Network Visualization of Cooperative Games

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This framework offers a systematic approach to visualizing cooperative game dynamics across various fields. The methodology may provide insights into collaborative interactions observed in disciplines such as economic analysis by mapping coalition formations and payoff distributions, social network research by analyzing collaborative behaviors, or resource management by optimizing collaborative project allocation.

Abstract

Over the last five decades, the modeling of characteristic functions has been the key focus of cooperative game theory research. Solutions for distributing goods among players have been extensively explored, while less emphasis has been placed on examining player interactions. This paper emphasizes the importance of player relationships, with the help of the ‘player interaction index’ to capture coalition formation needs. By representing interactions through a graph, this study deepens the understanding of player relationships and synergies. It elaborates a graphical representation using network analysis to visualize player interactions, enhancing the understanding of the dynamics within the game.

1. Introduction

A correct understanding of the factors that make up a problem is key when analyzing and solving such a problem, even more so when there is a high volume of information about the problem or its factors. This has led to the widespread adoption of graphical representation methods for analyzing complex problems. In this regard, Ref. [1] showed that information visualization can improve decision quality as well as speed, with more mixed effects on other variables, Ref. [2] presented a framework for thinking about how visual representations are likely to affect the decision processes, and Ref. [3] suggested that graphical risk information facilitates comprehension of risk information because it attracts and holds attention for a longer period of time than textual risk information.

To give a wider example, from the moment that the information contained in statistical analysis began to be visualized graphically, we have experienced an advent in its application to the understanding of real scenarios. The mere representation of the distribution of random variables in a histogram has led to a boom in the application of variable analysis in all sectors, as Ref. [4] showed in its history telling of the histogram. The foundational role of visualization in mathematical and economic analysis was pioneered by Von Neumann and Morgenstern in their seminal work [5], where they introduced game trees to represent sequential decision-making in games. This marked the first systematic effort to visualize strategic interactions, laying the groundwork for subsequent advances in both cooperative and non-cooperative game theory. Since then, visualization techniques have evolved to address diverse challenges across domains. For instance, Ref. [6] developed a visual language for non-cooperative game strategies, emphasizing traceability of player interactions to enhance educational and analytical clarity. Similarly, Ref. [7] applied social network visualizations to map collaboration patterns in evolutionary game theory, illustrating how network properties like centrality and tie strength reveal persistent research alliances. Beyond game theory itself, visualization has been integrated with game-theoretic concepts to solve broader problems, such as swarm intelligence-based clustering in high-dimensional data [8] and relation-algebraic modeling of tournament solutions [9].

We cannot deny the importance of the processing, adequacy, and availability of the information we manage. The generation of large volumes of data is the order of the day, and methods are urgently needed to help us convert this information into knowledge that is relevant to each problem.

In this context, cooperative game theory has recently become a fundamental tool for addressing many real-world problems, from [10], which uses this theory to optimize wind power and demand response coordination in energy markets, through [11], which employs the Shapley value and bankruptcy games to enhance bandwidth allocation among IBOC FM broadcasting nodes, to [12], which proposes a fair allocation strategy for the dispatching of a hybrid power system that integrates hot dry rock geothermal energy, thermal storage, and photovoltaic plants. Particularly noteworthy is the application of cooperative game theory to machine learning problems in [13,14]. The SHAP (SHapley Additive exPlanations) framework developed by [15] has garnered significant attention in explainable artificial intelligence (explainable AI) as it provides detailed insights into how machine learning models make decisions. Recent extensions, such as the Shapley variable importance cloud by [16], further integrate visualization to quantify uncertainty in feature importance across models, bridging game theory with interpretable machine learning. Earlier work by [17] also leveraged coalitional game theory to explain individual model predictions, demonstrating the symbiotic relationship between visualization and game-theoretic interpretability.

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.

This paper is organized as follows: In Section 2, we introduce some preliminaries for a better understanding of this paper, specifically some cooperative game theory concepts, along with the communication situations defined by [31]. In Section 3, we present the relationships between players through the interaction matrix among players, leading to the visualization of a cooperative game as a network in Section 4. In Section 5, we define certain properties for desirable games in this representation, and finally, numerous examples of well-known cooperative games are presented in Section 6, showing how representing interactions between players aids in comprehending and classifying the type of game under analysis. We conclude this paper with a final conclusion and research section.

2. Preliminaries

2.1. Cooperative Games

The definition of the following subjects can be found in [37].

A cooperative n-person game with transferable utility, or a TU-game, is a pair $(N,v)$, where $N=1,\dots ,n$ is the set of players and v is a real-valued function.

Given a TU-game $(N,v)$, the characteristic function v is a real-valued function defined by [38] on the power set of N, denoted as $2^N$. The function v assigns a value to each coalition $S\subseteq N$, representing the total worth or wealth that the coalition S can achieve by cooperating. Formally, the characteristic function v is defined as $v:2^N\to \mathbb{R}$.

The characteristic function v of a TU-game $(N,v)$ must satisfy $v(\varnothing )=0$, meaning that the value of the characteristic function for the empty coalition is zero.

The value $v\left(S\right)$ of a characteristic function v in a TU-game $(N,v)$ represents for any coalition $S\subseteq N$ the total worth that the members of S can achieve by cooperating.

We will denote the cardinality $\left|S\right|$ of S as s, and the vector space of all TU-games with the set of players N will be represented by $G^N$.

Given $(N,v)\in G^N$, $i\in N$, and $S\subseteq N\setminus \left\{i\right\}$, the marginal player contribution i to the coalition S in the game v is the difference $v(S\cup \{i\left\}\right)-v\left(S\right)$.

Considering the definition of a game $(N,v)\in G^N$, a player $i\in N$ is defined as a null player in $(N,v)$ if $v(S\cup \{i\left\}\right)-v\left(S\right)=0 for all S\subseteq N\setminus \left\{i\right\}$.

Once $(N,v)\in G^N$, two players $i,j\in N$ are defined to be symmetric in $(N,v)$ if $v(S\cup \{i\left\}\right)=v(S\cup \{j\left\}\right) for all S\subseteq N\setminus \{i,j\}$.

Symmetric games in $(N,v)\in G^N$ are those that fulfill $v\left(S\right)=v\left(T\right)$ for $S,T\subseteq N$ with $s=t$.

A point solution, or an allocation rule, for n-person TU-games is a function $\varphi :G^N\to {\mathbb{R}}^n$, where ${\varphi }_i(N,v)$ represents the outcome for player i in game $(N,v)\in G^N$.

The author of [18] introduced a very relevant point solution for TU-games, the so-called Shapley value, that is defined as in Equation (1).

$$S{h}_i(N,v)=\sum _{S\subseteq N\setminus \left\{i\right\}}{\displaystyle \frac{(n-s-1)!s!}{n!}}\left[v(S\cup \{i\left\}\right)-v\left(S\right)\right],i\in N$$

(1)

We will denote the vector of all Shapley values for a game $(N,v)\in G^N$ as $Sh=\{S{h}_i(N,v)\in \mathbb{R}|i\in N\}$.

Given that the exact calculation of the Shapley value is exponential, polynomial approximations can be used for large-scale problems, as defined in [19,20].

