# Identification of Players Ranking in E-Sport
## 5 Results
### Final Model  
CS: GO Players assessment model finally determines the uniqueness of the Counter-Strike Global: Offensive player by placing him in the final ranking, based on previous partial assessments. The final model for the players' assessment has three aggregated input variables. The output variable from the Effectiveness per round assessment, Frag gaining assessment, and the output variable from the Failures per round assessment were applied. The aggregated variables \(P_{1}\) and \(P_{2}\) are both profit type, whereas the \(P_{3}\) is cost type. The triangular fuzzy numbers of parameter \(P_{1}\) is presented in Figure 16, \(P_{2}\) in Figure 17, and \(P_{3}\) in Figure 18.  
Figure 16: Visualization of Effectiveness per round assessment model (\(P_{1}\)) and triangular fuzzy numbers 0.1 (\(P_{11}\)) and 0.9 (\(P_{12}\)). Figure 17: Visualization of Frag gaining assessment model (\(P_{2}\)) and triangular fuzzy numbers 0.2 (\(P_{21}\)) and 0.9 (\(P_{22}\)). Figure 15: The relation diagram of Average assists per round (\(C_{5}\)) for assessment \(P_{3}\) (**left side**) and Average deaths per round (\(C_{6}\)) for assessment \(P_{3}\) (**right side**). In the considered set of parameters there were players with: Effectiveness per round (\(P_{1}\)) with the values of the support of the triangular fuzzy number from 0.2 (\(P_{11}\)) to 1 (\(P_{12}\)), Frag gaining (\(P_{2}\)) with the values of the support of the triangular fuzzy number from 0.1 (\(P_{21}\)) to 0.9 (\(P_{22}\)), Failures per round (\(P_{3}\)) with the values of the support of the triangular fuzzy number from 0.2 (\(P_{31}\)) to 0.9 (\(P_{32}\)). The comparison of the 28 pairwise of the 8 characteristic objects were executed. Consequently, the Matrix of Expert Judgment (\(MEJ\)) was defined as (18), where each \(\alpha_{ij}\) value was calculated using Equation (11). \[MEJ=\left( \begin{array}{cccccccc}0.5&0&0&0&1&0&0&0\\ 1&0.5&1&0&1&1&1&0\\ 1&0&0.5&0&1&0&1&0\\ 1&1&1&0.5&1&1&1&1\\ 0&0&0&0&0.5&0&0&0\\ 1&0&1&0&1&0.5&1&0\\ 1&0&0&0&1&0&0.5&0\\ 1&1&1&0&1&1&1&0.5\end{array} \right). \tag{18}\]  
As a result, the vector of the Summed Judgements (\(SJ\)) was calculated using Equation (12), and it was employed to determine the final values of preference (\(P\)), which are presented in Table 9. The characteristic objects \(CO_{1}\)-\(CO_{8}\) presented in Table 9 are generated using the Cartesian product of the fuzzy numbers' cores of related models \(P_{1}\), \(P_{2}\), and \(P_{3}\). The highest value of preference \(P\) received \(CO_{4}\) with a triangular fuzzy number of parameter \(P_{1}\) valued 0.9 (\(P_{12}\)), with a triangular fuzzy number of parameter \(P_{2}\) valued 0.9 (\(P_{22}\)), and with a triangular fuzzy number of parameter \(P_{3}\) valued 0.2 (\(P_{32}\)). The lowest value of preference \(P\) fell to \(CO_{5}\) with a triangular fuzzy number of parameter \(P_{1}\) valued 0.1 (\(P_{11}\)), with a triangular fuzzy number of parameter \(P_{2}\) valued 0.2 (\(P_{21}\)), and with a triangular fuzzy number of parameter \(P_{3}\) valued 0.8 (\(P_{31}\)). With an increase in the value of the parameter \(P_{2}\), the preference increases more significantly than with an increase in the value of the parameters \(P_{1}\) and \(P_{3}\). It means that \(P_{2}\) has the greatest impact on the assessment of the \(P\) model compared to the other two parameters. \begin{table}
\begin{tabular}{l c c c c} \hline \hline \(CO_{i}\) & \(P_{1}\) & \(P_{2}\) & \(P_{3}\) & \(P\) \\ \hline \(CO_{1}\) & 0.1 & 0.2 & 0.2 & 0.1429 \\ \(CO_{2}\) & 0.1 & 0.9 & 0.2 & 0.7143 \\ \(CO_{3}\) & 0.9 & 0.2 & 0.2 & 0.4286 \\ \(CO_{4}\) & 0.9 & 0.9 & 0.2 & 1.0000 \\ \(CO_{5}\) & 0.1 & 0.2 & 0.8 & 0.0000 \\ \(CO_{6}\) & 0.1 & 0.9 & 0.8 & 0.5714 \\ \(CO_{7}\) & 0.9 & 0.2 & 0.8 & 0.2857 \\ \(CO_{8}\) & 0.9 & 0.9 & 0.8 & 0.8571 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Overview of characteristic objects \(CO\), vector \(P\) values for the Counter-Strike: Global Offensive (CS: GO) Players assessment model.