2. Example

Suppose a group of college friends, connected on Facebook, graduate and start their careers in different fields. Over time, they maintain their Facebook connections, sharing updates about their lives, jobs, and accomplishments. Take, for example, a member working as a graphic designer who posts on Facebook about a project they are currently working on. A member of the group with marketing expertise offers to help. They connect on LinkedIn to discuss potential collaborations. This connection generates a resource of their potential partnerships and professional collaborations which is dependent on both their Facebook and LinkedIn networks. Thus, by leveraging their Facebook connections, this group of individuals has created a valuable LinkedIn network that can generate various resources, through job opportunities, professional advice and collaborations. Suitable resource allocation within this network can enhance the benefits for all members.

Figure 1. Facebook network: $g_1=\{12,13, 23\}$ and LinkedIn network: $g_2=\{12,14,15\}$ .

Suppose players 1, 2, and 3 connect on Facebook and seek collaborations within their LinkedIn networks. Let 4 and 5 join them from 1’s LinkedIn network, and 2 already have a connection with 1 on LinkedIn. We consider this situation as an ordered pair of networks $(\{12, 13, 23\}, \{12, 14, 15\})$ as shown in Figure 1. There can be 64 possible collaborations in this way,Footnote ¹ viz., $(\{12, 13\}, \{12\}), (\{12, 13\}, \{14\})$ , $(\{12, 13\}, \{14, 15\})$ , $(\{12, 13, 23\}, \{15\})$ etc. All these possible collaborations generate resources, for example, in the form of market share through their networked efforts. Finally, based on all such possible interactions, the resource they generate needs to be shared among all the players. Existing theories on network games and cooperative games do not capture situations involving these multiple interconnected networks. Bi-cooperative network games are also limited in this regard, as they do not consider links like $\{12\}$ , which belong to both the Facebook and LinkedIn networks in this example. On the other hand, multigraphs treat such links as parallel links. However, in reality, links like $\{12\}$ are single links that share different information on different platforms, like a person adopting different roles in different contexts. In classical network games, as already mentioned, the Position value is one of the fairest allocation rules, taking into account player interactions in the networks. It aggregates the contributions of the subnetworks of a given network. It does not consider the influence of multiple networks. On the other hand, although the Position value in bi-cooperative network games takes into account the value generated by the network pairs, they only consider disjoint network pairs, viz., $(\{12, 13\}, \{14\})$ , $(\{12, 13\}, \{14, 15\})$ etc. Nevertheless, in case of multilayer network games, we also have to consider the values generated by pairs viz., $(\{12, 13\}, \{12, 15\})$ etc. Here, $\{12\}$ in the first component is a Facebook link and that in the second component is a LinkedIn link. Therefore, the Position value for multilayer network games needs to be defined in a more general setup such that the values of all possible ordered network pairs are taken into account. This further justifies that the study of link-based allocation rules in multilayer network games is more significant, as understanding the characteristics of various links across multiple networks helps designers identify the most valuable connections.

Government agencies can apply this concept for national accounting or tax and subsidy purposes when dealing with conglomerates. As conglomerates operate in multiple industries, the trade networks involving conglomerates can precisely be modeled by a multilayer network. The Position value naturally captures the value added by every trade link and thus gives good estimate for industrial policy perspective.

3. Preliminaries

In this section, we present the definitions and results from (Boruah et al., 2023; Grabisch and Labreuche, Reference Grabisch and Labreuche2002; Jackson and Wolinsky, Reference Jackson and Wolinsky1996; Jackson, Reference Jackson2005; Labreuche and Grabisch, Reference Labreuche and Grabisch2008) necessary for the development of our model. Specifically, definitions1, 2, 4, 5 and their consequences will be utilized in designing the value function for multilayer network games. Additionally, definitions3, 6, 7 will be used in the formulation of the Position value for multilayer network games.

Let $N=\{1,2,\ldots ,n\}$ be a finite player set. Every subset $S$ of $N$ is termed as a coalition and $N$ is referred to as the grand coalition. We will use lowercase letters, such as $s$ and $t$ , to represent the cardinality of sets $S$ and $T$ , respectively, i.e., ‘ $\#S$ ’ is equivalent to “ $s$ ” throughout the paper. Sometimes, depending upon the context, we will use “ $|S|$ ” to denote the cardinality of the set $S$ .

