Potentials in Variable Network Games ^†

Abstract

Variable network situations combine a network situation with a probability distribution on the set of all networks. In this framework, the decision maker does not know which particular network will materialize but has formed a well-defined probability distribution indicating the probabilities of the various networks materializing. I extend the concept of network potentials defined for network situations to variable network situations.

1. Introduction

Cooperative games introduced by von Neumann and Morgenstern (1944) specifies a value for every coalition, where a coalition is a subset of the player set. As opposed to non-cooperative games, it is meant to capture situations where binding contracts are feasible. The researchers focused on allocation rules or ways to divide the value of the grand coalition among all the players. One such allocation rule is the Shapley value (Shapley, 1953), and it has been characterized in various ways. Myerson (1977) introduce a network-restricted cooperative game where not all coalitions are feasible, but only those coalitions are feasible where there is a path between any pair of players in a feasible coalition. The network is referred to as a communication network, and the combination of a communication network and a cooperative game is a communication situation. The Shapley value of such restricted games, subsequently known as the Myerson value, has been characterized by component balance and equal bargaining power for the subclass of component efficient communication situations. Another allocation rule for communication situations that has gained a certain amount of popularity is the Position value due to Meessen (1988) which treats the links of a communication network as players and divides the Shapley value of a link equally among the two adjacent players.

In a communication situation, the precise architecture of the communication network is not important, in the sense that two different networks inducing the same partition of the player set will yield identical restricted games and hence, identical Myerson values. Jackson and Wolinsky (1996) address this issue by introducing network games where the value stems directly from the network itself rather than a coalition of players. A network game with a fixed network is referred to as a network situation and the Myerson value and Position value have been extended to such setting by Jackson and Wolinsky (1996) and Slikker (2007) respectively and characterized by similar properties.

A different direction to the literature emerged when Calvo et al. (1999) introduced the idea that links are formed with a certain probability. These link formation probabilities directly determine a probability distribution over the set of all networks in a communication situation. These were referred to as probabilistic communication situations. A more generalized approach specified an arbitrary probability distribution over the set of all networks rather than these being indirectly determined by a priori specified link probabilities. This approach taken by Gomez et al. (2008) is referred to as generalized probabilistic communication situations. Network situations have been extended to the probabilistic setting both using the framework of Calvo et al. (1999), referred to as a probabilistic network situation by Borkotokey et al. (2021), and the framework of Gomez et al. (2008), referred to as a variable network situation by Chakrabarti et al. (2024). The differences between variable network situations and the generalized probabilistic communication situations such as in Ghintran et al. (2012) mirror the differences between network situations and communication situations as all we are doing is adding a probability distribution over networks to these deterministic frameworks. In communication situations, links restrict coalition formation between players in what is otherwise just a cooperative game. In network situations, links themselves produce value. So, links become players and a network is essentially a coalition of such “players”. So, it is a completely different framework.

The concept of a (cooperative) potential was first discussed by Hart and Mas-Collel (1989) for cooperative games. The potential is a function that is defined on the space of all cooperative games such that the marginal contribution of all players according to this function adds up to the worth of the grand coalition. They showed that there is a unique such function and the vector of marginal payoffs coincides with the Shapley value. Slikker (2005) extended the idea to network situations and Ghintran et al. (2012) consider potentials in generalized probabilistic communication situations. With regard to network situations, two potential functions are defined, namely, the link potential function and the player potential function on the space of all network situations. The marginal contribution of the players defined differently for these functions add up to the value of the network. Slikker (2005) shows that these functions are defined uniquely and the vector of marginal contributions coincides with the network Myerson value for the player potential function and coincides with the network Position value for the link potential function. Ghintran et al. (2012) extends Slikker’s ideas to generalized probabilistic communication situations. They define the probabilistic link potential function and show that this function is uniquely defined and the vector of marginal payoffs of this function coincides with the probabilistic Position value.

The idea of probabilistic potential functions has already been explored in the context of generalized probabilistic communication situations by Ghintran et al. (2012). Therefore, it is natural to consider such a function in the context of variable network situations, and this is what I do here. In this paper, I discuss potentials in variable network situations. In doing so, I extend the results of Slikker (2005) and Ghintran et al. (2012). In variable network situations, one can define the probabilistic player potential function and the probabilistic link potential function as analogs of such functions defined by Slikker (2005) for deterministic network situations. The marginal contributions of players, defined appropriately, for these functions add up to the expected worth of the network. I show that these probabilistic potential functions are uniquely defined. Furthermore, the vector of marginal contributions of the probabilistic player potential function coincides with the Expected Myerson value and the vector of marginal contributions of the probabilistic link potential function coincides with the Expected Position value.

The rest of this paper proceeds as follows. Section 2 introduces facts, notation and terminology of potentials and consistency in the context of cooperative games, followed by a discussion of (deterministic) network situations. Section 3 introduces variable network situations and analyses potentials in such situations. Section 4 concludes.

2. Preliminaries

2.1. Consistency of the Shapley Value

In this subsection, I will discuss cooperative games and the cooperative potential. Let U be the universal set of potential players. A cooperative game $(N,\omega )$ where $N\subseteq U$1 is a mapping $\omega$: $2^N\to \mathbb{R}$ such that $\omega (\varnothing )=0$. A cooperative game assigns to every non-empty coalition of players $S\subseteq N$ some worth $\omega \left(S\right)$, representing a collectively generated wealth. The function $\omega$ is also referred to as a characteristic function. I will denote by ${\mathbb{C}}^N$ the set of all n player cooperative games where the player set is N. The set of all possible cooperative games will be referred to as $\mathbb{C}$. Given a cooperative game $(N,\omega )$, I will refer to $(S,\omega )$, where $S\subseteq N$ is the subgame of the original game, and is obtained by restricting $\omega$ to the subsets of S, or only to members of the set $2^S$.

