Seeding the UEFA Champions League participants: evaluation of the reforms

A.1 National association coefficient

All clubs participating in the Champions League and Europa League collect points for the national association they represent. In the qualifying rounds, each win adds 1 point to the association’s total, a draw – 0.5 points, a loss – 0 points. Starting from the group stage, these numbers are 2, 1 and 0, respectively. After the season, the total number of points is divided by the number of participants from this association. The resulting average is the association’s yearly coefficient. It is calculated over the last 5 seasons; the sum is an association’s 5-year coefficient.

A.2 Club coefficient

A UEFA team coefficient consists of two terms: number of points gained by this team in the Champions League and Europa League for the last 5 years and 20% of the coefficient of the association this club belongs to. Starting from the group stage, a win adds 2 points to the team’s coefficient, a draw – 1 point, a loss – 0 points. Bonus points are awarded for qualification into the latter stages of the tournaments. In the Champions League, it’s 4 points for reaching the group stage, plus 5 points for the Last-16 stage, plus 1 point for the quarterfinals, plus 1 point for the semifinals, plus 1 point for the final. In the Europa League, it’s 2 points for reaching the group stage, plus 1 point for the quarterfinals, plus 1 point for the semifinals, plus 1 point for the final. As an exception from all other bonus points, 2 points for reaching the group stage of the Europa League are partly awarded only if a team fails to obtain more than 2 points. More precisely, if a team gets x normal points during the group stage, it gets max(x,2) points instead. Thus, a total of 2 points is a guaranteed minimum number for the teams that reached the group stage of the Europa League. The participants that fail to reach the group stage get points on a different basis. Namely, a team that leaves the Champions League from the 1st qualifying round gets a fixed amount of 0.5 points, from the 2nd qualifying round – 1 point. A team that finishes its participation in the Europa League in the 1st qualifying round gets 0.25 points, in the 2nd qualifying round – 0.5 points, in the 3rd qualifying round – 1 point, in the 4th qualifying round – 1.5 points.

A.3 Monte-Carlo specification

The starting values of parameters υ¯ are random. On i-th Monte-Carlo step, the new set of parameters υ¯i+1 is tested. The new set of parameters υ¯i+1 is generated by adding a random normally distributed fluctuation to the previous set υ¯i:

where T_MC is Metropolis “temperature” (it regulates the degree of allowed fluctuations in the system), and σ is a coefficient, the value of which is chosen to obtain a mean acceptance ratio equal to 0.5. The step acceptance probability is equal to

During optimization, we decreased T_MC from 106 to 10−5 in 106 steps, which corresponds to a random walk in the parameter space at the beginning of optimization and negligibly small parameter fluctuations in the end.

Since scores follow a Poisson distribution, λ_h and λ_a must always be positive; parameters υ¯ that lead to negative λ_h or λ_a are considered invalid. During Monte-Carlo optimization, we used only valid sets of parameters υ¯. New fluctuations (and, thus, new sets of parameter values) were generated at each step until a valid set was obtained.