3.1 Overlapping coalitional games

The concept of an overlapping coalitional game is introduced in Boruah et al. (2023). Below, we formally present the model. Let there be two attributes, call them X and Z, where players can contribute independently.

Given a player set $N$ , let $2^N \times 2^N=\{(S, T)\,:\, S, T \subseteq N\}$ be the set of pairs of coalition. We call each $(S, T)\in 2^N \times 2^N$ an overlapping bi-coalition. For any $(S, T) \in 2^N \times 2^N$ , we assume that players in $S$ are X contributors, players in $T$ are Z contributors, players in $S \cap T$ contribute to both X and Z, call them X–Z contributors, and players in $N \setminus (S\cup T)$ absentees.

Definition 1. An overlapping coalitional game is a function $o\,:\,2^N \times 2^N \mapsto \mathbb{R}$ with $o(\emptyset , \emptyset )=0$ .

For each $(S, T) \in 2^N \times 2^N$ , $o(S, T)$ represents the worth (gain or loss) of $(S, T)$ , when the players in $S$ are X contributors, players in $T$ are Z contributors and $S\cap T$ may or may not be empty and finally the remaining $N\setminus (S \cup T)$ are absentees. Let $\mathbb{O}^N$ be the real vector space of all overlapping coalitional games on $N$ . Now, consider the order relation $\sqsubseteq$ on $2^N \times 2^N$ as follows: $(A, B)\sqsubseteq (C, D)\iff A \subseteq C\; \text{and} \; B \subseteq D.$ For $(S^{\prime},T^{\prime}) \in 2^N \times 2^N$ with $(S^{\prime},T^{\prime})\neq (\emptyset , \emptyset )$ , consider the following special games in $\mathbb{O}^N$ .

Definition 2. The superior unanimity game $\bar{u}_{(S^{\prime},T^{\prime})}\ : \ 2^N \times 2^N \mapsto \mathbb{R}$ is defined as follows

(1) \begin{align} \bar{u}_{(S^{\prime},T^{\prime})}(A,B)&=\begin{cases} 1& \ \text{if}\ (S^{\prime},T^{\prime}) \sqsubseteq (A,B), (A,B) \neq (\emptyset , \emptyset ),\\[3pt] 0& \ {otherwise.} \end{cases} \end{align}

The set of superior unanimity games forms a basis for $\mathbb{O}^N$ and so every $o \in \mathbb{O}^N$ can be expressed as a linear combination of the superior unanimity games as follows,

(2) \begin{equation} o=\sum _{(S^{\prime},T^{\prime})\in 2^N \times 2^N}c_{(S^{\prime},T^{\prime})}\bar{u}_{(S^{\prime},T^{\prime})} \end{equation}

where $c_{(S^{\prime},T^{\prime})}$ are the real constants. For every $o \in \mathbb{O}^N$ , we can associate a cooperative game $V$ defined on $S \cup T$ such that $V(P)=o(P \cap S, P \cap T)$ , for all $P \subseteq S \cup T$ . Consequently, $V$ possesses a corresponding representation in terms of unanimity cooperative games $\{U_P| P \subseteq N, P \neq \emptyset \}$ as follows (Harsanyi, Reference Harsanyi1963; Shapley, Reference Shapley, Kuhn and Tucker1953).

(3) \begin{equation} V=\sum _{\emptyset \neq P \subseteq S\cup T}c_P U_P \end{equation}

where $c_P=\sum _{M \subseteq P}({-}1)^{p-m}V(M)$ and the unanimity cooperative games $U_P$ are defined by

\begin{align*} U_P(S)&=\begin{cases} 1 & \text{if}\ P \subseteq S,\\ 0 & \text{otherwise.} \end{cases} \end{align*}

It follows that

(4) \begin{equation} \sum _{\emptyset \neq P \subseteq S \cup T}c_P=V(S \cup T)=o((S \cup T)\cap S, (S \cup T) \cap T)=o(S,T). \end{equation}

Definition 3. A one-point solution concept or a value for overlapping coalitional games is a function that assigns an $n$ -dimensional real vector to each overlapping coalitional game. This vector serves as a representation of a payoff distribution among the players.

Here, we introduce a value for overlapping coalitional games similar to the LG value (Labreuche and Grabisch, Reference Labreuche and Grabisch2008). This value is termed to as the Overlapping LG value, abbreviated as OLG, and is defined as follows.

