# Network Visualization of Cooperative Games
## 6 Examples
### Myerson Games
#### 6.5.7 Myerson Star Messages Game  
Finally, we show in this section the detailed information for the Myerson Star Message Game. In the [PERSON]'s message game on a star distribution, the central hub player is pivotal for message transmission between any two peripheral players. If any peripheral player wishes to send a message to another peripheral player, it must go through the central player. This gives the central player significant strategic importance and value, since they control the flow of communication. The information we use to describe this game is shown in Table 19.  
This cooperative game is characterized by a power imbalance where the central players have significantly more influence over the game's outcomes than the peripheral players, as represented in Figures 17 and 18.  
Compared with Table 18, Table 20 shows slightly lower dispersion values, with a maximum value five times greater than the minimum weighted degree. Nonetheless, the distribution of this game is also characterized by exactly alike values of \(I\), \(I^{+}\), and \(I^{|\cdot|}\), a concentration of data points less pronounced than that observed in normal distributions (kurtosis = 5.1111), and a positive skewness (skewness = 2.6666).  
\begin{table}
\begin{tabular}{l c c c c} \hline \hline \(s_{i}\) & \(R=I^{+}\) & \(R=I^{-}\) & \(R=I\) & \(R=I^{|\cdot|}\) \\ \hline Maximum & 9 & 0 & 9 & 9 \\ Q3 & 1 & 0 & 1 & 1 \\ Mean & 1.8 & 0 & 1.8 & 1.8 \\ Median & 1 & 0 & 1 & 1 \\ Q1 & 1 & 0 & 1 & 1 \\ Minimum & 1 & 0 & 1 & 1 \\ Standard Deviation & 2.4 & 0 & 2.4 & 2.4 \\ Skewness & 2.6666 & 0 & 2.6666 & 2.6666 \\ Kurtosis & 5.1111 & 0 & 5.1111 & 5.1111 \\ Gini Index & 0.4 & 0 & 0.4 & 0.4 \\ \hline \hline \end{tabular}
\end{table}
Table 18: Descriptive metrics for the Myerson Star Cost Game relationship matrices.  
Figure 16: Myerson Star Cost Game visualized as \((Sh,I)\).  
\begin{table}
\begin{tabular}{l c c c c c} \hline \hline \(\mathbf{s_{i}}\) & \(R=I^{+}\) & \(R=I^{-}\) & \(R=I\) & \(R=I^{|\cdot|}\) \\ \hline Maximum & 45 & 0 & 45 & 45 \\ Q3 & 9 & 0 & 9 & 9 \\ Mean & 12.6 & 0 & 12.6 & 12.6 \\ Median & 9 & 0 & 9 & 9 \\ Q1 & 9 & 0 & 9 & 9 \\ Minimum & 9 & 0 & 9 & 9 \\ Standard Deviation & 10.8 & 0 & 10.8 & 10.8 \\ Skewness & 2.6666 & 0 & 2.6666 & 2.6666 \\ Kurtosis & 5.1111 & 0 & 5.1111 & 5.1111 \\ Gini Index & 0.2571 & 0 & 0.2571 & 0.2571 \\ \hline \hline \end{tabular}
\end{table}
Table 20: Descriptive metrics for the Myerson Star Message Game relationship matrices.  
Figure 17: Myerson Star message Game visualized as \((Sh,I)\).  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \(Sh\) & \(I\) & \(\mathbf{P1}\) & \(\mathbf{P2}\) & \(\mathbf{P3}\) & \(\mathbf{P4}\) & \(\mathbf{P5}\) & \(\mathbf{P6}\) & \(\mathbf{P7}\) & \(\mathbf{P8}\) & \(\mathbf{P9}\) & \(\mathbf{P10}\) \\ \hline
16.5 & \(\mathbf{P1}\) & 0 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 \\
3.1666 & \(\mathbf{P2}\) & 5 & 0 & 0.5 & 0.5 & 0.5 & 0.5 & 0.5 & 0.5 \\
3.1666 & \(\mathbf{P3}\) & 5 & 0.5 & 0 & 0.5 & 0.5 & 0.5 & 0.5 & 0.5 \\
3.1666 & \(\mathbf{P4}\) & 5 & 0.5 & 0.5 & 0 & 0.5 & 0.5 & 0.5 & 0.5 \\
3.1666 & \(\mathbf{P5}\) & 5 & 0.5 & 0.5 & 0 & 0.5 & 0.5 & 0.5 & 0.5 \\
3.1666 & \(\mathbf{P6}\) & 5 & 0.5 & 0.5 & 0.5 & 0 & 0.5 & 0.5 & 0.5 \\
3.1666 & \(\mathbf{P7}\) & 5 & 0.5 & 0.5 & 0.5 & 0.5 & 0 & 0.5 & 0.5 \\
3.1666 & \(\mathbf{P8}\) & 5 & 0.5 & 0.5 & 0.5 & 0.5 & 0 & 0.5 & 0.5 \\
3.1666 & \(\mathbf{P9}\) & 5 & 0.5 & 0.5 & 0.5 & 0.5 & 0.5 & 0 & 0.5 \\
3.1666 & \(\mathbf{P10}\) & 5 & 0.5 & 0.5 & 0.5 & 0.5 & 0.5 & 0.5 & 0 \\ \hline \hline \end{tabular}
\end{table}
Table 19: Player relevance vector \(\Psi=Sh\) and player relations \(R=I\) in the Myerson Star Message Game.