## 7 Conclusions  
This paper investigated the performance of different ranking methods in the case of complete and incomplete comparisons. The investigated methods were the Thurstone method with ties, the analytic hierarchy process with logarithmic least squares method, and the RankNet neural network. Since one of the possible application areas of the investigated methods is the ranking of football teams, the performance of the methods and their characteristics were evaluated by ranking the teams of the top five European football leagues and constructing a composite ranking of the teams playing in these leagues. In addition, we ranked the national championships for the five nations under study based on their estimated strengths and compared the results obtained by the different methods. A further part of the research was to investigate the weighting factor for international matches in the case of aggregated ranking based on incomplete comparisons. The methods were compared through the rankings and ratings resulting from the different methods. During the evaluations, both the positions of the teams in the rankings and the correlation of the rankings were considered. Looking at the rankings of the teams and leagues, it was clear that in the 2020/2021 season, Manchester City was the best team in Europe. Regarding the national championshipships, the Premier League was the strongest among the five investigated championships (English, German, Italian, French, and Spanish). As a result of our analysis, we established that in the case of national championships, all three methods performed equally well. In the case of rank aggregation based on incomplete comparisons, the RankNet network performed worse than the other two methods tested. The RankNet method preferred the French teams to the German, Italian, and Spanish teams. We concluded that the RankNet method did not provide a trustworthy international ranking due to the few connections between the championships. A further characteristic of the RankNet network was that it tended to overweight the best teams compared to the other two methods. Furthermore, we found in our study that in the case of the aggregation of the rankings, the ranking method affected the ranking results more than the choice of weights for the linking comparisons. Larger weights caused noticeable modifications in the rankings in the case of the Thurstone method, but the result of the analytic hierarchy process with logarithmic least squares method was almost unaffected by the applied weights. In addition, we also compared the methods' properties (e.g., running time and stochastic nature). We concluded that RankNet was the least recommended method of the three investigated methods for combining rankings based on sparse data. Its application problems arose mainly from the small amount of data available for ranking. As it is a neural network, it would need more data for aggregation. This research had some obvious limitations. The results were instantaneous exposures, as they were based on the matches of a single (2020/2021) season and confined to the teams in the five investigated leagues. No temporal changes were taken into account. It would be interesting to analyze the changes from one season to another. Further important aspects (such as home-field advantage and differences in the number of goals) could also be considered. These could all be the topic of further research. The methods examined in this article have a wide range of applications. Ranking based on pairwise comparisons and the aggregation of rankings based on incomplete comparisons can be applied to the results of arbitrary sports competitions and championships, provided that there are some (in our study 3.84%) linking results between the participating competitors/teams. The extent to which the amount of comparisons affects the performance of the RankNet algorithm that produced the weakest results will be the subject of a subsequent study.