For any $o \in \mathbb{O}^N$ , $(S,T) \in 2^N \times 2^N$ such that for all $i \in N$

(5) \begin{equation} \Phi ^{OLG}_i(o)(S,T)=\sum _{P \subseteq (S\cup T) \setminus i} \dfrac{p!(s+t-st-p-1)!}{(s+t-st)!}\big (V(P \cup i)-V(P)\big ) \end{equation}

where for $ P \subseteq S \cup T$ , $V(P)=o(P\cap S, P \cap T)$ and $\# S\cap T\,:\!=\,st$ .

Applying Eq.s (2), (3), (4) on (5), we find the expression of the OLG value for $o \in \mathbb{O}^N$ in terms of the Harsanyi’s dividend (Harsanyi, Reference Harsanyi1963) for $\Phi ^{Sh}$ , the Shapley value (Shapley, Reference Shapley, Kuhn and Tucker1953) of the associated cooperative game $(S\cup T, V)$ as follows.

(6) \begin{equation} \Phi ^{OLG}_i(o)(S,T)=\Phi ^{Sh}_i(S \cup T, V)=\sum _{P \subseteq S \cup T\,:\, i \in P} \dfrac{c_P}{p}. \end{equation}

3.2 Network games

Given the player set $N$ and distinct players $i,j\in N$ , a link $ij$ is the pair $\{ i,j \}$ which represents an undirected relationship between $i$ and $j$ . Clearly, $ij$ is equivalent to $ji$ . The set of all possible links with the player set $N$ denoted by $g^{N}=\{ij$ $|$ $i,j\in N$ and $i\neq j\}$ is called the complete network. Let $G^N=\{g| g\subseteq g^N\}$ be the set of all possible networks on $N$ . The network $\varnothing$ is the network without any links, which we refer to as the empty network. Let the number of links in a network $g$ be denoted by $\ell (g)$ . We can consider the links in a network $g$ as hypothetical players. Thus $g$ can be thought of as a coalition of the players representing these links in $g$ . However, to distinguish the two representations, we use $[g]$ to denote the coalition. Obviously, $\ell (g^{N})= \binom{n}{2} = \tfrac{1}{2} n(n-1)$ and $\ell (\varnothing )=0$ . Let $L_i(g)=\{ij| \{i,j\} \in g\}$ be the set of links in which player $i$ is involved. Denote $\ell _i(g)$ , the number of links in player $i$ ’s link set, and $\#L_i(g)=\ell _i(g)$ . Therefore, it follows that $\ell (g)=\dfrac{1}{2} \sum _{i \in N} \ell _i(g)$ . Let $N(g)$ be the set of all players in $g$ . A value function $v$ for networks is such that $v\,:\, G^N \mapsto \mathbb{R}$ with $v(\varnothing )=0$ . For a network $g$ , $v(g)$ specifies the total worth generated by the network $g$ . Denote $\mathbb{V}^N=\{ v| v\,:\, G^N \mapsto \mathbb{R}, v(\varnothing )=0\}$ , the set of all value functions.

Definition 4. A network game is defined as a pair $(N, v)$ , where $N$ is the player set and $v$ is the value function.

When the player set $N$ is fixed and there is no ambiguity, we represent a network game simply as $v$ .

Definition 5. For $g \in G^N$ , the unanimity value function $u_{g}$ is given by

(7) \begin{equation} u_{g}(g^{\prime})=\begin{cases} 1 & {if} \ g \subseteq g^{\prime}\\ 0& \ {otherwise}. \end{cases} \end{equation}

The unanimity value functions form a basis for $\mathbb V^N$ . Thus $v \in \mathbb{V}^N$ can be written as a linear combination of unanimity value functions $u_g$ , i.e., $v=\sum _{g \subseteq g^N}c_g u_g$ , where $c_g\in \mathbb{R}$ are the unanimity coefficients of $v$ .

Definition 6. A network allocation rule is a function $Y\colon \mathbb G^N \times \mathbb V^N \to \mathbb R^n$ such that for every $g \in \mathbb G^N$ and every $v \in \mathbb V^N$ , $Y_i(g,v)$ represents the payoff to player $i$ with the condition that $Y_i(\varnothing , v)= 0$ .