Initially formulated in [36], and extended in [21,22], for a finite set $S\subseteq N$ with cardinality $\left|S\right|=s$ in a TU-game $(N,v)$, the interaction index for elements $i,j\in S$ is defined in Equation (2), with ${\zeta }_k=(s-k-2)!\left(k\right)!/(s-1)!$

$$I_{i,j}\left(v\right)=\sum _{k=0}^{s-2}{\zeta }_k\sum _{\begin{array}{c}K\subset S\setminus \{i,j\}\\ \left|K\right|=k\end{array}}\left(v\left(\right\{i,j\}\cup K)-v\left(\right\{i\}\cup K)-v\left(\right\{j\}\cup K)+v\left(K\right)\right)$$

(2)

2.2. Graphs

A weighted graph (weighted network) as reviewed in [39] is a pair $(N,A)$ in which $N=\{1,2,\dots ,n\}$ is the set of nodes and $A=\{a_{i,j}\in \mathbb{R}|i,j\in N,i\ne j\}$ represents all the possible links between two nodes $i,j$ in N. Each link $a_{i,j}\in A$ represents a direct communication between i and j, assigning a real number value to the weight of the communication between both.

We say that there is an edge between nodes i and j in the graph $(N,A)$ if there exists $i,j\in N$ with $i\ne j$ such that $a_{i,j}\ne 0; a_{i,j}\in A$. A set $\varnothing \ne S\subseteq N$ is connected in $(N,A)$ if it is a singleton or if there is an edge for every pair of nodes in S of graph $(S,A{|}_S)$ with $A{|}_S=\{a_{i,j}\in A|i,j\in S\}$.

Let $(N,A)$ be a weighted graph, where N is the set of nodes and A is the set of weighted edges. The weighted degree $s_i$ of a node $i\in N$ is given by $s_i={\sum }_{j\in N\setminus \left\{i\right\}}{a}_{i,j}$.

2.3. Communication Situations

A communication situation as introduced in [31] can be modeled using a triplet $(N,v,\gamma )$. Here, $(N,v)$ represents a transferable utility (TU) game and $(N,\gamma )$ signifies a communication network.

In this context, $\gamma$ represents the communication network, which is a graph indicating how players can communicate with each other. The graph $\gamma$ is a subset of the complete graph ${\gamma }_N$, which includes all possible communication links among players. The set of all communication situations with N as the set of nodes/players is denoted by Equation (3).

$$\mathcal{C}{\mathcal{S}}^N=\{(N,v,\gamma )\mid \gamma \subseteq {\gamma }_N\}$$

(3)

Refs. [31,40] defined the concept of a graph-restricted game $(N,v^{\gamma })$. In this model, the value of a coalition S under the communication constraints of $\gamma$ is given by Equation (4), where $S/\gamma$ represents the partition of the coalition S into connected components according to the graph $\gamma$.

$$v^{\gamma }\left(S\right)=\sum _{C\in S/\gamma }v\left(C\right)$$

(4)

Given $\mathcal{C}{\mathcal{S}}^N$, the set of all communication situations in a game $(N,v)$, Ref. [31] defined a value allocation rule $\mu$ for these situations. This rule assigns a value to each player in a communication situation, and is defined in Equation (5).

$${\mu }_i(N,v,\gamma )=S{h}_i(N,v^{\gamma })$$

(5)

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/criterion/variable i to an aggregation of all possible marginal contributions $v\left(\right\{i\}\cup S)-v\left(S\right)$ 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 or redundancies 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 $I_{i,j}\left(v\right)$ 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 ${\left|N\right|}^2$, where each element ${\iota }_{i,j}$ represents the interaction between players i and j. The matrix I is given by $I=\{{\iota }_{i,j}\in \mathbb{R}|i,j\in N\}$, where ${\iota }_{i,j}=I_{i,j}\left(v\right)$.

To help in further analyzing a given cooperative game, we may also define the Positive Matrix of the Interaction Index as $I^{+}=\{{\iota }_{i,j}^{+}\in {\mathbb{R}}^{+}|i,j\in N\}$ with ${\iota }_{i,j}^{+}=max(I_{i,j}\left(v\right),0)$, the Negative Matrix of the Interaction Index as $I^{-}=\{{\iota }_{i,j}^{-}\in {\mathbb{R}}^{+}|i,j\in N\}$ with ${\iota }_{i,j}^{-}=|min(I_{i,j}\left(v\right),0)|$, and the Matrix of Absolute Values of the Interaction Index as $I^{|·|}=\{{\iota }_{i,j}^{|·|}\in {\mathbb{R}}^{+}|i,j\in N\}$ with ${\iota }_{i,j}^{|·|}=\left|I_{i,j}\left(v\right)\right|$.

4. Network Visualization of a Cooperative Game

In social analysis, the network visualization of individuals’ interactions stands as a cornerstone for exploring the complex dynamics of collective behaviors. To use this instrument, we introduce the formal definition of a cooperative game representation as a network.

Definition 2 

(Network visualization of a cooperative game). Given a game $(N,v)\in G^N$, we define the network visualization of a cooperative game as $(\mathrm{\Psi },R)$, where the following hold:

• $\mathrm{\Psi }=\{{\psi }_i\in \mathbb{R}\mid i\in N\}$ is the vector representing each player’s relevance in the game. In this context, the player relevance ${\psi }_i$ is computed as an aggregated measure that combines the contributions of player i over all coalitions in which player i participates.
• $R=\{r_{i,j}\in \mathbb{R}\mid i,j\in N\}$ is the matrix that collects the information about the relationship between pairs of players.

Here, we propose the network visualization for a real cooperation situation where the size of a node corresponding to a player $i\in N$ will be represented by ${\psi }_i$ belonging to the relevance vector $\mathrm{\Psi }=Sh$.

The value ${\psi }_i$ is distinct from the characteristic function v, because $v\left(\right\{i\left\}\right)$ measures the wealth or payoff of player i when acting individually. In contrast, ${\psi }_i$ aggregates the various contributions player i makes when participating in different coalitions, thereby capturing the synergy and complementary effects inherent in joint cooperation.

As the magnitude of the share for a player increases, the size of the node representing the player will increase as its relevance increases. Depending on the sign of this value, the nodes will be colored differently, black for positive values of ${\psi }_i$ and red for negative ones.

The links of the network will show the relationship $r_{i,j}$ between player i and player j. To represent it, we will use the Matrix of the Interaction Index $R=I$. It is also possible to use the Positive Matrix of the Interaction Index $R=I^{+}$, the Negative Matrix of the Interaction Index $R=I^{-}$, or the Matrix of Absolute Values of the Interaction Index $R=I^{|·|}$ when needed.

Negative relationships will be painted in red, while black color will represent positive ones. We will depict the intensity of each pair of players’ relationship by the thickness of each link $(i,j)$ in the network. It will be proportional to the corresponding value $r_{i,j}\in R$, leading to thicker links in stronger relationships, and thinner for weaker connections.

A potential problem with the network representation of a cooperative game is that the number of links defined in the player relationship matrix may increase exponentially with the addition of more players to the game. To address this issue, we have incorporated a feature that filters the potential number of links and nodes in the network representation of the game. This is achieved through the use of the threshold parameters ${\alpha }^{+}$, ${\alpha }^{-}$, ${\beta }^{+}$, and ${\beta }^{-}$, which prevent an overabundance of links and nodes in the network and enable the visualization of significant relationships.

Definition 3 

(Filtered representation of a cooperative game). Given a network representation $(\mathrm{\Psi },R)$ of a game $(N,v)\in G^N$, and four threshold limit values named ${\alpha }^{+},{\alpha }^{-},{\beta }^{+},$ and ${\beta }^{-}\in {\mathbb{R}}^{+}$, we define the filtered representation of the cooperative game $(N,v)\in G^N$ as $({\mathrm{\Psi }}_{{\alpha }^{+},{\alpha }^{-}},R_{{\beta }^{+},{\beta }^{-}})$ in such a way that the following hold:

1.

