## 6 Comparison of the Different Methods  
In this section, the characteristics of the three applied evaluation methods are compared. All three methods (TH, AL, and RN) are based on optimization. In one case, the search for the maximum of a likelihood function, and in another case, a minimum search based on a least squares method solves the problem. The learning process of the RankNet neural network also requires optimization, which is generally carried out using variants of the stochastic gradient descent method. Each method can provide rankings and weights,  
Figure 4: Estimated values and real average Transfermarkt (TM) values of the teams by nation. Figure 3: Correlation between estimated values and real TM values of the English teams. i.e., in addition to the relative positions of the objects to each other, the difference in their strengths is also obtained. The TH method is an additive model, and AL is a multiplicative model. RN also works as an additive model. Due to the complexity of the functions to be maximized, the TH method requires numerical optimization, whereas the AL method does not. The RN method varies the scores representing the strength of the objects step-by-step through iterations. The scale of the weights resulting from the RankNet network is larger than the usual scale of the Thurstone method. This may be due to the fact that the last layer in the base network in RankNet has a linear activation function. As a result, the RankNet method attributed significantly larger weights to the top-ranked teams and extraordinarily small weights to the bottom-ranked teams. A fundamental feature of the methods is that TH and AL treat the comparisons as a global system, which is not true for RN. In the previous chapters, each of these methods demonstrated its excellent ranking ability in the case of national leagues. In the case of the rankings based on the results of national league matches, the comparison was complete, as each team played every other team. However, if teams from different leagues were aggregated into a single ranking, the comparison was not complete, as there were only a few links between teams from different nations. TH and AL performed adequately in this incomplete comparison case, but the RankNet method did not. RankNet seemed to require more connections among the national championships to provide an appropriate unified ranking. However, the result was not surprising. In the training phase, neural networks generally need a lot of training data to learn the optimal values for a large number of parameters. Focusing on the running time, due to the step-wise training, RN required the most computational time. This was followed by the Thurstone method, due to the complexity of multivariate numerical optimization, and AL required the shortest computational time. Comparing the evaluation results to the UEFA club coefficient rankings, we observed moderate correlations in the case of the TH and AL methods, similar to TM, but the RN method showed a weak correlation. Regarding the applicability of the methods, the most complex set of conditions belonged to the Thurstone method, where only sufficient conditions have been proven so far, and not necessary and sufficient ones. Condition checking requires the investigation of the connectivity of a graph for which an effective algorithm exists. For the AL method, a necessary and sufficient condition for applicability is known, and this condition is also related to graph connectivity, but the condition itself is more often satisfied than the graph connectivity required for the Thurstone method. Although both the TH and AL methods were applicable in the analyzed case, mathematically, the AL method might also produce results in cases where TH cannot. The reason is that when constructing the AHP matrix, a match result is converted into a number, whereas in the case of the Thurstone method, the result is just a relation (worse/tie/better).