Definition 7. For a given network $g$ , let $v$ be a value function with unanimity coefficients ${(c_{g^{\prime}})}_{g^{\prime}\subseteq g}$ . Then the Position value (Slikker, Reference Slikker2005), $Y^{PV}(g,v)$ is defined by

(8) \begin{equation} Y^{PV}_i(g,v)=\sum _{g^{\prime}\subseteq g} \dfrac{c_{g^{\prime}}{\ell }_i(g^{\prime})}{2{\ell }(g^{\prime})},\ \forall i \in N. \end{equation}

Let $v^g$ be the associated cooperative game with respect to the network game $v$ considering the links in $g^N$ as players. It follows that for every link $l$ of $g$ , the Shapley value $\Phi ^{Sh}$ (Shapley, Reference Shapley, Kuhn and Tucker1953) of $l$ with respect to $v^g$ is given by

(9) \begin{equation} \Phi ^{Sh}_{l}(g, v^g)=\sum _{g^{\prime}\subseteq g\,:\,l \subseteq g^{\prime}}\dfrac{c_{g^{\prime}}}{{\ell }(g^{\prime})}. \end{equation}

Now, from Eq.s (8) and (9), we obtain

(10) \begin{equation} Y^{PV}_i(g,v)=\sum _{l \in L_i(g)} \dfrac{1}{2}\Phi ^{Sh}_{l}(g, v^g),\ \forall i \in N. \end{equation}

4. Multilayer network games

In this section, we introduce the notion of our proposed multilayer network games. Let $g_1, g_2 \in G^N$ be two networks with possibly non-empty $g_1 \cap g_2$ . Then $(g_1,g_2)$ is called a multilayer network. We assume that players in $g_1$ and $g_2$ as X and Z contributors, respectively. Players belonging to $g_1 \cap g_2$ are X–Z contributors, and players in the set $N \setminus N(g_1 \cup g_2)$ are absentees. Following ideas similar to those of overlapping coalitions, we call a link in $g_1$ and $g_2$ an X-link and a Z link, respectively, and refer to them as single-layer links. On the other hand, a link in $g_1 \cap g_2$ exhibits both X and Z attributes is called an X–Z link, which we refer as multilayer link. In Figure 2, we present a multilayer network $(g_1, g_2)$ on the player set $N =\{1,2,3,4,5\}$ where $g_1=\{12,23\}$ and $g_2=\{23, 24, 34\}$ . Both $g_1$ and $g_2$ contains two common players $2$ and $3$ , and therefore, $(g_1, g_2)$ having an X–Z link $\{23\}$ forms a multilayer network. Let $2^{g^N} \times 2^{g^N}=\{(g_1,g_2)|g_1,g_2 \subseteq g^N\}$ be the set of all multilayer networks on $N$ .

For $(g_1,g_2) \in 2^{g^N} \times 2^{g^N}$ , let $L(g_1,g_2)$ denote the set of links in $g_1$ and $g_2$ , i.e., $L(g_1,g_2)=L(g_1 \cup g_2)$ and $\ell (g_1,g_2)$ represents the total number of links in $L(g_1,g_2)$ . Let $L_i(g_1,g_2)=L_i(g_1\cup g_2)$ be the set of links that involve player $i$ in $g_1$ and $g_2$ . Let $\ell _i(g_1,g_2)=\# L_i(g_1,g_2)$ . For any link $l$ , by $(g_1,g_2)\setminus l$ , we mean $(g_1 \setminus l, g_2 \setminus l)$ . We denote by $N(g_1, g_2)=N(g_1 \cup g_2)$ , the set of players in $g_1$ and $g_2$ .

Figure 2. An example of a multilayer network $(g_1,g_2)$ on $N=\{1,2,3,4,5\}$ .

A value function is a function $v^m\,:\,2^{g^N} \times 2^{g^N} \mapsto \mathbb{R}$ such that $v^m(\varnothing ,\varnothing )=0$ . Thus, a value function assigns each $(g_1,g_2) \in 2^{g^N} \times 2^{g^N}$ a real number, its generated worth, when players in $g_1$ are X contributors, players in $g_2$ are Z contributors, players in $g_1 \cap g_2$ act as X–Z contributors, and the remaining players are considered absentees. Let $ \mathcal{MG} $ denotes the set of all value functions on $2^{g^N} \times 2^{g^N}$ .

If the player set $N$ is fixed and there is no ambiguity, we simply write $v^m$ to denote a multilayer network game.

For each $i$ , $Y_i((g_1,g_2),v^m)$ represents the payoff to the player $i$ with respect to $v^m \in \mathcal{MG}$ and $(g_1,g_2) \in 2^{g^N} \times 2^{g^N}$ .

