# Network Visualization of Cooperative Games
## 6 Examples
### The Airport Runway Game  
As a first example, we examine how to fairly allocate the construction cost of an airport runway among the aircraft that use it. Let \(N\) denote the set of all aircraft scheduled to land, and let \(C(i)\) be the individual runway cost incurred by aircraft \(i\). For any coalition \(S\subseteq N\), the characteristic function is defined in Equation (8).  
\[v(S)=-\max_{i\in S}\{C(i)\} \tag{8}\]  
In this example, \(v(S)\) means that the cost for a group of aircraft is determined solely by the aircraft with the highest cost in that group. Essentially, \(v(S)\) assigns to each coalition the negative of its most expensive member's cost, reflecting the idea that the overall burden of constructing the runway is governed by the maximum individual expense among the coalition's members.  
The costs to construct a runway that gives service to each airplane \(N=\{\)SM1, SM2, SM3, SM4, SM5, MD1, MD2, MD3, LG1, LG2\(\}\), are, respectively \(C=\{100,100,100,100,\)\(100,150,150,150,200,200\}\). The relevance vector \(\Psi=Sh\) for this set is calculated in Table 1, with player relationship values \(R=I\).  
Analyzing the metrics of the weighted degree in Table 2, we can see that \(I\), \(I^{+}\), and \(I^{|\cdot|}\) values coincide. This result is expected, as there are no negative relationships in terms of the interaction index: all players are willing to establish cooperation with each other. Moreover, the variability is not huge, as we can see in the low value of the Gini Index and data dispersion. The shape of the weighted degree shows a slight right-skewed distribution (skewness = 0.5793) and a higher clustering than a normal distribution (kurtosis = \(-1.1357\)).  
The network of the game's relationships is depicted without any filter in Figure 1, and by a filtered representation \((Sh_{a^{+}=0A^{-}=0.25},I_{\beta^{+}=0,\beta^{-}=0})\) in Figure 2. The latter visualization helps us to filter aircraft types in terms of player relevance: we left behind small players and focused on the planes that require the top 75% of investments. For these cases, the black color and bold thickness of the link between LG1 and LG2 symbolize a great positive impact on \(r_{i,j}\). By applying such a filter that eliminates the nodes whose relevance is below the 25 th percentile in Figure 2, we gain clarity and focus on the nodes that are most relevant, and we can also visualize more clearly the relationships between those nodes. In a scenario where we seek to focus on these types of relationships, this visualization helps us eliminate information that is not relevant to our purpose. However, this filtering must be carried out carefully so as not to exceed the amount of filtered information. A situation may arise in which an inappropriate value of the filtering thresholds eliminates information necessary for the correct interpretation of the game as a whole. An example of this is discussed in Section 6.2.  
Figure 1, representing the network visualization of this game,  
for example, in the clique formed by LG1, LG2, and MD1, where we can see at a glance that the thickness of the edges that join them is greater than that of other cliques containing them. It is easily comparable with other cliques containing LG1, such as the one with SM1 and SM2, cliques containing LG2, such as the one with SM5 and SM4, or others containing MD1, such as the one with MD2 and SM3. All off them are linked with thinner edges. This helps us to understand that the larger the size of the nodes with which another node is associated, the greater the magnitude of their relationship.