We use the symbol # to denote the cardinality of a set. A value/allocation is a function $\varphi :{\mathbb{C}}^N\to {\mathbb{R}}^N$ such that ${\sum }_{i\in N}{\varphi }_i(N,\omega )=\omega \left(N\right)$. Shapley (1953) introduced his value as the function ${\varphi }^{Sh}:\mathbb{C}\to {\mathbb{R}}^N$ as follows:

$${\varphi }_i^{Sh}(N,\omega )={\displaystyle \sum }_{S\subseteq N-i}{\displaystyle \frac{\#S!(n-\#S-1)!}{n!}}\left[\omega (S+i)-\omega \left(S\right)\right]$$

for all $i\in N$.

Shapley (1953) demonstrated that every cooperative game $(N,\omega$) can be written as a unique linear combination of unanimity games $(N,u_T)$, i.e.,

$$\omega =\sum _{T\subseteq N}{\Delta }_{\omega }\left(T\right)u_T$$

where ${\Delta }_{\omega }\left(T\right)$ is referred to as the Harsanyi dividend of the coalition T and given by

$${\Delta }_{\omega }\left(T\right)=\sum _{S\subseteq T}{(-1)}^{\#T-\#S}\omega \left(S\right)$$

and

$$u_T\left(S\right)=\left\{\begin{array}{c}1\hfill \\ 0\hfill \end{array}\begin{array}{c} \mathrm{if} T\subseteq S;\hfill \\ \mathrm{otherwise}.\hfill \end{array}\right.$$

The Shapley value of the game $(N,\omega )$ can be described as

$${\varphi }_i^{Sh}(N,\omega )={\displaystyle \sum }_{T\subseteq N:i\in T}{\displaystyle \frac{{\Delta }_{\omega }\left(T\right)}{\#T}}$$

for all $i\in N$.

Hart and Mas-Collel (1989) define a potential function for cooperative games as follows:

A function $P:\mathbb{C}\to \mathbb{R}$ is a potential function if for all $(N,\omega )\in {\mathbb{C}}^N$

$${\displaystyle \sum }_{i\in N}{M}_i\left(P(N,\omega )\right)=\omega \left(N\right).$$

where

$$M_i\left(P(N,\omega )\right)=P(N,\omega )-P(N\setminus \left\{i\right\},\omega )$$

for all $i\in N$. Hart and Mas-Collel (1989) prove two key results, which I present below.

Theorem 1.

There exists a unique potential P. Further, $M_i\left(P(N,\omega )\right)={\varphi }_i^{Sh}(N,\omega )$ for all $i\in N$.

Hart and Mas-Collel (1989) have also demonstrated the consistency of the Shapley value. Here, I briefly state their results.

Let $\varphi$ be a value, $(N,\omega )$ be a cooperative game and $T\subseteq N$. The reduced game of $\omega$ with respect to T and $\varphi$ is defined as follows:

$${\omega }_T^{\varphi }\left(S\right)=\omega \left(S\cup T^c\right)-\sum _{i\in T^c}{\varphi }_i\left(S\cup T^c,\omega \right)$$

for every $S\subseteq T$, where $T^c$ denotes the complementarity of T in N.

Definition 1.

A value ϕ is consistent if for every coalition $T\subseteq N$,

$${\varphi }_i(T,{\omega }_T^{\varphi })={\varphi }_i(N,\omega ).$$

for all $i\in T$.

Hart and Mas-Collel (1989) next define the notion of a value being standard for two-person games.

Definition 2.

A value ϕ is standard for two-person games if

$${\varphi }_i(\{i,j\},\omega )=\omega \left(\left\{i\right\}\right)+{\displaystyle \frac{1}{2}}\left(\omega \left(\{i,j\}\right)-\omega \left(\left\{i\right\}\right)-\omega \left(\left[j\right\}\right)\right)$$

for all $i\ne j$ and all ω.

Theorem 2.

$\varphi ={\varphi }^{Sh}$ if and only if ϕ is consistent and standard for two-person games.

2.2. Networks

In this subsection, I will discuss the notation and terminology with regard to networks.

2.2.1. Definition

Let $N=\{1,2,\dots ,n\}$ be a finite set of players. With a slight abuse of notation, for every set of players $S\subseteq N$, I denote for any player $i\notin S$ the expanded set $S+i=S\cup \left\{i\right\}$ and for any player $i\in S$ the reduced set $S-i=S\setminus \left\{i\right\}$. In particular, the set $N-i$ for $i\in N$ is the set of all players other than i.2 Given the player set N, a link between two distinct players $i\in N$ and $j\in N$ is defined as the binary set $ij=\{i,j\}$, representing an undirected relationship between i and j. Clearly, $ij$ is equivalent to $ji$. The set of all possible links on N is denoted by $g_N=\{ij\mid i,j\in N$ and $i\ne j\}$.

A network on N is any set of links $g\subset g_N$. In particular, $g_N$ is called the complete network, while $g_0=⌀$ is the empty network. The class of all possible networks on N is given by ${\mathbb{G}}^N=\{g\mid g\subset g_N\}$. Clearly, $\#g_N={\displaystyle \frac{n(n-1)}{2}}$ and $\#{\mathbb{G}}^N=2^{{\displaystyle \frac{n(n-1)}{2}}}$.