5. The Position value for multilayer network games

Here, we initially present the link game associated with a multilayer network game, where the set of links serves as the set of players.

Let $[g^N]$ represent the hypothetical player set corresponding to the links in $g^N$ . For $(g_1, g_2) \in 2^{g^N} \times 2^{g^N}$ , let $[g_1]$ and $[g_2]$ refer to the set of all hypothetical players representing the links in $g_1$ and $g_2$ respectively. For a given multilayer network game $(N,v^m)$ , the associated link game $([g^N],{v^m}^*)$ (where ${v^m}^*\,:\, \mathbb O ^{[g^N]} \mapsto \mathbb{R}$ is a function) of $(N,v^m)$ is defined as follows.

For each $(S,T) \in 2^{[g^N]} \times 2^{[g^N]}=\{(S,T)| S, T \subseteq [g^N]\}$ there is a $(g_1,g_2) \in 2^{g^N} \times 2^{g^N}$ with $S=[g_1]$ , $T=[g_2]$ such that ${v^m}^*(S,T)=v^m(g_1,g_2)$ . Conversely, for each $(g_1, g_2) \in 2^{g^N} \times 2^{g^N}$ , there is a pair $(S, T) \in 2^{[g^N]} \times 2^{[g^N]}$ such that $S = [g_1]$ and $T = [g_2]$ .

Definition 10. An allocation rule $Y$ is called a link-based allocation rule if there is some $Y^L\,:\,\big({2^{g^N} \times 2^{g^N}}\big) \times \mathcal{MG} \mapsto \mathbb{R}^{\frac{n(n-1)}{2}}$ such that for all $(g_1,g_2) \in 2^{g^N} \times 2^{g^N}$ , $v^m \in \mathcal{MG}$ , $i \in N$ ,

\begin{equation*} \sum _{l \in ([g_1]\cup [g_2])} Y^L_l((g_1,g_2),v^m)=v^m(g_1,g_2), \end{equation*}

and

\begin{equation*} Y_i((g_1,g_2),v^m)=\sum _{l \in L_i(g_1,g_2)}\dfrac {Y^L_l((g_1,g_2),v^m)}{2}. \end{equation*}

Definition 11. The Position value in a multilayer network game, denoted as $Y^{MNPV}$ , is the allocation rule where each player $i \in N$ in a multilayer network game $(N,v^m)$ receives half of the OLG value for each of their links considered as hypothetical players in the associated link game $([g^N],{v^m}^*)$ , i.e.,

\begin{equation*} Y^{MNPV}_i((g_1,g_2),v^m)=\sum _{l \in L_i(g_1,g_2)}\dfrac {1}{2}\Phi _l^{OLG}({v^m}^*)([g_1],[g_2]). \end{equation*}

5.1 Characterization of the Position value

Here, we provide an axiomatic characterization of the Position value for the class of multilayer network games using axioms of Efficiency and Balanced Link Contribution.

Claim 1 (Efficiency). An allocation rule $Y$ satisfies Efficiency if for any $v^m \in \mathcal{MG}$ and $(g_1,g_2)\in 2^{g^N} \times 2^{g^N}$ we have,

\begin{equation*} \sum _{i \in N(g_1,g_2)}Y_i((g_1,g_2),v^m)=v^m(g_1,g_2). \end{equation*}

The Efficiency axiom implies that the payoffs received by players under the allocation rule should sum to the total worth generated by the multilayer network $(g_1, g_2)$ . This principle is consistent with the notion that an efficient allocation ensures the accurate distribution of the total worth within a multilayer network among its individual players in accordance with the specified allocation rules.

Lemma 1. The Position value in a multilayer network game satisfies Efficiency.

Proof. It directly follows from the definition11.

Claim 2 (Balanced Link Contribution). An allocation rule $Y$ satisfies Balanced Link Contribution if for any $v^m \in \mathcal{MG}$ and $(g_1,g_2) \in 2^{g^N} \times 2^{g^N}$ and $i,j \in N$ , we have

\begin{equation*} \sum _{l \in L_j(g_1,g_2)}\big ( Y_i((g_1,g_2),v^m)-Y_i((g_1,g_2)\setminus l,v^m) \big )= \sum _{l \in L_i(g_1,g_2)}\big ( Y_j((g_1,g_2),v^m)-Y_j((g_1,g_2)\setminus l,v^m) \big ). \end{equation*}

The Balanced Link Contribution axiom states that, within a multilayer network game and under an allocation rule, the impact of losing a link, whether multilayer or single-layer is identical for the involved players. Moreover, the net effects of removal of a X–Z link, remain balanced between players, ensuring fairness in the allocation process. The unique feature highlighted here is that the removal of a multilayer link results in its removal from both the networks $g_1$ and $g_2$ .