Nodes with positive sign in Ψ are assigned a value of zero if their absolute value does not exceed ${\alpha }^{+}$.

2.

Nodes with negative sign in Ψ are assigned a value of zero if their absolute value does not exceed ${\alpha }^{-}$.

3.

Links with positive sign in R are assigned a value of zero if their absolute value does not exceed ${\beta }^{+}$.

4.

Links with negative sign in R are assigned a value of zero if their absolute value does not exceed ${\beta }^{-}$.

Formally, we define the Filtered Players’ relevance representation as follows:

$${\psi }_{i,{\alpha }^{+},{\alpha }^{-}}=\left\{\begin{array}{cc}{\psi }_i& if {\psi }_i>0 and \left|{\psi }_i\right|>{\alpha }^{+}\hfill \\ {\psi }_i& if {\psi }_i<0 and \left|{\psi }_i\right|>{\alpha }^{-}\hfill \\ 0& otherwise\hfill \end{array}\right.$$

(6)

$$r_{i,j,{\beta }^{+},{\beta }^{-}}=\left\{\begin{array}{cc}{r}_{i,j}& if r_{i,j}>0 and \left|r_{i,j}\right|>{\beta }^{+}\hfill \\ r_{i,j}& if r_{i,j}<0 and \left|r_{i,j}\right|>{\beta }^{-}\hfill \\ 0& otherwise\hfill \end{array}\right.$$

(7)

The four threshold parameters ${\alpha }^{+}$, ${\alpha }^{-}$, ${\beta }^{+}$, and ${\beta }^{-}$ help filter out nodes and links that contribute little to the network. Choosing these values typically involves a data-driven approach that highlights only the most relevant nodes and interactions. One way to achieve this is by examining the distribution of player relevance scores ($\mathrm{\Psi }$) and interaction values (R), and then setting thresholds based on a specific quantile (e.g., above the first or second quartile). If the thresholds are too low, they will not filter out insignificant elements, making them ineffective. On the other hand, well-chosen thresholds create a clearer, more focused view of the network, making it easier to identify the most impactful elements in the game.

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, $S{h}_i(N,v)=0$. Also, by the definition of a null player, all marginal contributions to player i are zero, so $v(S\cup \{i,j\left\}\right)-v(S\cup \{j\left\}\right)=0$, $v(S\cup \{i\left\}\right)-v\left(S\right)=0$, and $I_{i,j}\left(v\right)=0$ 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 $v(S\cup \{i,k\left\}\right)=v(S\cup \{j,k\left\}\right)$ for all $k\in N\setminus \{i,j\}$, and $v(S\cup \{i\left\}\right)=v(S\cup \{j\left\}\right)$; therefore, $I_{i,k}\left(v\right)=I_{j,k}\left(v\right)$ 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 $v\left(\right\{i\left\}\right)$ 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 $v(S\cup \{i\left\}\right)-v\left(S\right)=v\left(\right\{i\left\}\right)$, and given that the Shapley value is a convex combination of the marginal contributions, the first result is proven. When considering R, given that $v(S\cup \{i,j\left\}\right)=v(S\cup \{i\left\}\right)+v\left(\right\{j\left\}\right)$, and also given $v(S\cup \{j\left\}\right)=v\left(S\right)+v\left(\right\{j\left\}\right)$, then $v(S\cup \{i,j\left\}\right)-v(S\cup \{i\left\}\right)-v(S\cup \{j\left\}\right)+v\left(S\right)=0$; therefore $I_{i,j}\left(v\right)=0$ 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 $I^{|·|}$.

6. Examples

This paper explores ten canonical examples to illustrate how their network representation displays relevant information contained in different types of cooperative games. Among these, the Airport Game example appears in 103 reviewed references, including a seminal discussion in [44]. The Bankruptcy Game example is noted in 129 references, with one prominent citation in [45]. The Voting Game example, referenced in 593 studies, is similarly acknowledged in [44]. Although the Shoe Game example is mentioned in only 40 references, it serves as an important illustration drawn from [46]. Additionally, Myerson Game examples are recognized in 163 studies, as discussed in [47]. Together, these examples substantiate this paper’s value proposition by demonstrating how network representations can capture essential insights into cooperative game dynamics.

Let us note that among many other things, the visualization of the interaction between players in a cooperative game allows us to identify clusters of players highly related by the need for cooperation. From the examples in this section, we can see that this visual representation simplifies the detection of interconnected players, revealing underlying patterns and structures within the cooperative game. By observing these clusters, researchers can better understand the relationships and dependencies among players, leading to more informed hypotheses and enhanced analytical insights.

This fact can be seen especially in the example in Section 6.1 (and also in the example in Section 6.2), where players LG1 and LG2 form a cluster with players MG1, MG2, and MG3. It is clear that LG1 and LG2 have a real need to be together with MG1, MG2, and MG3, which cannot be easily identified in other ways. Nodes SG1, SG2, SG3, and SG4 do not present the same interest or necessity in the large coalition as the cluster of LG1, LG2, MG1, MG2, and MG3.

Please be aware that the values of ${\psi }_i$ and $r_{i,j}$ shown in the $\mathrm{\Psi }$ and R vectors and matrices of the following examples are truncated to four decimal positions. This means that summing their column values may not result in the exact expected value.

We will use the weighted degree $s_i$ of the nodes in $R=I^{+}$, $R=I^{-}$, $R=I$, and $R=I^{|·|}$ to determine the statistical characteristics of each network. Please take the measures shown in each case as examples of the analytical potential offered by the proposed representation method, with the reader realizing that they could make use of other centrality measures depending on the needs they have.

In addition to the Shapley value, there are other values for measuring the relevance of the different players in cooperative games. Among these, we should highlight those calculated as weighted averages of the marginal contributions, such as the Banzhaf value [41] or the Owen value [48,49], and those arising from the concept of the core, such as the nucleolus [43]. If, by the definition of a game, a measure other than the Shapley value is considered to be the most appropriate, the graphs should be represented with that measure.

6.1. 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 $C\left(i\right)$ be the individual runway cost incurred by aircraft i. For any coalition $S\subseteq N$, the characteristic function is defined in Equation (8).

$$v\left(S\right)=-\underset{i\in S}{max}\left\{C\left(i\right)\right\}$$

(8)

In this example, $v\left(S\right)$ means that the cost for a group of aircraft is determined solely by the aircraft with the highest cost in that group. Essentially, $v\left(S\right)$ 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 $N=\{\mathrm{SM}1,\mathrm{SM}2,\mathrm{SM}3,\mathrm{SM}4,\mathrm{SM}5,\mathrm{MD}1,\mathrm{MD}2,\mathrm{MD}3,\mathrm{LG}1,\mathrm{LG}2\}$, are, respectively $C=\{100,100,100,100,100,150,150,150,200,200\}$. The relevance vector $\mathrm{\Psi }=Sh$ for this set is calculated in

Table 1. Airplane relevance vector $\mathrm{\Psi }=Sh$ and relationship matrix $R=I$ in the airport runway game.