For every network $g\in {\mathbb{G}}^N$ and every player $i\in N,$ I denote i’s neighborhood in g by $N_i\left(g\right)=\{j\in N\mid j\ne i$ and $ij\in g\}$ and her link neighborhood by $L_i\left(g\right)=\{ij\in g\mid j\in N_i\left(g\right)\}\subseteq g$. Furthermore, $n_i\left(g\right)=\#N_i\left(g\right)=\#L_i\left(g\right)$. In particular, I introduce $L_i=L_i\left(g_N\right)$ as the set of all potential links in which player i participates.

I also define $N\left(g\right)={\cup }_{i\in N}{N}_i\left(g\right)$ and let $n\left(g\right)=\#N\left(g\right)$ with the convention that if $N\left(g\right)=\varnothing$, I let $n\left(g\right)=1$.3 $n\left(g\right)$ will be referred to as size of the network g.

I say that player $i\in N$ is an isolated player in network $g\in {\mathbb{G}}^N$ if $i\notin N\left(g\right)$. This implies that for an isolated player i in g, it holds that $N_i\left(g\right)=⌀$. I denote the set of all isolated players in network g by $N_0\left(g\right)=N\setminus N\left(g\right)$.

2.2.2. Components in Networks

A path connecting $i\in N$ and $j\in N$ with $i\ne j$ in network $g\in {\mathbb{G}}^N$ is an ordered set of distinct players $P_{ij}=\{i_1,i_2,\dots ,i_p\}\subseteq N\left(g\right)$ with $p⩾2$ such that $i_1=i$, $i_p=j$, and $i_k{i}_{k+1}\in g$ for all $k=1,\dots ,p-1$. I say that $i,j\in N$ with $i\ne j$ are connected in g if there exists a path $P_{ij}\subseteq N\left(g\right)$ between i and j and disconnected otherwise. A network is connected if all pairs of players $i,j\in N$ with $i\ne j$ are connected. In particular, for a connected network g, it holds $N\left(g\right)=N$ and $N_0\left(g\right)=⌀$.

The network $h\in {\mathbb{G}}^N$ is a component of $g\in {\mathbb{G}}^N$ if $h\subseteq g$ is connected and for any $i\in N\left(h\right)$ and $j\in N\left(g\right)$, $ij\in g$ implies $ij\in h$. In other words, a component is simply a maximally connected subnetwork of g. I denote the set of network components of the network g by $C\left(g\right)$. Note that for any connected network $g\in {\mathbb{G}}^N$ it holds that $C\left(g\right)=\left\{g\right\}$. In particular, $C\left(g_N\right)=\left\{g_N\right\}$ and $C\left(g_0\right)=⌀$.

2.2.3. Restrictions of Networks

Let $g\in {\mathbb{G}}^N$ and let $S\subseteq N$ be some set of players. Then, the restriction of g to S is the network defined as $g|S=g\cap g_S$, where $g_S=\{ij\mid i,j\in S$ and $i\ne j\}$. The restricted network $g|S$ is obviously a subnetwork of g.

For a subnetwork $h\subseteq g$, I denote by $g-h=g\setminus h$ the network that results after the removal from g of all links in the subnetwork h. Similarly, for any network $h\subset g_N-g=g_N\setminus g$, I denote by $g+h=g\cup h$ the network that results from g after adding all links in h to the network g. Clearly, for $h\subseteq g$ it holds that $(g-h)+h=g$.

2.3. Network Situations

In this subsection, I discuss network situations and allocation rules on network situations. Jackson and Wolinsky (1996) introduced the notion of a network game on the player set N. It is assumed that every cooperation among players intermediated through some configuration of relationships between these players creates a level of wealth that is determined by the architecture of the network that is formed. This leads to the introduction of a function that assigns a wealth level to every network that can be formed on N.

Denote a network value function by $v:{\mathbb{G}}^N\to \mathbb{R}$ such that $v\left(g_0\right)=0$. A value function $v:{\mathbb{G}}^N\to \mathbb{R}$ is component additive if

$$v\left(g\right)=\sum _{h\in C\left(g\right)}v\left(h\right).$$

This implies that there are no externalities across components and isolated nodes do not create any value. A value function is monotonic if $v\left(g\right)⩾v\left(g^{\prime }\right)$ for $g\supseteq g^{\prime }$. The class of all network value functions on N is defined as

$${\mathbb{V}}^N=\{v\mid v:{\mathbb{G}}^N\to \mathbb{R}\mathrm{such}\mathrm{that}v\left(g_0\right)=0\}.$$

The class ${\mathbb{V}}^N$ has been the subject of study of numerous contributions to game theoretic network analysis. It is clear that ${\mathbb{V}}^N$ is a $\left(2^{\frac{1}{2}n(n-1)}-1\right)$ dimensional Euclidean space. A pair $(N,v)$ with set of players N and network value function $v\in {\mathbb{V}}^N$ is called a network game.4

Following the notation introduced by Jackson and Wolinsky (1996), an allocation rule on the class of network games is a mapping $Y:{\mathbb{G}}^N\times {\mathbb{V}}^N\to {\mathbb{R}}^N$ that for every network value function $v\in {\mathbb{V}}^N$ assigns to every player $i\in N$ in a network $g\in {\mathbb{G}}^N$ an allocated value $Y_i(g,v)$, such that $Y_i(g,v)=0$ for every isolated player $i\in N_0\left(g\right)$.5 A pair $(g,v)\in {\mathbb{G}}^N\times {\mathbb{V}}^N$ is called a network situation.