Lemma 2. The Position value in a multilayer network game satisfies Balanced Link Contribution.

Proof. For a given $v^m \in \mathcal{MG}$ , $(g_1,g_2) \in 2^{g^N} \times 2^{g^N}$ and $i, j \in N$ , let $Y^{MNPV}_i$ and $Y^{MNPV}_j$ be the $i^{th}$ and $j^{th}$ components of the Position value. Now,

\begin{align*} &\sum _{l \in L_j(g_1,g_2)}\big ( Y^{MNPV}_i((g_1,g_2),v^m)-Y^{MNPV}_i((g_1,g_2)\setminus l,v^m) \big )\\ &= \sum _{l \in L_j(g_1,g_2)}\bigg ( \sum _{l_1 \in L_i(g_1,g_2)} \frac{1}{2} \Phi _{l_1}^{OLG}({v^m}^*)([g_1],[g_2])-\sum _{l^{\prime}_1 \in L_i(g_1\setminus l,g_2\setminus l)} \frac{1}{2} \Phi _{l^{\prime}_1}^{OLG}({v^m}^*)([g_1 \setminus l],[g_2\setminus l ])\bigg )\\ &=\sum _{l \in L_j(g_1,g_2)}\bigg ( \sum _{l_1 \in L_i(g_1,g_2)} \frac{1}{2} \Phi _{l_1}^{Sh}([g_1]\cup [g_2], V)-\sum _{l^{\prime}_1 \in L_i(g_1\setminus l,g_2\setminus l)} \frac{1}{2} \Phi _{l^{\prime}_1}^{Sh}([g_1 \setminus l] \cup [g_2\setminus l ], V)\bigg )\\ &=\frac{1}{2}\sum _{l \in L_j(g_1,g_2)}\bigg ( \sum _{l_1 \in L_i(g_1,g_2)}\ \sum _{P \subseteq ([g_1] \cup [g_2])\,:\,l_1 \in P} \dfrac{c_P}{|P|}-\sum _{l^{\prime}_1 \in L_i(g_1\setminus l,g_2\setminus l)} \ \sum _{P^{\prime} \subseteq ([g_1 \setminus l] \cup [g_2 \setminus l])\,:\,l^{\prime}_1 \in P^{\prime}} \dfrac{c_{P^{\prime}}}{|P^{\prime}|}\bigg )\\ &=\frac{1}{2}\bigg ( \sum _{P \subseteq [g_1] \cup [g_2]}|P_j| \dfrac{c_P}{|P|}|P_i|- \sum _{P^{\prime} \subseteq [g_1 \setminus l] \cup [g_2 \setminus l]} |P^{\prime}_j|\dfrac{c_{P^{\prime}}}{|P^{\prime}|}|P^{\prime}_i| \bigg )\\ &=\sum _{l \in L_i(g_1,g_2)}\big ( Y^{MNPV}_j((g_1,g_2),v^m)-Y^{MNPV}_j((g_1,g_2)\setminus l,v^m) \big ), \end{align*}

where the first equality follows by definition, the second and third equalities are derived from Eq.(6), and the fourth equality is obtained through the rearrangement of terms. Here, for each $S \subseteq N$ , $|S_i|$ denotes the number of hypothetical players in S corresponding to the links involving player $i \in N$ . Note that the expression after the fourth equality sign is symmetric in both $i$ and $j$ .

Next, we will establish that the Position value for the class of multilayer network games is the unique allocation rule that satisfies these two axioms: Efficiency and Balanced Link Contribution.

Theorem 1. The Position value $Y^{MNPV}$ is the unique allocation rule determined by the axioms of Efficiency and Balanced Link Contribution.

Proof. The proof proceeds through induction on $\ell (g_1, g_2)$ , where $(g_1, g_2) \in 2^{g^N} \times 2^{g^N}$ . Let $N=\{i,j\}$ such that $\ell (g_1,g_2)=1$ . Then, one of the following scenarios arises:

Case (i) $g_1=\{ij\}$ , $g_2=\varnothing$ ,

Case (ii) $g_1=\varnothing$ , $g_2=\{ij\}$ and

Case (iii) $g_1=\{ij\}$ , $g_2=\{ij\}$ .