$\mathit{Sh}$	I	SM1	SM2	SM3	SM4	SM5	MD1	MD2	MD3	LG1	LG2
−10	SM1	0	11.1111	11.1111	11.1111	11.1111	11.1111	11.1111	11.1111	11.1111	11.1111
−10	SM2	11.1111	0	11.1111	11.1111	11.1111	11.1111	11.1111	11.1111	11.1111	11.1111
−10	SM3	11.1111	11.1111	0	11.1111	11.1111	11.1111	11.1111	11.1111	11.1111	11.1111
−10	SM4	11.1111	11.1111	11.1111	0	11.1111	11.1111	11.1111	11.1111	11.1111	11.1111
−10	SM5	11.1111	11.1111	11.1111	11.1111	0	11.1111	11.1111	11.1111	11.1111	11.1111
−20	MD1	11.1111	11.1111	11.1111	11.1111	11.1111	0	23.6111	23.6111	23.6111	23.6111
−20	MD2	11.1111	11.1111	11.1111	11.1111	11.1111	23.6111	0	23.6111	23.6111	23.6111
−20	MD3	11.1111	11.1111	11.1111	11.1111	11.1111	23.6111	23.6111	0	23.6111	23.6111
−45	LG1	11.1111	11.1111	11.1111	11.1111	11.1111	23.6111	23.6111	23.6111	0	73.6111
−45	LG2	11.1111	11.1111	11.1111	11.1111	11.1111	23.6111	23.6111	23.6111	73.6111	0

, with player relationship values $R=I$.

Analyzing the metrics of the weighted degree in

Table 2. Descriptive metrics for the Airport Game relationship matrices.

${\mathit{s}}_{\mathit{i}}$	$\mathit{R}={\mathit{I}}^{+}$	$\mathit{R}={\mathit{I}}^{-}$	$\mathit{R}=\mathit{I}$	$\mathit{R}={\mathit{I}}^{|·|}$
Maximum	200	0	200	200
Q3	150	0	150	150
Mean	135	0	135	135
Median	125	0	125	125
Q1	100	0	100	100
Minimum	100	0	100	100
Standard Deviation	39.0512	0	39.0512	39.0512
Skewness	0.5793	0	0.5793	0.5793
Kurtosis	−1.1357	0	−1.1357	−1.1357
Gini Index	0.1518	0	0.1518	0.1518

, we can see that I, $I^{+}$, and $I^{|·|}$ 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 $(S{h}_{{\alpha }^{+}=0,{\alpha }^{-}=0.25},I_{ {\beta }^{+}=0,{\beta }^{-}=0})$ 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 25th 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, also helps us to see quickly that the edges of a three-node clique to which a node belongs always add up to a higher absolute value when larger nodes are also part of the three-node clique. We see this, 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.

6.2. 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 $N=\{\mathrm{SL}1,\mathrm{SL}2,\mathrm{SL}3,\mathrm{SL}4,\mathrm{SL}5,\mathrm{ML}1,\mathrm{ML}2,\mathrm{LL}1,\mathrm{LL}2\}$ has provided a loan of $L=\{50,50,50,50,50,200,200,200,400,400\}$, respectively, to the borrower.

Table 3. Lender relevance vector $\mathrm{\Psi }=Sh$ and relationship matrix $R=I$ in the Bankruptcy Game.

$\mathit{Sh}$	I	SL1	SL2	SL3	SL4	SL5	ML1	ML2	ML3	LL1	LL2
29.9404	SL1	0	1.2896	1.2896	1.2896	1.2896	5.8730	5.8730	5.8730	13.6111	13.6111
29.9404	SL2	1.2896	0	1.2896	1.2896	1.2896	5.8730	5.8730	5.8730	13.6111	13.6111
29.9404	SL3	1.2896	1.2896	0	1.2896	1.2896	5.8730	5.8730	5.8730	13.6111	13.6111
29.9404	SL4	1.2896	1.2896	1.2896	0	1.2896	5.8730	5.8730	5.8730	13.6111	13.6111
29.9404	SL5	1.2896	1.2896	1.2896	1.2896	0	5.8730	5.8730	5.8730	13.6111	13.6111
119.8611	ML1	5.8730	5.8730	5.8730	5.8730	5.8730	0	27.4801	27.4801	57.8373	57.8373
119.8611	ML2	5.8730	5.8730	5.8730	5.8730	5.8730	27.4801	0	27.4801	57.8373	57.8373
119.8611	ML3	5.8730	5.8730	5.8730	5.8730	5.8730	27.4801	27.4801	0	57.8373	57.8373
245.3571	LL1	13.6111	13.6111	13.6111	13.6111	13.6111	57.8373	57.8373	57.8373	0	158.4325
245.3571	LL2	13.6111	13.6111	13.6111	13.6111	13.6111	57.8373	57.8373	57.8373	158.4325	0

shows the relevance vector $\mathrm{\Psi }=Sh$ and the relationship matrix $R=I$ for this game.

$$v\left(S\right)=max\left\{0,E-\sum _{j\in N\setminus S}{L}_j\right\}$$

(9)

When reviewing the weighted degree metrics in

Table 4. Descriptive metrics for the Bankruptcy Game relationship matrices.

${\mathit{s}}_{\mathit{i}}$	$\mathit{R}={\mathit{I}}^{+}$	$\mathit{R}={\mathit{I}}^{-}$	$\mathit{R}=\mathit{I}$	$\mathit{R}={\mathit{I}}^{|·|}$
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

, we find equal values of I, $I^{+}$, and $I^{|·|}$ 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 here is noticeably greater when compared to cliques where a larger node pairs with smaller ones, such as the clique comprising LL1 with SL3 and SL4, the one with LL2 alongside SL5 and SL1, or the clique containing ML1 with SL1 and SL2. These discrepancies make it clear that as the size of the nodes increases, so too does the magnitude of their relationships, underscoring the powerful influence of larger nodes in the network.

6.3. Voting Game

This third example describes the Voting Game for the parties of a parliament. Consider a chamber composed of ten parties, each with the following vote distribution: party A with 29 votes, B with 26, C with 16, D with 13, E with 6, F with 3, G with 3, H with 2, I with 1, and J with 1.

This game is modeled as a simple weighted cooperative game, represented as $v\left(S\right)\equiv [q;w_1,w_2,w_3,\dots ,w_s]$, where each party i has a weight $w_i$ corresponding to its number of votes, the s value refers to the cardinal $\left|S\right|$ of the coalition S, and q denotes the quota required to form a majority. The characteristic function $v\left(S\right)$, given in Equation (10), determines whether a coalition S is winning or losing. Specifically, $v\left(S\right)=1$ if the total weight of the coalition meets or exceeds the quota, indicating that the coalition can pass decisions; otherwise, $v\left(S\right)=0$.

$$v\left(S\right)=\left\{\begin{array}{cc}1,\hfill & {\displaystyle \mathrm{if}\sum _{i\in S}{w}_i\ge q}\hfill \\ 0,\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

(10)

The relevance vector $\mathrm{\Psi }=Sh$ and the relationship matrix $R=I$ are provided in

Table 5. Party relevance vector $\mathrm{\Psi }=Sh$ and player relationship matrix $R=I$ in the Voting Game.

$\mathit{Sh}$	I	A	B	C	D	E	F	G	H	I	J
0.3226	A	0	0.0142	−0.0048	−0.0024	0.0047	−0.0024	−0.0024	−0.0072	0	0
0.2615	B	0.0142	0	−0.0024	0	−0.0096	0	0	−0.0024	0	0
0.1559	C	−0.0048	−0.0024	0	0.0142	0.0047	−0.0024	−0.0024	−0.0072	0	0
0.0948	D	−0.0024	0	0.0142	0	−0.0096	0	0	−0.0024	0	0
0.0753	E	0.0047	−0.0096	0.0047	−0.0096	0	0.0071	0.0071	−0.0048	0	0
0.0281	F	−0.0024	0	−0.0024	0	0.0071	0	0	−0.0024	0	0
0.0281	G	−0.0024	0	−0.0024	0	0.0071	0	0	−0.0024	0	0
0.0206	H	−0.0072	−0.0024	−0.0072	−0.0024	−0.0048	−0.0024	−0.0024	0	0.0142	0.0142
0.0063	I	0	0	0	0	0	0	0	0.0142	0	−0.0142
0.0063	J	0	0	0	0	0	0	0	0.0142	−0.0142	0

.