Allocation rules in network situations can satisfy a number of standard properties that have been introduced and investigated in the literature. I list the most relevant of these properties below.

• An allocation rule Y on ${\mathbb{G}}^N\times {\mathbb{V}}^N$ is balanced if for every network value function $v\in {\mathbb{V}}^N$ and every network $g\in {\mathbb{G}}^N:{\sum }_{i\in N}{Y}_i(g,v)=v\left(g\right)$.
• An allocation rule Y on ${\mathbb{G}}^N\times {\mathbb{V}}^N$ is component balanced if for every network value function $v\in {\mathbb{V}}^N$, every network $g\in {\mathbb{G}}^N$ and all of its components $h\in C\left(g\right):{\sum }_{i\in N\left(h\right)}{Y}_i(g,v)=v\left(h\right)$.
  It is clear that component balancedness implies balancedness. Two well-known component balanced allocation rules are the Myerson value and the Position value.

Jackson and Wolinsky (1996) introduced the Myerson Value as an allocation rule on the class of network situations as an extension of the allocation rule for communication situations introduced by Myerson (1977). Formally, the Myerson value on the class of network situations is the allocation rule $Y^m:{\mathbb{G}}^N\times {\mathbb{V}}^N\to {\mathbb{R}}^N$ defined by

$$Y_i^m(g,v)=\sum _{S\subseteq N-i}{\displaystyle \frac{\#S!(n-\#S-1)!}{n!}}[v\left(g\right|S+i)-v\left(g\right|S)]$$

for all $i\in N$.

The Position value was introduced by Meessen (1988) as an allocation rule for allocation of value in communication situations and subsequently developed by Borm et al. (1992). Slikker (2007) introduce the Position value as an allocation rule on the class of network situations by extending the earlier definition to the new framework. Formally, the Position value on the class of network situations is the allocation rule $Y^p:{\mathbb{G}}^N\times {\mathbb{V}}^N\to {\mathbb{R}}^N$ defined by

$$Y_i^p(g,v)={\displaystyle \frac{1}{2}}\sum _{ij\in g}{\displaystyle \sum }_{h\subset g-ij}{\displaystyle \frac{\left(\#h!\right)\left(\left(\#g-\#h-1\right)!\right)}{(\#g!)}}\left(v(h+ij)-v\left(h\right)\right)$$

for all $i\in N$.

2.4. Link Games

Denote the restriction of v to subsets of g as ${v|}_g$. I can look upon links as players and a coalition of such players as a network. Under this interpretation, the value function of the network game ${v|}_g$ can be looked upon as the characteristic function of the cooperative game $(g,v{|}_g)$. Following Slikker (2005), I refer to this cooperative game as the link game. Therefore, the value function ${v|}_g$ can be written as the unique linear combination of the unanimity link games, i.e.,

$${v|}_g=\sum _{h\subseteq g}{\Delta }_v\left(h\right)u_h$$

where

$$u_h\left(h^{\prime }\right)=\left\{\begin{array}{c}1\hfill \\ 0\hfill \end{array}\begin{array}{c} \mathrm{if} fh\subseteq h^{\prime };\hfill \\ \mathrm{otherwise}.\hfill \end{array}\right.$$

The Shapley value of a link $ij\in g$ can be expressed as

$${\varphi }_{ij}^{Sh}\left({g,v|}_g\right)={\displaystyle \sum }_{h\subseteq g:ij\in h}{\displaystyle \frac{{\Delta }_v\left(h\right)}{\#h}}.$$

Following Slikker (2007), I can express the Position value of a player in a network situation in terms of Shapley value of links in the link game as

$$Y_i^p(g,v)={\displaystyle \frac{1}{2}}{\displaystyle \sum }_{ij\in L_i\left(g\right)}{\varphi }_{ij}^{Sh}\left({g,v|}_g\right)$$

for all $i\in N$.

Slikker (2007) also shows that

$$Y_i^m(g,v)={\displaystyle \sum }_{h\subseteq g:i\in N\left(h\right)}{\displaystyle \frac{{\Delta }_v\left(h\right)}{\#N\left(h\right)}}$$

for all $i\in N$.

2.5. Network Potentials

Network potentials were introduced by Slikker (2005). I give the definitions and a brief summary of the results. Consider a function $P:{\mathbb{G}}^N\times {\mathbb{V}}^N\to \mathbb{R}$ that assigns to every network situation $(g,v)$ a real number. Define the marginal contribution of a player in two ways:

$$M_i^L\left(P(g,v)\right)=\sum _{ij\in L_i\left(g\right)}\left[P\left(g,v\right)-P\left(g-ij,v\right)\right]$$

for all $i\in N$ and

$$M_i^P\left(P(g,v)\right)=\left[P\left(g,v\right)-P\left(g-L_i\left(g\right),v\right)\right]$$

for all $i\in N$.

P is called a link potential function if for all $g\in {\mathbb{G}}^N$

$${\displaystyle \sum }_{i\in N}{M}_i^L\left(P(g,v)\right)=v\left(g\right)$$

and P is called a player potential function if for all $g\in {\mathbb{G}}^N$

$${\displaystyle \sum }_{i\in N}{M}_i^P\left(P(g,v)\right)=v\left(g\right).$$

Lemma 1

(Theorem $5.1$ of Slikker, 2005). There exists a unique player potential P. Further,

$$M_i^P\left(P(g,v)\right)=Y_i^m(g,v)$$

for all $i\in N$ and for all $g\in {\mathbb{G}}^N,v\in {\mathbb{V}}^N.$

Lemma 2

(Theorem $5.2$ of Slikker, 2005). There exists a unique link potential P. Further,

$$M_i^L\left(P(g,v)\right)=Y_i^p(g,v)$$

for all $i\in N$ and for all $g\in {\mathbb{G}}^N,v\in {\mathbb{V}}^N$.