Case (i): If possible let, $Y,\ Y^{\prime}$ be two allocation rules that satisfy Efficiency and Balanced Link Contribution. It follows directly from Balanced Link Contribution that,

Similarly, $Y^{\prime}_i((g_1,g_2),v^m)=Y^{\prime}_j((g_1,g_2),v^m)$ . Therefore, by Efficiency, we have $Y=Y^{\prime}$ .

Cases (ii) and (iii) are similar to Case (i), and so the proofs are omitted here. Thus, the result holds for a single link.

Let $\ell (g_1, g_2)\gt 1$ . For $\ell (g_1,g_2)\leq k-1$ , suppose the allocation rule for $v^m$ which satisfies Efficiency and Balanced Link Contribution is unique. From Balanced Link Contribution, we have

\begin{align*} \sum _{l \in L_2(g_1,g_2)}\big (Y_1((g_1,g_2),v^m)-Y_1((g_1,g_2)\setminus l,v^m) \big )&=\sum _{l \in L_1(g_1,g_2)}\big (Y_2((g_1,g_2),v^m)-Y_2((g_1,g_2)\setminus l,v^m) \big ),\\ \sum _{l \in L_3(g_1,g_2)}\big (Y_1((g_1,g_2),v^m)-Y_1((g_1,g_2)\setminus l,v^m) \big )&=\sum _{l \in L_1(g_1,g_2)}\big (Y_3((g_1,g_2),v^m)-Y_3((g_1,g_2)\setminus l,v^m) \big ),\\ \vdots \hspace{95pt} &\vdots \hspace{105pt}\vdots \\ \sum _{l \in L_n(g_1,g_2)}\big (Y_1((g_1,g_2),v^m)-Y_1((g_1,g_2)\setminus l,v^m) \big )&=\sum _{l \in L_1(g_1,g_2)}\big (Y_n((g_1,g_2),v^m)-Y_n((g_1,g_2)\setminus l,v^m) \big ). \end{align*}

This would further imply,

\begin{align*} &\ell _2(g_1,g_2)Y_1((g_1,g_2),v^m)- \ell _1(g_1,g_2)Y_2((g_1,g_2),v^m) &=& \sum _{l \in L_2(g_1,g_2)}Y^{MNPV}_1((g_1,g_2) \setminus l,v^m)\\ &&&-\sum _{l \in L_1(g_1,g_2)}Y^{MNPV}_2((g_1,g_2) \setminus l,v^m), \\ &\ell _3(g_1,g_2)Y_1((g_1,g_2),v^m)- \ell _1(g_1,g_2)Y_3((g_1,g_2),v^m) &=& \sum _{l \in L_3(g_1,g_2)}Y^{MNPV}_1((g_1,g_2) \setminus l,v^m)\\ &&& - \sum _{l \in L_1(g_1,g_2)}Y^{MNPV}_3((g_1,g_2) \setminus l,v^m), \\ &\hspace{125pt}\vdots & \vdots &\hspace{80pt} \vdots \hspace{120pt}\\ &\ell _n(g_1,g_2)Y_1((g_1,g_2),v^m)- \ell _1(g_1,g_2)Y_n((g_1,g_2),v^m) &=& \sum _{l \in L_n(g_1,g_2)}Y^{MNPV}_1((g_1,g_2) \setminus l,v^m)\\ &&& - \sum _{l \in L_1(g_1,g_2)}Y^{MNPV}_n((g_1,g_2) \setminus l,v^m), \end{align*}

and by Efficiency,

\begin{align*} \sum _{i \in N(g_1,g_2)}Y_i((g_1,g_2),v^m)=v^m(g_1,g_2). \end{align*}

Here, we have a system of $n$ linearly independent equations with $n$ variables: $Y_1$ , $Y_2$ , …, $Y_n$ . The coefficient matrix is of full rank, $n$ . Hence, the system possesses a unique solution. Therefore, for $\ell (g_1, g_2) = k$ , the allocation rule that upholds Efficiency and Balanced Link Contribution is unique. In conclusion, the Position value stands as the unique allocation rule meeting the axioms of Efficiency and Balanced Link Contribution.

6. Concluding discussions

In this section, we explore the distinctions and parallels of multilayer network games, bi-cooperative network games, and multigraphs. By understanding how links and nodes operate within these different frameworks, we can better address various network challenges.

