## 5 Results and Discussion
### Comparison of Different National Championships
Finally, we considered the best 20 teams in the international rankings. According to the Thurstone method using W = 4, we found eight English, four German, four Spanish, two Italian, and two French teams among them. According to the AL method, the top 20 teams included eleven English, four Spanish, two German, two French, and one Italian team. The RankNet method distributed the places uniformly: seven English, four German, three Spanish, three French, and three Italian teams were contained in the best 20 teams in the international rankings. Summarizing the observations, we could unequivocally declare that, concerning the performances, the English championship was the strongest. If we look at the financial ranking of the teams according to Transfermarkt data, the English teams were the most expensive, followed by ITA, GER, ESP, and FRA, respectively (see Table 11). Figure 3 shows an example for evaluating the correlation between the teams' real and estimated financial values. The horizontal axis presents the teams' values estimated by their performances. On the vertical axis, the TM values are shown. In this example, the evaluation was performed for the English teams, and the estimated values were computed by the AL method. For the calculations, only the match results in the national championship were taken into account. Figure 3 clearly demonstrates that in the case of poor performances, the teams were rather undervalued, and in the case of good performances, the teams were frequently overvalued in England. This phenomenon could also be observed in the case of other national championships. Let us consider the financial values of the teams averaged by country. Figure 4 shows that the Transfermarkt data presented higher values than those computed on the basis of the performance weights produced by the TH and AL methods in the case of ENG and ITA teams. In the case of GER and ESP, the TH and AL methods estimated larger values than the Transfermarkt values. Conversely, the RankNet method overestimated the English and French teams while underestimating the other countries compared to the TM values. This phenomenon aligned with the observation that RN allocated multiple weights for the top teams and only fractional weights for the bottom teams. Overall, the average values computed by the TH and AL methods were closer to the average values of the Transfermarkt data than those computed by the RN method. \begin{table}
\begin{tabular}{c c c c c c c} \hline  & \multicolumn{3}{c}{**International W = 1**} & \multicolumn{3}{c}{**International W = 4**} \\ \cline{2-7}  & **TH** & **AL** & **RN** & **TH** & **AL** & **RN** \\ \hline ENG & 39.000 & 20.800 & 43.700 & 31.600 & 20.850 & 38.400 \\ FRA & 67.850 & 65.200 & 47.900 & 71.750 & 66.100 & 50.150 \\ GER & 36.389 & 46.167 & 50.778 & 38.722 & 46.278 & 52.556 \\ ITA & 55.500 & 65.800 & 51.600 & 58.150 & 65.000 & 52.250 \\ ESP & 47.450 & 49.200 & 53.650 & 46.200 & 48.950 & 54.450 \\ \hline \end{tabular}
\end{table} 
Table 10: Average ranking positions of the teams by nation. \begin{table}
\begin{tabular}{c c} \hline  & **TM (c)** \\ \hline ENG & 457.653 m \\ FRA & 201.006 m \\ GER & 266.406 m \\ ITA & 272.250 m \\ ESP & 261.403 m \\ \hline \end{tabular}
\end{table} 
Table 11: Average values of the teams by nation. Finally, for the sake of completeness, we also evaluated the UEFA country coefficients in the 2020/21 season, downloaded from the website [56]. According to this ranking, the best championship, based on international matches, in the 2020/21 season was ENG, the second was ESP, the third was ITA, the fourth was GER, and the last was FRA among the investigated championships. The first and last countries were the same as in our evaluations. Furthermore, FRA was definitely lower, being far behind GER in terms of points.