This game is characterized by a perfect balance in the influence of positive and negative interactions. The third column in

Table 6. Descriptive metrics for the Voting Game relationship matrices.

${\mathit{s}}_{\mathit{i}}$	$\mathit{R}={\mathit{I}}^{+}$	$\mathit{R}={\mathit{I}}^{-}$	$\mathit{R}=\mathit{I}$	$\mathit{R}={\mathit{I}}^{|·|}$
Maximum	0.0286	0.0286	0	0.0571
Q3	0.019	0.019	0	0.0381
Mean	0.0161	0.0161	0	0.0323
Median	0.0143	0.0143	0	0.0286
Q1	0.0143	0.0143	0	0.0286
Minimum	0.0071	0.0071	0	0.0143
Standard Deviation	0.0063	0.0063	0	0.0127
Skewness	0.3383	0.3383	0	0.3359
Kurtosis	−0.5317	−0.5317	0	−0.5453
Gini Index	0.2150	0.2150	0	0.2143

shows null values, meaning that the sum of positive and negative interactions is zero in all cases. If we focus on either the positive or negative relationships, we can see identical distributions, with a moderate variability (Gini Index 0.21 and a maximum value four times higher than the minimum). The shape of the weighted degree distribution is slightly right-skewed, with a higher clustering than a normal distribution.

The relationships of the Voting Game $(Sh,I)$ are represented in Figure 5, and those of $(S{h}_{{\alpha }^{+}=0,{\alpha }^{-}=0},I_{{\beta }^{+}=1,{\beta }^{-}=0.1})$ in Figure 6.

In the visual representation of the Voting Game in Figure 5, successful relationships between two players are clearly visible. However, positive relationships between three players, or positive relationship cliques, are not abundant. The only clique of positive relationships is the one formed by parties A, C, and E. The rest of the relationships do not form cliques of positive edges. This is because the large parties, normally necessary to form a majority, will not seek alliances with small parties that do not grant such majorities, but they will do so with medium or large parties that enable access to power.

6.4. Shoe Game

In the Shoe Game, each coalition will obtain as much profit as the number of complete pairs of shoes it can form. To form a complete pair of shoes, two players must join with one shoe on each foot. This game has ten players, five of them with the left shoe, and another five of them with the right shoe, with the set of players being $N=\{lef{t}_1,lef{t}_2,lef{t}_3,lef{t}_4,lef{t}_5,righ{t}_1,righ{t}_2,righ{t}_3,righ{t}_4,righ{t}_5\}$. For any coalition $S\subseteq N$, denote $S_{left}=S\cap \{lef{t}_1,lef{t}_2,lef{t}_3,lef{t}_4,lef{t}_5\}\mathrm{and}{S}_{right}=S\cap \{righ{t}_1,righ{t}_2,righ{t}_3,righ{t}_4,righ{t}_5\}$. The characteristic function $v\left(S\right)$ of this game is defined in Equation (11).

$$v\left(S\right)=min\{\left|S_{left}\right|,\left|S_{right}\right|\}$$

(11)

Equation (11) reflects the fact that a complete pair of shoes requires one left and one right shoe. Hence, the maximum number of pairs (and the profit for the coalition) is determined by the smallest number of available shoes between the two types. Any surplus in one category does not contribute to forming additional pairs.

The negative relationships’ representation of this game is shown in Figure 7 and its positive relationships’ representation in Figure 8. The relevance vector $\mathrm{\Psi }=Sh$ and the relationship matrix $R=I$ are shown in

Table 7. Relevance vector $\mathrm{\Psi }=Sh$ and player relationship matrix $R=I$ in the Shoe Game.

$\mathit{Sh}$	I	LS1	LS2	LS3	LS4	LS5	RS1	RS2	RS3	RS4	RS5
0.5	LS1	0	−0.2580	−0.2580	−0.2580	−0.2580	0.4063	0.4063	0.4063	0.4063	0.4063
0.5	LS2	−0.2580	0	−0.2580	−0.2580	−0.2580	0.4063	0.4063	0.4063	0.4063	0.4063
0.5	LS3	−0.2580	−0.2580	0	−0.2580	−0.2580	0.4063	0.4063	0.4063	0.4063	0.4063
0.5	LS4	−0.2580	−0.2580	−0.2580	0	−0.2580	0.4063	0.4063	0.4063	0.4063	0.4063
0.5	LS5	−0.2580	−0.2580	−0.2580	−0.2580	0	0.4063	0.4063	0.4063	0.4063	0.4063
0.5	RS1	0.4063	0.4063	0.4063	0.4063	0.4063	0	−0.2580	−0.2580	−0.2580	−0.2580
0.5	RS2	0.4063	0.4063	0.4063	0.4063	0.4063	−0.2580	0	−0.2580	−0.2580	−0.2580
0.5	RS3	0.4063	0.4063	0.4063	0.4063	0.4063	−0.2580	−0.2580	0	−0.2580	−0.2580
0.5	RS4	0.4063	0.4063	0.4063	0.4063	0.4063	−0.2580	−0.2580	−0.2580	0	−0.2580
0.5	RS5	0.4063	0.4063	0.4063	0.4063	0.4063	−0.2580	−0.2580	−0.2580	−0.2580	0

.

This cooperative game is characterized by its equal player influence and high level of connectivity, making it an exemplary model for cooperative and inclusive gameplay, as evidenced in

Table 8. Descriptive metrics for the Shoe Game relationship matrices.

${\mathit{s}}_{\mathit{i}}$	$\mathit{R}={\mathit{I}}^{+}$	$\mathit{R}={\mathit{I}}^{-}$	$\mathit{R}=\mathit{I}$	$\mathit{R}={\mathit{I}}^{|·|}$
Maximum	2.0317	1.0317	1	3.0634
Q3	2.0317	1.0317	1	3.0634
Mean	2.0317	1.0317	1	3.0634
Median	2.0317	1.0317	1	3.0634
Q1	2.0317	1.0317	1	3.0634
Minimum	2.0317	1.0317	1	3.0634
Standard Deviation	0	0	0	0
Skewness	0	0	0	0
Kurtosis	0	0	0	0
Gini Index	0	0	0	0

. It also shows that the third column, representing the sum of all interactions in each node, is greater than zero in all quartiles. Such a positive value means that all positive influence performed by players in the game is greater than the negative one. We will see that this also occurs in cyclic Myerson games.

The absolute symmetry shown in the bipartite graph of the Shoe Game is clearly visible in the visualization of the game network in Figure 8. This is not so evident in the adjacency matrix R of Table 7.

6.5. Myerson Games

6.5.1. Introduction to Myerson Games

Traditional cooperative games assign a value $v\left(S\right)$ to every coalition $S\subseteq N$ without considering any underlying structure that might restrict cooperation. In contrast, a Myerson game incorporates a communication network $\gamma$ that restricts which subsets of players can feasibly cooperate. Specifically, the value of any coalition S is computed by first partitioning S into its connected components with respect to $\gamma$, and then summing the contributions of these components as in Equation (4).

This formulation reflects real-world scenarios where cooperation is limited by physical connectivity, leading to strategic interactions that depend on the network topology. Therefore, solution concepts, such as the Myerson value in Equation (5), are developed to allocate the total game value by accounting for the structural constraints imposed by $\gamma$.

