# Network Visualization of Cooperative Games
## 6 Examples
### Voting Game  
This third example describes the Voting Game for the parties of a parliament. Consider a chamber composed of ten parties, each with the following vote distribution: party A with 29 votes, B with 26, C with 16, D with 13, E with 6, F with 3, G with 3, H with 2, I with 1, and J with 1.  
This game is modeled as a simple weighted cooperative game, represented as \(v(S)\equiv[q;w_{1},w_{2},w_{3},\ldots,w_{s}]\), where each party \(i\) has a weight \(w_{i}\) corresponding to its number of votes, the \(s\) value refers to the cardinal \(|S|\) of the coalition \(S\), and \(q\) denotes the quota required to form a majority. The characteristic function \(v(S)\), given in Equation (10), determines whether a coalition \(S\) is winning or losing. Specifically, \(v(S)=1\) if the total weight of the coalition meets or exceeds the quota, indicating that the coalition can pass decisions; otherwise, \(v(S)=0\).  
Figure 3: Bankruptcy Game visualized as \((Sh,I)\).  
\[v(S)=\begin{cases}1,&\text{if }\sum_{i\in S}w_{i}\geq q\\ 0,&\text{otherwise}\end{cases} \tag{10}\]  
The relevance vector \(\Psi=Sh\) and the relationship matrix \(R=I\) are provided in Table 5.  
This game is characterized by a perfect balance in the influence of positive and negative interactions. The third column in Table 6 shows null values, meaning that the sum of positive and negative interactions is zero in all cases. If we focus on either the positive or negative relationships, we can see identical distributions, with a moderate variability (Gini Index 0.21 and a maximum value four times higher than the minimum). The shape of the weighted degree distribution is slightly right-skewed, with a higher clustering than a normal distribution.  
The relationships of the Voting Game \((Sh,I)\) are represented in Figure 5, and those of \((Sh_{\alpha^{+}=0,\alpha^{-}=0},I_{\beta^{+}=1,\beta^{-}=0.1})\) in Figure 6.  
In the visual representation of the Voting Game in Figure 5, successful relationships between two players are clearly visible. However, positive relationships between three players, or positive relationship cliques, are not abundant. The only clique of positive relationships is the one formed by parties A, C, and E. The rest of the relationships do not form cliques of positive edges. This is because the large parties, normally necessary to form a majority, will not seek alliances with small parties that do not grant such majorities, but they will do so with medium or large parties that enable access to power.  
\begin{table}
\begin{tabular}{l c c c c} \hline \hline \(s_{i}\) & \(R=I^{+}\) & \(R=I^{-}\) & \(R=I\) & \(R=I^{|\cdot|}\) \\ \hline Maximum & 0.0286 & 0.0286 & 0 & 0.0571 \\ Q3 & 0.019 & 0.019 & 0 & 0.0381 \\ Mean & 0.0161 & 0.0161 & 0 & 0.0323 \\ Median & 0.0143 & 0.0143 & 0 & 0.0286 \\ Q1 & 0.0143 & 0.0143 & 0 & 0.0286 \\ Minimum & 0.0071 & 0.0071 & 0 & 0.0143 \\ Standard Deviation & 0.0063 & 0.0063 & 0 & 0.0127 \\ Skewness & 0.3383 & 0.3383 & 0 & 0.3359 \\ Kurtosis & \(-0.5317\) & \(-0.5317\) & 0 & \(-0.5453\) \\ Gini Index & 0.2150 & 0.2150 & 0 & 0.2143 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Descriptive metrics for the Voting Game relationship matrices.  
\begin{table}
\begin{tabular}{l l l l l l l l l l l} \hline \hline \(Sh\) & \(I\) & \(\mathbf{A}\) & \(\mathbf{B}\) & \(\mathbf{C}\) & \(\mathbf{D}\) & \(\mathbf{E}\) & \(\mathbf{F}\) & \(\mathbf{G}\) & \(\mathbf{H}\) & \(\mathbf{I}\) & \(\mathbf{J}\) \\ \hline
0.3226 & \(\mathbf{A}\) & 0 & 0.0142 & \(-0.0048