# Network Visualization of Cooperative Games
## 5 Properties of the Network Visualization of a Cooperative Game  
We may note that the representation of players who have no influence in the outcomes of a game, known as _null players_, are essential in understanding it. To provide insights into the visualization of such players in specific network representations, we highlight the following:  
**Proposition 1** (Isolated null player with empty area).: _Given a game \((N,v)\in G^{N}\), with \(i\in N\), a null player in the game \(i\) is then represented as an isolated and area-less point in \((Sh,I)\)._Proof.: Ref. [42] proved that the Shapley value fulfills the _null player_ property; therefore, \(Sh_{i}(N,v)=0\). Also, by the definition of a _null player_, all marginal contributions to player \(i\) are zero, so \(v(S\cup\{i,j\})-v(S\cup\{j\})=0\), \(v(S\cup\{i\})-v(S)=0\), and \(I_{i,j}(v)=0\) for all \(S\subseteq N\setminus\{i,j\}\), resulting in null values associated with this player in the matrix \(I\).  
The symmetries between players in a cooperative game often manifest in consistent influence across various scenarios. Such equivalence leads to similar network representations for the symmetric players, implying consistent visualization in terms of both area and links within the network. The following proposition delineates this notion of symmetry between players in specific game structures.  
**Proposition 2** (Symmetric players representation).: _Given the games \((N,v)\in G^{N}\), and with \(i,j\in N\) being two symmetric players, players \(i,j\) will have the same node area (i.e., \(\psi_{i}=\psi_{j}\)), same links (i.e., \(r_{i,j}=r_{j,i}\)), and symmetric graph representation in \((Sh,I)\)._  
Proof.: Ref. [18] already proved that symmetric players may receive an identical share using the Shapley value, with \(v(S\cup\{i,k\})=v(S\cup\{j,k\})\) for all \(k\in N\setminus\{i,j\}\), and \(v(S\cup\{i\})=v(S\cup\{j\})\); therefore, \(I_{i,k}(v)=I_{i,k}(v)\) for all \(k\in N\setminus\{i,j\}\), resulting in symmetric matrices of \(I\).  
In games termed as _inessential_, the global value of the coalitions might not differ significantly from the sum of individual values of the players. This results in a unique representation wherein each player's weight in the network directly mirrors their standalone contribution. The following proposition provides insights into the representation of players within the network for inessential games.  
**Proposition 3** (Inessential game).: _If a game \((N,v)\in G^{N}\) is inessential, then \(\psi_{i}\) will only depend on \(v(\{i\})\) and therefore \(R\) will be null for all \(i,j\in N\) in \((Sh,I)\)._  
Proof.: By definition, in an _inessential_ game the marginal contributions for all \(i\in N\) are \(v(S\cup\{i\})-v(S)=v(\{i\})\), and given that the Shapley value is a convex combination of the marginal contributions, the first result is proven. When considering \(R\), given that \(v(S\cup\{i,j\})=v(S\cup\{i\})+v(\{j\})\), and also given \(v(S\cup\{j\})=v(S)+v(\{j\})\), then \(v(S\cup\{i,j\})-v(S\cup\{i\})-v(S\cup\{j\})+v(S)=0\); therefore \(I_{i,j}(v)=0\) for all \(i,j\in N\), resulting in null values of the matrix \(I\).  
Propositions 1 and 2 can be extrapolated to the main solution concepts, specifically to all those defined as a weighted average of the marginal contributions (regardless of the weightings) and to the nucleolus defined in [43]. The same is true of Proposition 3; it can be extrapolated to any interaction measure defined as a generalization of the definition shown in Equation (2).  
If the reader needs a deeper explanation of the different scenarios proposed in this study, we should note that it is possible to implement similar demonstrations with matrices \(I^{+}\), \(I^{-}\), and \(I^{|\cdot|}\).