To illustrate the network representation of a Myerson cooperative game $(N,v,\gamma )\in \mathcal{C}{\mathcal{S}}^N$, we present examples using three graphical representations (cycle, chain, and star), two symmetric characteristic functions (a unitary cost share game, as shown in Equation (12) and a message game, as in Equation (13), and ten players ($n=10$).

$$v^{\gamma }\left(S\right)=-\sum _{C\in S{/}_{\gamma }}1$$

(12)

In Equation (12), each connected component C of S incurs a unit cost. Thus, if S is connected (i.e., $S{/}_{\gamma }$ consists of one component), then $v^{\gamma }\left(S\right)=-1$, and if S splits into k connected components, then $v^{\gamma }\left(S\right)=-k$.

$$v^{\gamma }\left(S\right)=\sum _{C\in S{/}_{\gamma }}\left({\displaystyle \genfrac{}{}{0pt}{}{\left|C\right|}{2}}\right)$$

(13)

Meanwhile, in Equation (13) $\left({\displaystyle \genfrac{}{}{0pt}{}{\left|C\right|}{2}}\right)$ counts the number of unordered pairs of players in C, so that $v^{\gamma }\left(S\right)$ equals the total number of potential pairwise interactions within all connected components of S.

6.5.2. Myerson Cycle Cost Game

This example deals with the Myerson communication situation of ten players in a unitary cost share game and communications arranged in a cycle layout. This situation favors players when they are directly connected, and penalizes them when they join a coalition without connecting directly to the existing players. When two players are not connected by the communication situation $\gamma$, the network representation will exhibit negative relationships, expressed as a red-filled link connecting both nodes in Figure 9, where we use $\mathrm{\Psi }=Sh$ and $R=I$ filtered with ${\alpha }^{+}=1, {\alpha }^{-}=0, {\beta }^{+}=1,$ and ${\beta }^{-}=0$. On the other hand, when two players are connected in $\gamma$, they will display positive relationships in the network representation of the game, as shown in Figure 10, where we use $\mathrm{\Psi }=Sh$ and $R=I$ filtered with ${\alpha }^{+}=1, {\alpha }^{-}=0, {\beta }^{+}=0,$ and ${\beta }^{-}=1$.

Relationship and relevance data from

Table 9. Player relevance $\mathrm{\Psi }=Sh$ and player relationship matrix $R=I$ in the Myerson Cycle Cost Game.

$\mathit{Sh}$	I	P1	P2	P3	P4	P5	P6	P7	P8	P9	P10
−0.1	P1	0	0.8888	−0.1112	−0.1112	−0.1112	−0.1112	−0.1112	−0.1112	−0.1112	0.8888
−0.1	P2	0.8888	0	0.8888	−0.1112	−0.1112	−0.1112	−0.1112	−0.1112	−0.1112	−0.1112
−0.1	P3	−0.1112	0.8888	0	0.8888	−0.1112	−0.1112	−0.1112	−0.1112	−0.1112	−0.1112
−0.1	P4	−0.1112	−0.1112	0.8888	0	0.8888	−0.1112	−0.1112	−0.1112	−0.1112	−0.1112
−0.1	P5	−0.1112	−0.1112	−0.1112	0.8888	0	0.8888	−0.1112	−0.1112	−0.1112	−0.1112
−0.1	P6	−0.1112	−0.1112	−0.1112	−0.1112	0.8888	0	0.8888	−0.1112	−0.1112	−0.1112
−0.1	P7	−0.1112	−0.1112	−0.1112	−0.1112	−0.1112	0.8888	0	0.8888	−0.1112	−0.1112
−0.1	P8	−0.1112	−0.1112	−0.1112	−0.1112	−0.1112	−0.1112	0.8888	0	0.8888	−0.1112
−0.1	P9	−0.1112	−0.1112	−0.1112	−0.1112	−0.1112	−0.1112	−0.1112	0.8888	0	0.8888
−0.1	P10	0.8888	−0.1112	−0.1112	−0.1112	−0.1112	−0.1112	−0.1112	−0.1112	0.8888	0

analyzed in

Table 10. Descriptive metrics for the Myerson Cycle Cost Game relationship matrices.

${\mathit{s}}_{\mathit{i}}$	$\mathit{R}={\mathit{I}}^{+}$	$\mathit{R}={\mathit{I}}^{-}$	$\mathit{R}=\mathit{I}$	$\mathit{R}={\mathit{I}}^{|·|}$
Maximum	1.7778	0.7778	1	2.5556
Q3	1.7778	0.7778	1	2.5556
Mean	1.7778	0.7778	1	2.5556
Median	1.7778	0.7778	1	2.5556
Q1	1.7778	0.7778	1	2.5556
Minimum	1.7778	0.7778	1	2.5556
Standard Deviation	0	0	0	0
Skewness	0	0	0	0
Kurtosis	0	0	0	0
Gini Index	0	0	0	0

show that the dynamics are equitable due to the uniform distribution of influence among players. This parity encourages strategic cooperation, as all players have equal power and benefit from direct connections in the cycle network. Its third columns also tell us that the positive influence fulfilled by players in the game is greater than the negative one.

6.5.3. Myerson Cycle Messages Game

The focus of this game is on rewarding coalitions that can create the most communication pairs. When a player who does not communicate with any other player joins the game, it poses a disadvantage to the existing players. This is because they will not be able to form a communication pair with the newcomer, and consequently, the benefits will have to be divided among more players. However, if a player joins a coalition that already has a member, their contribution is advantageous. This is because they can form at least one additional communication pair.

We see this reflected in Figure 11, where two players will have a positive relationship if they share a link with a third one. Per contra, when two players do not share any link with a third common node in $\gamma$, their relationship is negative, as shown in Figure 12.

The game relationships and players’ relevance are represented in

Table 11. Player relevance vector $\mathrm{\Psi }=Sh$ and player relationship matrix $R=I$ in the Myerson Cycle Message Game.

$\mathit{Sh}$	I	P1	P2	P3	P4	P5	P6	P7	P8	P9	P10
4.5	P1	0	2.6710	0.9531	−0.1397	−0.7564	−0.9564	−0.7564	−0.1397	0.9531	2.6710
4.5	P2	2.6710	0	2.6710	0.9531	−0.1397	−0.7564	−0.9564	−0.7564	−0.1397	0.9531
4.5	P3	0.9531	2.6710	0	2.6710	0.9531	−0.1397	−0.7564	−0.9564	−0.7564	−0.1397
4.5	P4	−0.1397	0.9531	2.6710	0	2.6710	0.9531	−0.1397	−0.7564	−0.9564	−0.7564
4.5	P5	−0.7564	−0.1397	0.9531	2.6710	0	2.6710	0.9531	−0.1397	−0.7564	−0.9564
4.5	P6	−0.9564	−0.7564	−0.1397	0.9531	2.6710	0	2.6710	0.9531	−0.1397	−0.7564
4.5	P7	−0.7564	−0.9564	−0.7564	−0.1397	0.9531	2.6710	0	2.6710	0.9531	−0.1397
4.5	P8	−0.1397	−0.7564	−0.9564	−0.7564	−0.1397	0.9531	2.6710	0	2.6710	0.9531
4.5	P9	0.9531	−0.1397	−0.7564	−0.9564	−0.7564	−0.1397	0.9531	2.6710	0	2.6710
4.5	P10	2.6710	0.9531	−0.1397	−0.7564	−0.9564	−0.7564	−0.1397	0.9531	2.6710	0

. As it has the same structure as the Myerson Cycle Cost Game in the example in Section 6.5.2, we conclude that players cooperate in a parity-based environment, where all players have equal influence and benefit from their connections. As in the Myerson Cycle Cost Game, we see that the third column of

Table 12. Descriptive metrics for the Myerson Cycle Message Game relationship matrices.