3. Probabilistic Potentials

3.1. Network Formation Probability Distributions

Variable network situations were introduced by Chakrabarti et al. (2024). I present the definition below. They begin by introducing a network formation probability distribution that assigns to every network a probability that it forms.

Definition 3.

A network formation probability distribution on N is a map $\rho :{\mathbb{G}}^N\to [0,1]$ such that ${\sum }_{g\in {\mathbb{G}}^N}\rho \left(g\right)=1$.

The class of all network formation probability distributions on N is denoted by

$${\mathbb{P}}^N=\left\{\left.\rho \in {[0,1]}^{\#{\mathbb{G}}^N}\right|{\displaystyle \sum _{g\in {\mathbb{G}}^N}}\rho \left(g\right)=1\right\}.$$

Note that $\dim {\mathbb{P}}^N=\#{\mathbb{G}}^N-1=2^{{\displaystyle \frac{1}{2}}n(n-1)}-1$.

A network formation probability distribution naturally results in the following conceptions.

Definition 4.

Let $\rho \in {\mathbb{P}}^N$ be some network formation probability distribution on N.

A network $g\in {\mathbb{G}}^N$ is formable under ρ if $\rho \left(g\right)>0$. The class of formable networks under ρ is given by $\mathbb{G}\left(\rho \right)=\{g\in {\mathbb{G}}^N\mid \rho \left(g\right)>0\}$, which can also be denoted as the support of ρ.

The extent of the distribution ρ is the union of the class of formable networks given by $g\left(\rho \right)=\cup \mathbb{G}\left(\rho \right)\in {\mathbb{G}}^N$.

A network $h\in {\mathbb{G}}^N$ is a component of ρ if $h\in C\left(g\right(\rho \left)\right)$ is a component of the extent $g\left(\rho \right)$ and a player $i\in N$ is isolated in ρ if $i\in N_0\left(g\left(\rho \right)\right)$ is an isolated player in its extent $g\left(\rho \right)$.

Formable networks are assigned a positive formation probability. The extent of the network formation distribution is simply the collection of all links that are part of a formable network. Hence, a link is not in the extent if it is not part of any formable network and, as such, will form with zero probability. Therefore, a player is isolated if there is a zero probability that she is linked to any other player under the given network formation probability distribution.

I denote by ${\rho }_0$ the network probability distribution whose extent is the empty network. Namely, ${\rho }_0\left(g_0\right)=1$ and ${\rho }_0\left(g\right)=0$ for all $g\ne g_0$.

3.2. Restrictions of Network Formation Probability Distributions

Chakrabarti et al. (2024) discuss the notion of restricting a network formation probability distribution to a certain given network. This implies that its extent is limited to the given network. The following definition formalizes this by transferring network formation probabilities of network extending beyond the imposed restriction to the networks that form within the imposed extent. The applied conception is due to Gomez et al. (2008).

Definition 5.

Let $\rho \in {\mathbb{P}}^N$ be some network formation probability distribution and let $g\in {\mathbb{G}}^N$ be some given network on N. Then, the restriction of $\rho$ to g is the modified network formation probability distribution ${\rho }_g\in {\mathbb{P}}^N$, defined by

$${\rho }_g\left(h\right)=\left\{\begin{array}{cc}\sum _{h^{\prime }\subset g_N\setminus g}\rho (h\cup h^{\prime })& \mathit{if}h\subseteq g;\\ 0& \mathit{otherwise}.\end{array}\right.$$

In this definition, the formation probabilities of subnetworks of $g_N\setminus g$ are transferred to subnetworks of g itself. It should be clear that the extent of the restriction of $\rho$ to some network g is determined as $g\left({\rho }_g\right)=g\left(\rho \right)\cap g$.

As a special case of this notion of a restriction of a network formation probability distribution to a given network, Chakrabarti et al. (2024) introduce devices for removing individual links and players. For the computation of these restrictions, they prove the following proposition.

Proposition 1.

Let $\rho \in {\mathbb{P}}^N$ and let $i,j\in N$ with $i\ne j$.

(a) 

Let ${\rho }^{-ij}={\rho }_{g_N-ij}$ be the restriction of ρ to $g_N-ij$. Then, for every network $g\in {\mathbb{G}}^N:$

$${\rho }^{-ij}\left(g\right)=\left\{\begin{array}{c}\rho \left(g\right)+\rho (g+ij)\hfill \\ 0\hfill \end{array}\begin{array}{c} if ij\notin g;\hfill \\ if ij\in g.\hfill \end{array}\right.$$

Furthermore, it holds that $g\left({\rho }^{-ij}\right)=g\left(\rho \right)-ij$ and $\mathbb{G}\left({\rho }^{-ij}\right)=\{g\in {\mathbb{G}}^N\mid ij\notin g$ and $\mathbb{G}\left(\rho \right)\cap \{g,g+ij\}\ne \varnothing \}$.

