw mechanisms in sports tournaments that involve perfect matchings in bipartite graphs, revealing biases and proposing optimal and less distorted methods for better fairness.

Contribution

It compares biases of existing draw procedures, identifies an optimal mechanism for quarterfinals, and evaluates real-world UEFA draws for fairness improvements.

Findings

Existing draw mechanisms are biased and non-uniform.

An optimal draw mechanism exists under certain restrictions.

UEFA's Round of 16 draw is among the least biased options.

Abstract

The draw of some knockout tournaments requires finding a perfect matching in a balanced bipartite graph. The problem becomes challenging with draw constraints: the two draw procedures used in sports are known to be non-uniformly distributed (the feasible matchings are not equally likely), which may threaten fairness. We compare the biases of both mechanisms, each of them having two forms, for reasonable subsets of balanced bipartite graphs up to 16 nodes. An optimal mechanism exist in the draw of quarterfinals under reasonable restrictions. The UEFA Champions League Round of 16 draw is verified to apply the least distorted design among the four available options between the 2003/04 and 2023/24 seasons. However, a considerable scope remains to improve these randomisation procedures.