In a multilayer network, a link acts as an X–Z link, displaying both X and Z characteristics simultaneously. It is possible that the set of links for both multilayer network games and bi-cooperative network games are the same. However, in a bi-cooperative network, a link can either be exclusively X or Z but not both at the same time, see Figure 2. Hence, even though the set of links are same, their roles may differ in each case. For instance, consider $(h_1, h_2) = (\{12,23\}, \{23\})$ and $(g_1, g_2) = (\{12\}, \{23\})$ . In this scenario, $L(h_1, h_2) = L(g_1, g_2) = \{12, 23\}$ . However, $\{23\}$ acts as a X–Z link in $(h_1, h_2)$ and as a Z link in $(g_1, g_2)$ . Consequently, $(h_1, h_2)$ belongs to a multilayer network, while $(g_1, g_2)$ belongs to a bi-cooperative network.

Multigraph is a graph in which multiple links (arcs) between two given nodes and loops (links ending at the same nodes) can occur (Forlicz et al., Reference Forlicz, Mercik, Stach and Ramsey2018). Consider, $g_1=\{12,23\}$ and $g_2=\{23, 24, 34\}$ on $N=\{1,2,3,4,5\}$ . Here, the pair $(g_1,g_2)$ forms a multilayer network (see Figure 2). However, if we combine the two networks, viz., $g_1+g_2=\{ 12,23,23,24, 34\}$ , then $g_1+g_2$ represents a multigraph as shown in Figure 3.

Figure 3. Multigraph: $g_1 + g_2=\{ 12,23, 23, 24, 34\}$ .

Further, an allocation rule (or a point-value solution) in a multigraph (Forlicz et al., Reference Forlicz, Mercik, Stach and Ramsey2018) or multilayer network allows for determining the role of each player within it. For example, players (2, 3, and 4 in Figure 3) connected by more links are considered more important in multigraphs than players with fewer links (such as player 1 in Figure 3). Similarly, in multilayer networks, players connected by X–Z links are more significant than those connected only by X or Z links. Therefore, an allocation rule in both multigraphs and multilayer networks should prioritize these players. However, the distinction between a multigraph and a multilayer network lies in how we deal with the links. In a multigraph, all links between nodes are considered to be distinct, and there are no overlaps among them (Forlicz et al., Reference Forlicz, Mercik, Stach and Ramsey2018). On the contrary, in the multilayer network games, we have “multilayer links” that belong to two different networks simultaneously. Although a multilayer link belongs to two different networks, it acts as a single link with two distinct attributes (say, X and Z). For example, in Figure 2 multilayer link $\{23\}$ acts as a single link. Therefore, when determining the Position value, we count multilayer links only once. Forlicz et al. (Reference Forlicz, Mercik, Stach and Ramsey2018) define the Shapley value for multigraphs that considers every individual link between two given nodes, including both single and multiple links, as well as loops. To illustrate, in the multigraph $g_1+g_2$ depicted in Figure 3, nodes 2 and 3 are connected through two links. In the model proposed by Forlicz et al. (Reference Forlicz, Mercik, Stach and Ramsey2018), each link between nodes 2 and 3 is evaluated separately when computing the Shapley value for each player in $g_1+g_2$ .

The multilayer network games, bi-cooperative network games, and multigraphs offer valuable insights into managing, optimizing, and decision-making in complex network systems. Multilayer network games highlight the interconnectedness across different network layers, acknowledging nodes’ multiple roles in managing internal data and external communications simultaneously. This understanding supports integrated network management and optimizes resource allocation based on critical multilayer connections. Bi-cooperative network games simplify interactions within distinct network structures (X or Z), clarifying responsibilities in simpler contexts. Multilayer network games address decision-making complexities to comprehensively assess risks and balance security, efficiency, and operational continuity in dynamic environments. Multigraphs provide detailed insights into network structure and the significance of nodes, contributing to enhanced network security and efficiency and ensuring resilience across interconnected layers of network infrastructures.

In conclusion, we have studied the allocation rule of players in a network in presence of an another network having common or shared players. Our multilayer network game in the cooperative framework has proven to be more effective in addressing real life scenarios arising in the government and private sectors. The proposed framework is free from the limitations of bi-cooperative network game and classical network game. Our model offers more flexibility in terms of network connections while calculating utilities of players, enabling us to solve real life complexities.