# Network Visualization of Cooperative Games
## 3 Players' Relationship Representation  
Understanding how to capture the interaction between variables, individuals, or criteria in models and information systems--beyond their individual effects--has been a subject of study for more than 50 years. Within the frameworks of game theory, fuzzy measures, and capacity measures, interaction indexes are widely used to quantify the joint influence of two or more variables/individuals in a system and have been applied in multiple contexts, including multi-criteria decision-making, game theory, and machine learning, where understanding variable interdependence is crucial for improving predictive models and decision-making processes.  
Based on the same principle as well-known power indices like the Shapley value in [18] or/and the Banzhaf index in [41]--which attribute the importance of an individual/criteria/variable \(i\) to an aggregation of all possible marginal contributions \(v(\{i\}\cup S)-v(S)\) for any subset \(S\)--the interaction indexes reflect the added value of two variables/individuals/criteria \(i\) and \(j\) when considered together, compared to when they are considered separately. They are defined as a combination of the individuals marginal contributions in Equation (2).  
Just as in the one-dimensional case, where different ways of aggregating marginal contributions lead to various importance measures (such as the Shapley, Banzhaf, or similar values), different ways of combining these interaction contributions result in different interaction indexes. Among them, the one used in this work is the most widely adopted due to its strong mathematical properties.  
In this study we will use the interaction index introduced by [36], and extended in [21; 22], which measures the dynamics between elements within the context of cooperative game theory. It quantifies the average contribution of a coalition that includes elements \(i\) and \(j\), considering all possible subsets. The interaction index extends the Shapley value, which measures the individual importance of each element, by capturing synergies orredundancies between pairs of elements. To employ it in the context of this paper, we will host the interaction index in the matrix \(I\) defined in this section.  
The value of the interaction index can be positive, negative, or zero. A positive value indicates that the players in the subsets under consideration enhance each other's ability to contribute to the coalition, thereby increasing its value. A negative value shows that the interaction between the players has a detrimental effect on the coalition's value. A zero value implies that the subsets operate independently, with no significant influence on each other's contributions.  
**Definition 1** (Matrix of the Interaction Index \(I\)).: _Given a cooperative game \((N,v)\in G^{N}\), and with \(I_{i,j}(v)\) as the interaction index between two players \(i,j\in N\), the Matrix of the Interaction Index of a game \((N,v)\) is defined as a square matrix \(I\) of order \(|N|^{2}\), where each element \(i_{i,j}\) represents the interaction between players \(i\) and \(j\). The matrix \(I\) is given by \(I=\{t_{i,j}\in\mathbb{R}|i,j\in N\}\), where \(t_{i,j}=I_{i,j}(v)\)._  
To help in further analyzing a given cooperative game, we may also define the Positive Matrix of the Interaction Index as \(I^{+}=\{t^{+}_{i,j}\in\mathbb{R}^{+}|i,j\in N\}\) with \(t^{+}_{i,j}=\max(I_{i,j}(v),0)\), the Negative Matrix of the Interaction Index as \(I^{-}=\{t^{-}_{i,j}\in\mathbb{R}^{+}|i,j\in N\}\) with \(t^{-}_{i,j}=|\min(I_{i,j}(v),0)|\), and the Matrix of Absolute Values of the Interaction Index as \(I^{|\cdot|}=\{t^{|\cdot|}_{i,j}\in\mathbb{R}^{+}|i,j\in N\}\) with \(t^{|\cdot|}_{i,j}=|I_{i,j}(v)|\).