${\mathit{s}}_{\mathit{i}}$	$\mathit{R}={\mathit{I}}^{+}$	$\mathit{R}={\mathit{I}}^{-}$	$\mathit{R}=\mathit{I}$	$\mathit{R}={\mathit{I}}^{|·|}$
Maximum	7.2484	2.7484	4.5	9.9968
Q3	7.2484	2.7484	4.5	9.9968
Mean	7.2484	2.7484	4.5	9.9968
Median	7.2484	2.7484	4.5	9.9968
Q1	7.2484	2.7484	4.5	9.9968
Minimum	7.2484	2.7484	4.5	9.9968
Standard Deviation	0	0	0	0
Skewness	0	0	0	0
Kurtosis	0	0	0	0
Gini Index	0	0	0	0

, representing the sum of all interactions in each node, is greater than zero in all quartiles.

Visualizing Myerson’s Cycle Cost Game in Figure 10 and the current Cycle Message Game in Figure 11 reveals that although both games share the same communication topology governed by the same $\gamma$ in the form of a cycle, the interaction of the players is different. We see that in the case of the cost game, the players only maintain positive relationships with those who are immediate neighbors in the $\gamma$ topology. On the other hand, in the case of the message game, the players maintain positive relationships with the players who are immediate neighbors in the $\gamma$ topology, and also with those who are two hops away in this topology.

In the next examples, we can also see this behavior more clearly. In the case of cost games, we do not find positive relationships between players who are not direct neighbors in the $\gamma$ topology, as shown in Figure 13. In the case of $\gamma$ topologies in the form of a chain or a star, the message scenarios relate neighboring players who are further away in each of the topologies, as seen in Figure 14, with the specificity that the relationship is attenuated as the distance between them in the topology increases.

6.5.4. Myerson Chain Cost Game

This example deals with the Myerson Chain Cost Game in a communication situation represented by a chain of ten players. In this cost share game, the worth of each player in the chain is influenced by their position and the importance of forming continuous coalitions. Players positioned in the middle of the chain have a higher strategic value because they act as bridges or connectors. This behavior is reflected in Figure 13 and

Table 13. Player relevance vector $\mathrm{\Psi }=Sh$ and player relationship matrix $R=I$ in the Myerson Chain Cost Game.

$\mathit{Sh}$	I	P1	P2	P3	P4	P5	P6	P7	P8	P9	P10
−0.5	P1	0	1	0	0	0	0	0	0	0	0
0	P2	1	0	1	0	0	0	0	0	0	0
0	P3	0	1	0	1	0	0	0	0	0	0
0	P4	0	0	1	0	1	0	0	0	0	0
0	P5	0	0	0	1	0	1	0	0	0	0
0	P6	0	0	0	0	1	0	1	0	0	0
0	P7	0	0	0	0	0	1	0	1	0	0
0	P8	0	0	0	0	0	0	1	0	1	0
0	P9	0	0	0	0	0	0	0	1	0	1
−0.5	P10	0	0	0	0	0	0	0	0	1	0

.

In the context of this Myerson Chain Cost Game, the analysis of the metric of weighted degree presented in

Table 14. Descriptive metrics for the Myerson Chain Cost Game relationship matrices.

${\mathit{s}}_{\mathit{i}}$	$\mathit{R}={\mathit{I}}^{+}$	$\mathit{R}={\mathit{I}}^{-}$	$\mathit{R}=\mathit{I}$	$\mathit{R}={\mathit{I}}^{|·|}$
Maximum	2	0	2	2
Q3	2	0	2	2
Mean	1.8	0	1.8	1.8
Median	2	0	2	2
Q1	2	0	2	2
Minimum	1	0	1	1
Standard Deviation	0.4	0	0.4	0.4
Skewness	−1.5	0	−1.5	−1.5
Kurtosis	0.25	0	0.25	0.25
Gini Index	0.0888	0	0.0888	0.0888

reveals an alignment among the I, $I^{+}$, and $I^{|·|}$ values. This alignment is obvious, given the game’s structure, where negative relationships are absent, showing a collective preference towards establishing cooperation. The observed variability is minimal, underscored by a low Gini Index and limited data dispersion. Contrary to previous observations, the weighted degree distribution exhibits a negative skewness, suggesting a left-skewed distribution, alongside a clustering tendency that surpasses that of a normal distribution (kurtosis = 0.25).

6.5.5. Myerson Chain Messages Game

In this example section, we are using a chain of ten players as an example of a communication situation. In this case, the value of a coalition is based on the number of adjacent pairs within it, as each pair signifies a direct communication link. For instance, the coalition of players P2, P3, P4 has two adjacent pairs, (P2, P3) and (P3, P4), giving it a value of 2. More details about the relationships and player relevance of this game are given in

Table 15. Player relevance vector $\mathrm{\Psi }=Sh$ and player relationship matrix $R=I$ in the Myerson Chain Message Game.

$\mathit{Sh}$	I	P1	P2	P3	P4	P5	P6	P7	P8	P9	P10
1.9289	P1	0	2.8289	1.8289	1.3289	0.9956	0.7456	0.5456	0.3789	0.2361	0.1111
3.7579	P2	2.8289	0	4.5468	3.0468	2.2134	1.6301	1.1801	0.8134	0.5039	0.2361
4.9757	P3	1.8289	4.5468	0	5.6396	3.8063	2.7230	1.9396	1.3230	0.8134	0.3789
5.7353	P4	1.3289	3.0468	5.6396	0	6.2563	4.1730	2.8896	1.9396	1.1801	0.5456
6.1019	P5	0.9956	2.2134	3.8063	6.2563	0	6.4563	4.1730	2.7230	1.6301	0.7456
6.1019	P6	0.7456	1.6301	2.7230	4.1730	6.4563	0	6.2563	3.8063	2.2134	0.9956
5.7353	P7	0.5456	1.1801	1.9396	2.8896	4.1730	6.2563	0	5.6396	3.0468	1.3289
4.9757	P8	0.3789	0.8134	1.3230	1.9396	2.7230	3.8063	5.6396	0	4.5468	1.8289
3.7579	P9	0.2361	0.5039	0.8134	1.1801	1.6301	2.2134	3.0468	4.5468	0	2.8289
1.9289	P10	0.1111	0.2361	0.3789	0.5456	0.7456	0.9956	1.3289	1.8289	2.8289	0

.

Analyzing this game in

Table 16. Descriptive metrics for the Myerson Chain Message Game relationship matrices.

${\mathit{s}}_{\mathit{i}}$	$\mathit{R}={\mathit{I}}^{+}$	$\mathit{R}={\mathit{I}}^{-}$	$\mathit{R}=\mathit{I}$	$\mathit{R}={\mathit{I}}^{|·|}$
Maximum	29	0	29	29
Q3	27	0	27	27
Mean	21	0	21	21
Median	23	0	23	23
Q1	17	0	17	17
Minimum	9	0	9	9
Standard Deviation	7.2663	0	7.2663	7.2663
Skewness	−0.5504	0	−0.5504	−0.5504
Kurtosis	−1.1060	0	−1.1060	−1.1060
Gini Index	0.1904	0	0.1904	0.1904

, it shows identical values across I, $I^{+}$, and $I^{|·|}$. As in the Myerson Chain Cost Game in Section 6.5.4, negative interactions are absent, meaning there is no interest in breaking collaboration. We also see that players in the middle of the chain tend to have a higher participation, as they possess more opportunities to form valuable coalitions with their adjacent neighbors. This is depicted in Figure 14 and Figure 15. The game’s variability remains reduced, as indicated by a Gini Index of 0.1904 and a packed range of data values. The distribution tilts towards a left skew (skewness = −0.5504), and displays a concentration of data points more pronounced than that observed in normal distributions (kurtosis = −1.1060).

