# Network Visualization of Cooperative Games
## 6 Examples
### Bankruptcy Game  
In this second example, we consider a set of players \(i\in N\), who may divide the remaining capital from a company that has gone bankrupt and was in debt to them. The characteristic function in Equation (9) defines how a coalition \(S\subseteq N\) recovers its debt: it might be the remaining amount after the rest of the lenders \(j\in N\setminus S\) have previously collected their corresponding cash \(L_{j}\) from the equity \(E\). In this example, each of the lenders \(N=\{\text{SL1},\text{SL2},\text{SL3},\text{SL4},\text{SL5},\text{ML1},\text{ML2 },\text{LL1},\text{LL2}\}\) has provided a loan of \(L=\{50,50,50,50,50,200,200,400,400\}\), respectively, to the borrower. Table 3 shows the relevance vector \(\Psi=Sh\) and the relationship matrix \(R=I\) for this game. Figure 1: Airport Game represented as \((Sh,I)\). \[v(S)=\max\left\{0,E-\sum_{j\in N\setminus S}L_{j}\right\} \tag{9}\]  
When reviewing the weighted degree metrics in Table 4, we find equal values of \(I\), \(I^{+}\), and \(I^{|\cdot|}\) due to the lack of existence of negative relationships among players in the network. In other words, all players want to meet the others in the game. In this case, the variability is higher than in the example in Section 6.1, depicted by a higher Gini Index and greater dispersion. Regarding the shape of the weighted degree distribution, it exhibits right skewness and greater clustering than a normal distribution. A lender who analyzes the game could pick the whole representation of the relationships in Figure 3, or better use its filtered visualization in Figure 4 when representing the cooperative game to outline relationships with an intensity above 25% of the maximum recorded. The filtered version would depict the top seven strongest relationships in terms of intensity. Filtering the seven most relevant relationships in terms of intensity within the graph in Figure 4 has a direct consequence, which is to leave the nodes SL1, SL2, SL3, SL4, and SL5 without any connection to any other node, thus appearing isolated in the filtered visualization of the game network. This can be an unwanted behavior, which can be addressed with filtering by nodes as well, and not only by relationships in the network. The Bankruptcy Game network visualization in Figure 3 reveals an even more evident pattern compared to the Airport Game visualization in Figure 1: here, every three-node clique that includes larger nodes displays edges with substantially higher cumulative values. For example, consider the clique formed by LL1, LL2, and ML1. The edge thickness \begin{table}
\begin{tabular}{l c c c c} \hline \(\mathbf{s_{i}}\) & \(\mathbf{R=I^{+}}\) & \(\mathbf{R=I^{-}}\) & \(\mathbf{R=I}\) & \(\mathbf{R=I^{|\cdot|}}\) \\ \hline Maximum & 400 & 0 & 400 & 400 \\ Q3 & 200 & 0 & 200 & 200 \\ Mean & 165 & 0 & 165 & 165 \\ Median & 125 & 0 & 125 & 125 \\ Q1 & 50 & 0 & 50 & 50 \\ Minimum & 50 & 0 & 50 & 50 \\ Standard Deviation & 134.2572 & 0 & 134.2572 & 134.2572 \\ Skewness & 0.7636 & 0 & 0.7636 & 0.7636 \\ Kurtosis & \(-0.8520\) & 0 & \(-0.8520\) & \(-0.8520\) \\ Gini Index & 0.4212 & 0 & 0.4212 & 0.4212 \\ \hline \end{tabular}
\end{table} 
Table 4: Descriptive metrics for the Bankruptcy Game relationship matrices.