(b) 

Let ${\rho }^{-i}={\rho }_{g_{N-i}}$ be the restriction of ρ to $g_{N-i}=g_N\setminus L_i\left(g_N\right)$. Then, for every network $g\subseteq g_{N-i}$, it holds that

$${\rho }^{-i}\left(g\right)=\rho \left(g\right)+\sum _{⌀\ne h\subseteq L_i\left(g\left(\rho \right)\right)}\rho (g\cup h)$$

and for any $g⊈g_{N-i}:{\rho }^{-i}\left(g\right)=0$.

Furthermore, it holds that $g\left({\rho }^{-i}\right)=g\left(\rho \right)\setminus L_i\left(g_N\right)$ and

$$\mathbb{G}\left({\rho }^{-i}\right)=\{g\in {\mathbb{G}}^N|g\cap L_i\left(g_N\right)=\varnothing \mathit{and}\mathbb{G}\left(\rho \right)\cap \{g+h|h\subseteq L_i\left(g_N\right)\}\ne \varnothing \}.$$

3.3. Variable Network Situations

In this subsection, I introduce variable network situations.

Definition 6.

A variable network situation is a pair $(\rho ,v)\in {\mathbb{P}}^N\times {\mathbb{V}}^N$ consisting of a network formation probability distribution ρ describing the probabilities with which networks form and a network value function v describing the potential wealth levels created through the formed networks.

Next, I define allocation rules for variable network situations.

Definition 7.

An allocation rule on the class of variable network situations is a mapping $\Psi :{\mathbb{P}}^N\times {\mathbb{V}}^N\to {\mathbb{R}}^N$, such that for every variable network situation $(\rho ,v)\in {\mathbb{P}}^N\times {\mathbb{V}}^N$, it holds that ${\Psi }_i(\rho ,v)=0$ for every isolated player $i\in N_0\left(g\left(\rho \right)\right)$.

The definition of an allocation rule is clearly a straightforward conceptual extension from the definition of allocation rules on the class of network situations to the class of variable network situations. In particular, all allocation rules on ${\mathbb{V}}^N$ can be extended to the larger class of variable network situations by taking their expected payoffs. This is formalized as follows.

Definition 8.

Let $Y:{\mathbb{G}}^N\times {\mathbb{V}}^N\to {\mathbb{R}}^N$ be an allocation rule on the class of network situations. Then, its extension to the class of variable network situations is the allocation rule ${\Psi }^Y:{\mathbb{P}}^N\times {\mathbb{V}}^N\to {\mathbb{R}}^N$ defined by

$${\Psi }^Y(\rho ,v)=\sum _{g\in {\mathbb{G}}^N}\rho \left(g\right)·Y(g,v)=\sum _{g\in \mathbb{G}\left(\rho \right)}\rho \left(g\right)·Y(g,v)$$

for every variable network situation $(\rho ,v)\in {\mathbb{P}}^N\times {\mathbb{V}}^N$.

The extension of an allocation rule simply assigns the expected payoff under that rule for the given network formation probability distribution. It should be clear that the extension of an allocation rule satisfies most of the properties of the original rule.

For instance, balancedness and component balance extend in a straightforward way to variable network situations.

The expected wealth that is created in the variable network situation $(\rho ,v)\in {\mathbb{P}}^N\times {\mathbb{V}}^N$ is defined as

$$\mathbb{E}(\rho ,v)=\sum _{g\in {\mathbb{G}}^N}\rho \left(g\right)·v\left(g\right)$$

An allocation rule $\Psi :{\mathbb{P}}^N\times {\mathbb{V}}^N\to {\mathbb{R}}^N$ satisfies balancedness—or is balanced—if for every variable network situation $(\rho ,v)\in {\mathbb{P}}^N\times {\mathbb{V}}^N$ it holds that ${\sum }_{i\in N}{\Psi }_i(\rho ,v)=\mathbb{E}(\rho ,v)$, i.e., the wealth allocated through the rule exactly covers the expected wealth that is created in the variable network situation.

An allocation rule $\Psi :{\mathbb{P}}^N\times {\mathbb{V}}^N\to {\mathbb{R}}^N$ satisfies component balancedness—or is component-balanced—if for every variable network situation $(\rho ,v)\in {\mathbb{P}}^N\times {\mathbb{V}}^N$, such that $v:{\mathbb{G}}^N\to \mathbb{R}$ is component additive, it holds that for every component $h\in C\left(g\right(\rho \left)\right):$

$$\sum _{i\in N\left(h\right)}{\Psi }_i(\rho ,v)=\sum _{g\in \mathbb{G}\left(\rho \right)}\rho \left(g\right)·v(g\cap h)$$

Component balancedness implies that all wealth that can be attributed to a certain component to the extent of the network formation probability distribution is allocated to the constituting members of that component. Hence, the wealth allocated through the rule exactly covers the expected wealth that is created in that component in the given variable network situation.

Chakrabarti et al. (2024) show the following two propositions.

Proposition 2.

Let $Y:{\mathbb{G}}^N\times {\mathbb{V}}^N\to {\mathbb{R}}^N$ be a balanced allocation rule on the class of network situations. Then, its extension ${\Psi }^Y:{\mathbb{P}}^N\times {\mathbb{V}}^N\to {\mathbb{R}}^N$ is balanced.

Proposition 3.

Let $Y:{\mathbb{G}}^N\times {\mathbb{V}}^N\to {\mathbb{R}}^N$ be a component-balanced allocation rule on the class of network situations. Then, its extension ${\Psi }^Y:{\mathbb{P}}^N\times {\mathbb{V}}^N\to {\mathbb{R}}^N$ is component-balanced.