6.5.6. Myerson Star Cost Game

Here, we describe the unitary cost share game in a Myerson communication situation represented by the star of ten players. A star distribution differs from a chain in its structure. In a star, there is a central player to whom all other players are directly connected, as defined in

Table 17. Player relevance vector $\mathrm{\Psi }=Sh$ and player relations $R=I$ in the Myerson Star Cost Game.

$\mathit{Sh}$	I	P1	P2	P3	P4	P5	P6	P7	P8	P9	P10
3.5	P1	0	1	1	1	1	1	1	1	1	1
−0.5	P2	1	0	0	0	0	0	0	0	0	0
−0.5	P3	1	0	0	0	0	0	0	0	0	0
−0.5	P4	1	0	0	0	0	0	0	0	0	0
−0.5	P5	1	0	0	0	0	0	0	0	0	0
−0.5	P6	1	0	0	0	0	0	0	0	0	0
−0.5	P7	1	0	0	0	0	0	0	0	0	0
−0.5	P8	1	0	0	0	0	0	0	0	0	0
−0.5	P9	1	0	0	0	0	0	0	0	0	0
−0.5	P10	1	0	0	0	0	0	0	0	0	0

. When studying the Myerson Unitary Cost Share Game, we see that the central player holds significant value due to its unique position connecting all peripheral players. In consequence, peripheral players have value in relation to their direct link to the center. This game highlights how central figures can dramatically influence cooperative outcomes and the strategic importance of the central player in star networks, as shown in Figure 16.

When analyzing the metrics of the weighted degree in

Table 18. Descriptive metrics for the Myerson Star Cost Game relationship matrices.

${\mathit{s}}_{\mathit{i}}$	$\mathit{R}={\mathit{I}}^{+}$	$\mathit{R}={\mathit{I}}^{-}$	$\mathit{R}=\mathit{I}$	$\mathit{R}={\mathit{I}}^{|·|}$
Maximum	9	0	9	9
Q3	1	0	1	1
Mean	1.8	0	1.8	1.8
Median	1	0	1	1
Q1	1	0	1	1
Minimum	1	0	1	1
Standard Deviation	2.4	0	2.4	2.4
Skewness	2.6666	0	2.6666	2.6666
Kurtosis	5.1111	0	5.1111	5.1111
Gini Index	0.4	0	0.4	0.4

, we can verify that the network is characterized by a global interest in collaborating, represented by identical I, $I^{+}$, and $I^{|·|}$ values. This time kurtosis tells us about a distribution with a concentration of data points less pronounced than that observed in normal distributions (kurtosis = 5.1111). We may also highlight the fact that the central player relevance is clearly described by the Gini Index of 0.4, which is higher than the previous examples. The asymmetry of the distribution is also evident, as indicated by a positive skewness (skewness = 2.6666).

6.5.7. Myerson Star Messages Game

Finally, we show in this section the detailed information for the Myerson Star Message Game. In the Myerson’s message game on a star distribution, the central hub player is pivotal for message transmission between any two peripheral players. If any peripheral player wishes to send a message to another peripheral player, it must go through the central player. This gives the central player significant strategic importance and value, since they control the flow of communication. The information we use to describe this game is shown in

Table 19. Player relevance vector $\mathrm{\Psi }=Sh$ and player relations $R=I$ in the Myerson Star Message Game.

$\mathit{Sh}$	I	P1	P2	P3	P4	P5	P6	P7	P8	P9	P10
16.5	P1	0	5	5	5	5	5	5	5	5	5
3.1666	P2	5	0	0.5	0.5	0.5	0.5	0.5	0.5	0.5	0.5
3.1666	P3	5	0.5	0	0.5	0.5	0.5	0.5	0.5	0.5	0.5
3.1666	P4	5	0.5	0.5	0	0.5	0.5	0.5	0.5	0.5	0.5
3.1666	P5	5	0.5	0.5	0.5	0	0.5	0.5	0.5	0.5	0.5
3.1666	P6	5	0.5	0.5	0.5	0.5	0	0.5	0.5	0.5	0.5
3.1666	P7	5	0.5	0.5	0.5	0.5	0.5	0	0.5	0.5	0.5
3.1666	P8	5	0.5	0.5	0.5	0.5	0.5	0.5	0	0.5	0.5
3.1666	P9	5	0.5	0.5	0.5	0.5	0.5	0.5	0.5	0	0.5
3.1666	P10	5	0.5	0.5	0.5	0.5	0.5	0.5	0.5	0.5	0

.

This cooperative game is characterized by a power imbalance where the central players have significantly more influence over the game’s outcomes than the peripheral players, as represented in Figure 17 and Figure 18.

Compared with Table 18,

Table 20. Descriptive metrics for the Myerson Star Message Game relationship matrices.

${\mathit{s}}_{\mathit{i}}$	$\mathit{R}={\mathit{I}}^{+}$	$\mathit{R}={\mathit{I}}^{-}$	$\mathit{R}=\mathit{I}$	$\mathit{R}={\mathit{I}}^{|·|}$
Maximum	45	0	45	45
Q3	9	0	9	9
Mean	12.6	0	12.6	12.6
Median	9	0	9	9
Q1	9	0	9	9
Minimum	9	0	9	9
Standard Deviation	10.8	0	10.8	10.8
Skewness	2.6666	0	2.6666	2.6666
Kurtosis	5.1111	0	5.1111	5.1111
Gini Index	0.2571	0	0.2571	0.2571

shows slightly lower dispersion values, with a maximum value five times greater than the minimum weighted degree. Nonetheless, the distribution of this game is also characterized by exactly alike values of I, $I^{+}$, and $I^{|·|}$, a concentration of data points less pronounced than that observed in normal distributions (kurtosis = 5.1111), and a positive skewness (skewness = 2.6666).

7. Conclusions and Future Research Directions

In this paper, we introduce a new method to visualize the relationships among players in a cooperative game. This visualization and the descriptive analysis of the associated network enhance the understanding of the cooperative game under analysis. It is important to note that the interaction between two players can be positive (when there is an interest and a benefit in staying together), negative (when there is a detriment to interacting together), or neutral, when there is no motivation either to form or not to form a coalition. So, we have distinguished among three different interaction matrices from a game. Additionally, by examining the degree of these interaction matrices, we can distinguish between players inclined towards cooperation and those who are not, establishing a ranking among them.

Let us also note that it is possible to establish general metrics for the entire network of relationships among players, enabling the comparison of different games based on overall characteristics of the entire graph, as we have carried out in the examples. This not only allows for a better understanding of the game but also makes it possible to establish differences between games that may not be easily perceptible from the solution concepts. For example, through an analysis of the ‘weighted degree distribution’ of the network associated with the cooperative game, we can discern games with greater density or variability in cooperation among many other network characteristics.

Finally, we want to conclude this section emphasizing that it is possible to continue this work in many ways. For example, we can highlight the potential for alternative definitions of $\mathrm{\Psi }$ and R with different properties and applications. Moreover, the properties of the definitions $\mathrm{\Psi }=Sh$ and $R=I$ can be analyzed in greater depth. Finally, characterizations for $\mathrm{\Psi }=Sh$ and $R=I$, or for other definitions in this paper, could be explored.