Random two-sided matching

A random two-sided matching is a process by which members of two groups are matched to each other in a random way. It is often used in sports in order to match teams in knock-out tournaments. In this context, it is often called a draw, as it is implemented by drawing balls at random from a bowl, each ball representing the name of a team.

Examples

The UEFA Champions League draw

A random two-sided matching occurs in the UEFA Champions League Round of 16. ^{[1] [2]} After some games are done within 8 groups, the group winner and the group runner-up proceed to the champions league. The UEFA rules say that each winner should be paired with a runner-up. Without further constraints, this problem could easily be solved by finding a random permutation of the winners. But UEFA rules impose two additional constraints: two teams from the same group cannot be paired, and two teams from the same association cannot be paired. Thus, the goal is to choose a random matching in an incomplete bipartite graph.

The UEFA mechanism makes several draws from different bowls. At the beginning, there are:

• Bowl 1, containing identical balls each of which represents one group runner-up;
• Bowl 2, initially empty, to be filled and refilled later.
• Bowls A to H, each of which represents a group winner and contains 7 balls with the winner's name on it.

The draw proceeds as follows:

• A ball is drawn from bowl 1, and the runner-up's name is displayed;
• A computer program shows all winners that can — according to the UEFA rules — be paired with the drawn runner-up. This takes into account not only current constraints, but also constraints for future runners-up.
• From some of the bowls A to H, representing the potential winners, a single ball is taken and put in bowl 2;
• The balls in bowl 2 are shuffled. One ball is drawn, and it represents the winner matched to the previously drawn runner-up.
• Bowl 2 is emptied, and the process repeats for 8 rounds.

This procedure yields probabilities that are different than just choosing a matching at random; this creates a distortion in the matching probalities of different groups, which raises suspicion and conspiracy theories. ^{[1] [2]}

The FIFA draw

Another two-sided matching occurs in the FIFA World Cup. ^{[3] [4]} First, the runners-up are drawn in a random order. Then, each winner in turn is drawn, and it is matched to the first runner-up in the order, to which it can be matched according to the constraints.

This draw, too, produces distorted probabilities relative to the uniform-random matching. ^{[3] [4]}

See also

• Fair random assignment - one-sided matching - allocating items to agents with different preferences.

References

1. Klößner, Stefan; Becker, Martin (2013-09-01). "Odd odds: The UEFA Champions League Round of 16 draw". Journal of Quantitative Analysis in Sports. 9 (3): 249–270. doi:10.1515/jqas-2013-0016. ISSN   1559-0410.
2. Boczoń, Marta; Wilson, Alistair J. (2023-06-01). "Goals, Constraints, and Transparently Fair Assignments: A Field Study of Randomization Design in the UEFA Champions League". Management Science. 69 (6): 3474–3491. doi:10.1287/mnsc.2022.4528. ISSN   0025-1909.
3. Csató, László (2023-08-31), On the fairness of the restricted group draw in the 2018 FIFA World Cup, arXiv: 2103.11353
4. Roberts, Gareth O.; Rosenthal, Jeffrey S. (2023-01-25), Football Group Draw Probabilities and Corrections, arXiv: 2205.06578