As discussed earlier, there are traditionally two principal allocation rules in the class of network situations, namely the Myerson value and the Position value. Both of these values can be extended to the class of variable network situations through the consideration of their standard extension, reflecting the expected allocation of the generated wealth under these two values.

Definition 9.

The Expected Myerson value is the allocation rule ${\Psi }^m:{\mathbb{P}}^N\times {\mathbb{V}}^N\to {\mathbb{R}}^N$ defined as the standard extension ${\Psi }^m={\Psi }^{Y^m}$ of the Myerson value $Y^m$ on the class of network situations to the class of variable network situations.

The Expected Myerson value is simply the Expected Myerson payoff to a player that arises in every possible network that can form under the imposed network formation probability distribution.

As applied to the Myerson value, I can also base an allocation rule on the class of variable network situations on the formulated Position value. This introduces the Expected Position value.

Definition 10.

The Expected Position value is the allocation rule ${\Psi }^p:{\mathbb{P}}^N\times {\mathbb{V}}^N\to {\mathbb{R}}^N$ defined as the standard extension ${\Psi }^p={\Psi }^{Y^p}$ of the Position value $Y^p$ on the class of network situations to the class of variable network situations.

3.4. Probabilistic Potentials

In this subsection, I introduce probabilistic (network) potentials and prove their existence and properties.

Let $\mathcal{P}:{\mathbb{P}}^N\times {\mathbb{V}}^N\to \mathbb{R}$ be a function defined on the class of variable network situations.

Define

$$M_i^L\left(\mathcal{P}(\rho ,v)\right)=\sum _{ij\in L_i\left(g\left(\rho \right)\right)}\left[\mathcal{P}\left(\rho ,v\right)-\mathcal{P}\left({\rho }^{-ij},v\right)\right]$$

(1)

for all $i\in N$ and

$$M_i^P\left(\mathcal{P}(\rho ,v)\right)=\left[\mathcal{P}\left(\rho ,v\right)-\mathcal{P}\left({\rho }^{-i},v\right)\right]$$

(2)

for all $i\in N$.

Definition 11.

$\mathcal{P}$ is called the probabilistic link potential function if $\mathcal{P}({\rho }_0,v)=0$ and

$${\displaystyle \sum }_{i\in N}{M}_i^L\left(\mathcal{P}(\rho ,v)\right)=\mathbb{E}(\rho ,v).$$

(3)

Definition 12.

$\mathcal{P}$ is called the probabilistic player potential function if $\mathcal{P}({\rho }_0,v)=0$ and

$${\displaystyle \sum }_{i\in N}{M}_i^P\left(\mathcal{P}(\rho ,v)\right)=\mathbb{E}(\rho ,v).$$

(4)

Lemma 3.

There exists a unique probabilistic link potential function. Moreover, for each $(\rho ,v)\in {\mathbb{P}}^N\times {\mathbb{V}}^N$, $M_i^L\left(\mathcal{P}(\rho ,v)\right)={\Psi }_i^p(\rho ,v)$ holds.

Proof. 

First, I shall prove that a probabilistic link potential function exists. First, define $\mathcal{P}(\rho ,v)={\displaystyle {\displaystyle \sum }_{g\subseteq g\left(\rho \right)}}\rho \left(g\right)·P(g,v)$, where $P:{\mathbb{G}}^N\times {\mathbb{V}}^N\to \mathbb{R}$ is the link potential function.

Now,

$$\begin{array}{cc}& {\displaystyle \sum }_{i\in N}{M}_i^L\left(\mathcal{P}(\rho ,v)\right)\hfill \\ & ={\displaystyle \sum }_{i\in N}\sum _{ij\in L_i\left(g\left(\rho \right)\right)}\left[\mathcal{P}\left(\rho ,v\right)-\mathcal{P}\left({\rho }^{-ij},v\right)\right]\hfill \\ & ={\displaystyle \sum }_{i\in N}\sum _{ij\in L_i\left(g\left(\rho \right)\right)}\left[{\displaystyle \sum }_{g\subseteq g\left(\rho \right)}\rho \left(g\right)·P(g,v)-{\displaystyle \sum }_{g\subseteq g\left({\rho }^{-ij}\right)}{\rho }^{-ij}\left(g\right)·P(g,v)\right].\hfill \end{array}$$

$$\begin{array}{cc}& ={\displaystyle \sum }_{i\in N}\sum _{ij\in L_i\left(g\left(\rho \right)\right)}\left[{\displaystyle \sum }_{g\subseteq g\left(\rho \right):ij\in g}\rho \left(g\right)·P(g,v)+{\displaystyle \sum }_{g\subseteq g\left({\rho }^{-ij}\right)}\rho \left(g\right)·P(g,v)-{\displaystyle \sum }_{g\subseteq g\left({\rho }^{-ij}\right)}{\rho }^{-ij}\left(g\right)·P(g,v)\right]\hfill \\ & ={\displaystyle \sum }_{i\in N}\sum _{ij\in L_i\left(g\left(\rho \right)\right)}\left[{\displaystyle \sum }_{g\subseteq g\left(\rho \right):ij\in g}\rho \left(g\right)·P(g,v)+{\displaystyle \sum }_{g\subseteq g\left({\rho }^{-ij}\right)}P(g,v)\left(\rho \left(g\right)-{\rho }^{-ij}\left(g\right)\right)\right]\hfill \end{array}$$

since by Proposition 1, $g\left({\rho }^{-ij}\right)=g\left(\rho \right)-ij$, and hence, $g\subseteq g\left({\rho }^{-ij}\right)=g\subseteq g\left(\rho \right):ij\notin g$. By Proposition 1, ${\rho }^{-ij}\left(g\right)=\rho \left(g\right)+\rho (g+ij)$ if $g\subseteq g\left({\rho }^{-ij}\right)$, and hence, $\rho \left(g\right)-{\rho }^{-ij}\left(g\right)=-\rho (g+ij)$.

