## 4 Experiment
### Results and Analysis
#### 4.4.2 Multiple Game Analysis  
We determined if these results exist across starting conditions by mapping out the rewards over 50 different starting networks and statistically analyzing the resulting reward distribution. Table 2 shows the mean, standard deviation, and 99% confidence interval for the scalarization of the cumulative game rewards and the cosine value of the game rewards. The table also includes the result of a \(t\)-test comparing the game rewards of the Pareto-Q-learning with that of the other approaches; and the results of a \(t\)-test comparing the cosine value of each approach to the desired value of 1. A graphical representation of these distributions is visible in Figures 7 and 8.  
The results show that Pareto-Q-learning and linear scalarization consistently outperform the other approaches for the cumulative rewards. A standard \(t\)-test, with a \(p\)-value of 0.05, confirms this difference. Linear scalarization performs comparable to Pareto-Q-learning with a t-statistic of 0.853. STOM and GUESS perform very poorly with results comparable to a random strategy selection. If the defender aims to merely maximize the game rewards, regardless of which goals are prioritized, the Pareto-Q-learning approach or linear scalarization give the best results.  
The cosine results, however, show that Pareto-Q-learning and linear scalarization achieve this high cumulative result by ignoring the defender's desired goal prioritization. The STOM and GUESS approaches meanwhile result in a goal distribution which is close to the optimum. The GUESS distribution is statistically not different from the desired goal prioritization. The standard \(t\)-test confirms this with GUESS \(t\)-statistic of 0.057, being the only one to exceed the \(p\)-value of 0.05. It is noteworthy that the STOM method led to a ratio that was still close to the desired goal prioritization, while the other approaches were nowhere near the distribution.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & **Mean** & **Std. Dev.** & **99\% C.I.** & \(t\)**-Test** \\ \hline \multirow{4}{*}{Game rewards} & Random & 2827.32 & 294.96 & \(\pm\)107.45 & 2.03 \(\times\) 10\({}^{-14}\) \\  & Pareto-Q-learning & 3707.52 & 627.66 & \(\pm\)228.66 & - \\  & Linear scalarization & 3727.16 & 405.20 & \(\pm\)147.61 & 0.853 \\  & STOM & 2754.14 & 317.25 & \(\pm\)115.58 & 9.60 \(\times\) 10\({}^{-16}\) \\  & GUESS & 2733.68 & 302.934 & \(\pm\)110.36 & 2.20 \(\times\) 10\({}^{-16}\) \\ \hline \multirow{4}{*}{\(f^{(1)}/f^{(2)}\) ratio} & Random & 0.831 & 0.066 & \(\pm\)0.024 & 2.77 \(\times\) 10\({}^{-17}\) \\  & Pareto-Q-learning & 0.894 & 0.032 & \(\pm\)0.012 & 1.71 \(\times\) 10\({}^{-16}\) \\ \cline{1-1}  & Linear scalarization & 0.870 & 0.036 & \(\pm\)0.013 & 2.24 \(\times\) 10\({}^{-21}\) \\ \cline{1-1}  & STOM & 0.939 & 0.051 & \(\pm\)0.019 & 0.016 \\ \cline{1-1}  & GUESS & 0.945 & 0.050 & \(\pm\)0.018 & 0.057 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Multiple game analysis results.  
Figure 7: Distribution of scalarized game rewards.  
The results clearly show that if the defender wishes to have control over goal prioritization, the GUESS approach is the best. However the fairer distribution of rewards may not be worth the loss in cumulative rewards. The Pareto-Q-learning approach still appears to strike the best balance between the conflicting goals, with a high cumulative reward and a respectable goal distribution.