# Game Theory-Inspired Evolutionary Algorithm for Global Optimization
## 4 Performance Comparison and Experimental Results
### Results and Comparision Analysis
It is a remarkable fact that the StGA and the DPGA do not converge to the global optimal solution with the same standard deviation of zero in all the trials for function \(f_{13}\), which indicates that those algorithms stalled at the local optimum points, whereas the GameEA escapes and continually evolves to the global minimum. We also observed the final solutions of the GameEA with respect to the \(f_{13}\), although its average optimums are not perfect, thirty-nine parts in fifty obtain the global optimum. With respect to function \(f_{3}\), the GameEA is obviously worse than other four algorithms. When we scrutinize the 50 experimental outputs, we find that the best solutions are set to 4.43 \(\times\) 10\({}^{-12}\) and the worst optimum is set to 0.0122, which means that the final results have a wide range and, thus, a poor average and standard deviation. Furthermore, based on those results, the Friedman test [51; 52; 53], which is a non-parametric alternative to ANOVA, is conducted with SPSS 22 to check whether the StGA, the IMGA, the RTS, the DPGA, and GameEA have similar performances. The null hypothesis is set to: \"the distributions of StGA, IMGA, RTS, DPGA, and GameEA are the same\". This null hypothesis states that the observed algorithms output the same distribution and, therefore, have the same mean ranking. Table 4 shows the results of the pairwise comparisons in the hypothesis test summary. Especially, all the observed algorithms reported the optimal solution for functions \(f_{5},f_{7},f_{8}\), and \(f_{9}\), so we do not check the Friedman test on those functions. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Iteration**} & \multicolumn{2}{c}{**Optimum**} & \multicolumn{2}{c}{**StGA**} & \multicolumn{2}{c}{**IMCA**} & \multicolumn{2}{c}{**RTS**} & \multicolumn{2}{c}{**DPGA**} & \multicolumn{2}{c}{**GameEA**} \\ \cline{3-12}  & \multirow{2}{*}{**Solution**} & \multicolumn{2}{c}{**Average**} & \multicolumn{2}{c}{**Standard**} & \multicolumn{2}{c}{**Average**} & \multicolumn{2}{c}{**Standard**} & \multicolumn{2}{c}{**Average**} & \multicolumn{2}{c}{**Standard**} & \multicolumn{2}{c}{**Average**} & \multicolumn{2}{c}{**Standard**} \\  & & & & & & & & & & & & & \\ \hline \(f_{1}\) & \(15\times 10^{6}\) & 0 & 6.12 \(\times\) 10\({}^{-34}\) & 1.27 \(\times  
As seen from Table 4, all the algorithms could offer different performance for all the used functions, except\(f_{6}\) and\(f_{13}\), on the whole. It is clear that the GameEA has given the best performance amongst all four algorithms according to Table 3. For function\(f_{2}\), the decision rejected the null hypothesis according to Table 4, and the GameEA has the best performance according to Table 3, but the results of the pairwise comparison was only rejected in the case of the StGA. In addition, for function\(f_{3}\), it indicates that the GameEA has given the worst performance according to Table 3, but the Friedman test results shows that the distribution of IMGA, RTS, DPGA, and GameEA are the same according to Table 4. The null hypothesis of pairwise comparison was only rejected in the case of StGA. Additionally, for functions \(f_{6}\) and\(f_{13}\), although all the algorithms show different distributions according to the statistic results shown in Table 3, their null hypothesizes were not rejected, which means that all the algorithms could offer similar performances. In addition, for function\(f_{1}\), we checked its detailed Friedman test results, and Figure 1 and Table 5 show the detailed statistics of the Friedman test. According to Figure 1, it can be found that the GameEA states the smallest average ranking of 1.00 and the StGA reports the largest mean ranking of 4.82. According to the definition of the Friedman test, the mean ranking can be employed as an indicator as to how successful the considered algorithm