## 4 Data and Research Methodology  
In our study, the top five European national leagues in the 2020/21 season were analyzed, and this dataset was augmented with the results of the UEFA Champions League (UCL), the UEFA Europa League (UEL), and the UEFA Super Cup (USC) from the same season.  
\begin{table}
\begin{tabular}{l c c c} \hline \hline
**Layer** & **Neurons** & **Activation** & **Dropout Regularization** \\ \hline Input layer & 2 & Leaky ReLU & Yes, dropout rate = 0.1 \\ Hidden layer 1 & 32 & Leaky ReLU & Yes, dropout rate = 0.1 \\ Hidden layer 2 & 16 & Leaky ReLU & No \\ Output layer 1 & 1 & Linear & No \\ Output layer 2 & 1 & Sigmoid & No \\ \hline \hline \end{tabular}
\end{table}
Table 1: The number of neurons and the applied activation functions in the layers of the RankNet network designed for the study.  
Figure 2: The sequential block diagram of the RankNet network used in the study.  
In the study, the following national championships were enrolled: the English Premier League (ENG), the French Ligue 1 (FRA), the German Bundesliga (GER), the Italian Serie A (ITA) and the Spanish La Liga (ESP). We chose these leagues because they were the top 5 leagues in the season 2020/21 (and in some other seasons, too); furthermore, the numbers of teams playing in these domestic leagues and the systems for running these championships were similar. The ENG, FRA, ITA, and ESP national leagues comprise 20-20 teams, and every team plays with the others twice; therefore, these national leagues have 380 matches. In the Bundesliga, there are only 18 teams; hence, the number of matches is only 306. Altogether, we considered in this study 1826 matches from national leagues.  
In the season considered, teams from the mentioned five leagues played 53 international matches against each other in the UCL, 19 matches in the UEL, and 1 match in the USC. This resulted in 73 international matches that provided the basis for merging the individual rankings arising from the match results achieved in the national leagues. The number of matches for concatenating the national leagues' teams was not small (73), but if we compare it to the total number of matches, it did not represent a significant portion. The ratio was \(73/(1826+73)=0.0384\), which was under 4%. We note that the comparisons in the national leagues were always complete, but the unified ranking was based on incomplete comparisons. This explains the necessity of using the LLSM method instead of the eigenvector method. We checked that in the case of each national league and also in the case of all matches, the assumptions of Theorem 1 were satisfied. Therefore, the evaluations by the Thurstone and AHP with LLSM methods were mathematically established in the study. It is also worth noting that, although many teams did not play any international matches, we could fit them into the international ranking with the help of their results against teams who played in international cups.  
In [48], the authors argued for considering international matches with multiple weights. Following this recommendation, the merging was performed both without weights and with weights equal to 4. The reason for weighting, as explained by [PERSON] and [PERSON], was the following: as international matches are usually more important, they are given priority attention by teams. The ranking and weights of the teams for international matches could also serve as a basis for comparing the national championships themselves, in addition to the comparison of the teams.  
For our study, all the match data mentioned above were downloaded with the help of [52]. Additionally, Transfermarkt data [35] containing the financial values of the teams were also downloaded and were used to evaluate the results more extensively. The UEFA club coefficients were downloaded from [53].