# Network Visualization of Cooperative Games
## 6 Examples
### Myerson Games
#### 6.5.3 Myerson Cycle Messages Game  
The focus of this game is on rewarding coalitions that can create the most communication pairs. When a player who does not communicate with any other player joins the game, it poses a disadvantage to the existing players. This is because they will not be able to form a communication pair with the newcomer, and consequently, the benefits will have to be divided among more players. However, if a player joins a coalition that already has a member, their contribution is advantageous. This is because they can form at least one additional communication pair. We see this reflected in Figure 11, where two players will have a positive relationship if they share a link with a third one. Per contra, when two players do not share any link with a third common node in \(\gamma\), their relationship is negative, as shown in Figure 12.  
The game relationships and players' relevance are represented in Table 11. As it has the same structure as the Myerson Cycle Cost Game in the example in Section 6.5.2, we conclude that players cooperate in a parity-based environment, where all players have equal influence and benefit from their connections. As in the Myerson Cycle Cost Game, we see that the third column of Table 12, representing the sum of all interactions in each node, is greater than zero in all quartiles. \begin{table}
\begin{tabular}{l c 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
4.5 & \(\mathbf{P1}\) & 0 & 2.6710 & 0.9531 & \(-0.1397\) & \(-0.7564\) & \(-0.9564\) & \(-0.7564\) & \(-0.1397\) & 0.9531 & 2.6710 \\
4.5 & \(\mathbf{P2}\) & 2.6710 & 0 & 2.6710 & 0.9531 & \(-0.1397\) & \(-0.7564\) & \(-0.9564\) & \(-0.7564\) & \(-0.1397\) & 0.9531 \\
4.5 & \(\mathbf{P3}\) & 0.9531 & 2.6710 & 0 & 2.6710 & 0.9531 & \(-0.1397\) & \(-0.7564\) & \(-0.7564\) & \(-0.1397\) \\
4.5 & \(\mathbf{P4}\) & \(-0.1397\) & 0.9531 & 2.6710 & 0 & 2.6710 & 0.9531 & \(-0.1397\) & \(-0.7564\) & \(-0.7564\) \\
4.5 & \(\mathbf{P5}\) & \(-0.7564\) & \(-0.1397\) & 0.9531 & 2.6710 & 0 & 2.6710 & 0.9531 & \(-0.1397\) & \(-0.7564\) \\
4.5 & \(\mathbf{P6}\) & \(-0.7564\) & \(-0.7564\) & \(-0.1397\) & 0.9531 & 2.6710 & 0 & 2.6710 & 0.9531 & \(-0.1397\) \\
4.5 & \(\mathbf{P7}\) & \(-0.7564\) & \(-0.9564\) & \(-0.7564\) & \(-0.1397\) & 0.9531 & 2.6710 & 0 & 2.6710 & 0.9531 & \(-0.1397\) \\
4.5 & \(\mathbf{P8}\) & \(-0.1397\) & \(-0.7564\) & \(-0.9564\) & \(-0.7564\) & \(-0.1397\) & 0.9531 & 2.6710 & 0 & 2.6710 & 0.9531 \\
4.5 & \(\mathbf{P9}\) & 0.9531 & \(-0.1397\) & \(-0.7564\) & \(-0.9564\) & \(-0.7564\) & \(-0.1397\) & 0.9531 & 2.6710 & 0 & 2.6710 \\
4.5 & \(\mathbf{P10}\) & 2.6710 & 0.9531 & \(-0.1397\) & \(-0.7564\) & \(-0.9564\) & \(-0.7564\) & \(-0.1397\) & 0.9531 & 2.6710 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Player relevance vector \(\Psi=Sh\) and player relationship matrix \(R=I\) in the Myerson Cycle Message Game. Figure 11: Myerson Cycle Message Game visualized as \((Sh_{\pi^{+}=0,\alpha^{-}=0},I_{\beta^{+}=0,\delta^{-}=1})\). \begin{table}
\begin{tabular}{l c c c c} \hline \hline \(s_{i}\) & \(R=I^{+}\) & \(R=I^{-}\) & \(R=I\) & \(R=I^{|\cdot|}\) \\ \hline Maximum & 7.2484 & 2.7484 & 4.5 & 9.9968 \\ Q3 & 7.2484 & 2.7484 & 4.5 & 9.9968 \\ Mean & 7.2484 & 2.7484 & 4.5 & 9.9968 \\ Median & 7.2484 & 2.7484 & 4.5 & 9.9968 \\ Q1 & 7.2484 & 2.7484 & 4.5 & 9.9968 \\ Minimum & 7.2484 & 2.7484 & 4.5 & 9.9968 \\ Standard Deviation & 0 & 0 & 0 & 0 \\ Skewness & 0 & 0 & 0 & 0 \\ Kurtosis & 0 & 0 & 0 & 0 \\ Gini Index & 0 & 0 & 0 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Descriptive metrics for the Myerson Cycle Message Game relationship matrices. players who are immediate neighbors in the \(\gamma\) topology, and also with those who are two hops away in this topology. In the next examples, we can also see this behavior more clearly.