# Network Visualization of Cooperative Games
## 1 Introduction
It is important to note that most of the applications of cooperative game theory to problems share a common denominator: the solution concepts of cooperative game theory models as in [18]. In contrast to non-cooperative games, cooperative game theory represents situations where individual players form coalitions for mutual benefit. In this cooperative context, a natural question arises: how to distribute the benefits obtained among all participants. Solution concepts, both single-valued and set-based, aim to address this notion of a 'fair distribution' that has often been applied in real-world problems, with improvements applied to their computation through approaches such as the ones offered by [19; 20]. However, understanding and exploring all the information associated with a cooperative game should not only focus on solution concepts. This aspect is notably addressed by some authors, such as [21; 22], who emphasize the surprisingly restricted progress in the examination of interactions among players in a cooperative game, in contrast to the advancements in solution concepts or in comparison with the significance that interaction has gained in other similar disciplines. The study of interactions between items has proven to be indispensable and pertinent in various contexts for understanding problems of a nature not necessarily additive,deviating from the typical structure of most game theory models. In scenarios like multi-criteria decision problems studied in [23; 24], interaction indexes facilitate the modeling of a reality that surpasses the concept of independence between criteria (a highly simplified version of real-life situations that is inadequate for addressing many practical issues). This necessity is underscored in [24]. Models such as the Analytic Network Process described in [25], Decision-Making Trial and Evaluation Laboratory developed in [26], Interpretive Structural Modeling crafted in [27], Fuzzy Measures or Choquet Integral models designed in [28], among others, explicitly integrate this interaction as a fundamental tool in their analyses. And, as with the Shapley value, there are approximations in the calculation of this interaction index, such as the one described in [29]. This recognition is crucial for a more realistic representation of complex decision-making scenarios, contributing to a better understanding and resolution of practical issues. Even within the applications of the Shapley value to the realm of SHAP explainable models, recent research incorporates the study of interaction among 'players/variables' for variable selection problems, as in [30]. One of the fundamental objectives of social network analysis is the study of relationships between agents beyond their intrinsic characteristics. In this field, its connection with cooperative game theory is not a novel concept, as evidenced by numerous works in the literature, such as [31; 32; 33; 34]. These studies assume the existence of a graph among players that somehow impacts their relationships. Utilizing this information, cooperative games are formulated to address various network analysis problems, such as identifying the most relevant nodes (centrality) or detecting structures and groups, as in [35]. It is essential to note that in these works, the network is a crucial part of the available information of the problem, and the aim of them is applying cooperative game theory tools and concepts to solve network problems. In contrast with these approaches, the focus of this work is on how network analysis can contribute to visualizing, understanding, and comprehending a cooperative game. For this reason, we aim to represent cooperation needs between players, measured using the interaction index defined in [21; 22; 36], through different networks. The visualization and description of these networks will enhance the understanding of a cooperative game. Additionally, it will enable an interesting classification of the game type and player characteristics in a cooperative setting based on some network properties.