Hence,

$$\begin{array}{cc}& {\displaystyle \sum }_{i\in N}{M}_i^L\left(\mathcal{P}(\rho ,v)\right)\hfill \\ & ={\displaystyle \sum }_{i\in N}\sum _{ij\in L_i\left(g\left(\rho \right)\right)}\left[{\displaystyle \sum }_{g\subseteq g\left(\rho \right):ij\in g}\rho \left(g\right)·P(g,v)-{\displaystyle \sum }_{g\subseteq g\left({\rho }^{-ij}\right)}\rho (g+ij)·P(g,v)\right].\hfill \end{array}$$

Now, the set of networks $g\subseteq g\left(\rho \right):ij\in g$ is the same as the set of networks $g+ij:g\subseteq g\left({\rho }^{-ij}\right)$.

Hence,

$$\begin{array}{cc}& {\displaystyle \sum }_{i\in N}{M}_i^L\left(\mathcal{P}(\rho ,v)\right)\hfill \\ & ={\displaystyle \sum }_{i\in N}\sum _{ij\in L_i\left(g\left(\rho \right)\right)}\left[{\displaystyle \sum }_{g\subseteq g\left(\rho \right):ij\in g}\rho \left(g\right)·\left(P(g,v)-P(g-ij,v)\right)\right]\hfill \\ & ={\displaystyle \sum }_{i\in N}{\displaystyle \sum }_{g\subseteq g\left(\rho \right)}\rho \left(g\right)\left[{\displaystyle \sum }_{ij\in L_i\left(g\right)}\left(P(g,v)-P(g-ij,v)\right)\right].\hfill \end{array}$$

By Lemma 2, ${\displaystyle {\displaystyle \sum }_{ij\in L_i\left(g\right)}}\left(P(g,v)-P(g-ij,v)\right)=Y_i^p(g,v)$.

Hence,

$$\begin{array}{cc}& {\displaystyle \sum }_{i\in N}{M}_i^L\left(\mathcal{P}(\rho ,v)\right)\hfill \\ & ={\displaystyle \sum }_{i\in N}{\displaystyle \sum }_{g\subseteq g\left(\rho \right)}\rho \left(g\right)\left[{\displaystyle \sum }_{ij\in L_i\left(g\right)}\left(P(g,v)-P(g-ij,v)\right)\right]\hfill \\ & ={\displaystyle \sum }_{i\in N}\left[{\displaystyle \sum }_{g\subseteq g\left(\rho \right)}\rho \left(g\right)·Y_i^p(g,v)\right]\hfill \end{array}$$

$$\begin{array}{cc}& ={\displaystyle \sum }_{i\in N}{\Psi }_i^p(\rho ,v)\hfill \\ & =\mathbb{E}(\rho ,v)\hfill \end{array}$$

given that the Expected Position value is balanced by of Proposition 2. It is also clear from the above steps that $M_i^L\left(\mathcal{P}(\rho ,v)\right)={\Psi }_i^p(\rho ,v)$ for all $i\in N$.

Our proof of uniqueness closely follows Ghintran et al. (2012). From (1) and (3), I get

$${\displaystyle \sum }_{i\in N}\sum _{ij\in L_i\left(g\left(\rho \right)\right)}\left[\mathcal{P}\left(\rho ,v\right)-\mathcal{P}\left({\rho }^{-ij},v\right)\right]=\mathbb{E}(\rho ,v)$$

$$\begin{array}{cc}& \Rightarrow 2{\displaystyle \sum }_{ij\in g\left(\rho \right)}\left[\mathcal{P}\left(\rho ,v\right)-\mathcal{P}\left({\rho }^{-ij},v\right)\right]=\mathbb{E}(\rho ,v)\hfill \\ & \Rightarrow 2\left(\#g\left(\rho \right)\right)\mathcal{P}\left(\rho ,v\right)-2{\displaystyle \sum }_{ij\in g\left(\rho \right)}\mathcal{P}\left({\rho }^{-ij},v\right)=\mathbb{E}(\rho ,v)\hfill \\ & \Rightarrow 2\left(\#g\left(\rho \right)\right)\mathcal{P}\left(\rho ,v\right)=\mathbb{E}(\rho ,v)+2{\displaystyle \sum }_{ij\in g\left(\rho \right)}\mathcal{P}\left({\rho }^{-ij},v\right)\hfill \\ & \Rightarrow \mathcal{P}\left(\rho ,v\right)={\displaystyle \frac{1}{2\left(\#g\left(\rho \right)\right)}}\left[\mathbb{E}(\rho ,v)+2{\displaystyle \sum }_{ij\in g\left(\rho \right)}\mathcal{P}\left({\rho }^{-ij},v\right)\right].\hfill \end{array}$$

(5)

Starting with $\mathcal{P}\left({\rho }_0,v\right)=0$ for every $({\rho }_0,v)\in {\mathbb{P}}^N\times {\mathbb{V}}^N$, using (5), I can recursively define $\mathcal{P}\left(\rho ,v\right)$ in a unique way. □

Lemma 4.

There exists a unique probabilistic player potential function. Moreover, for each $(\rho ,v)\in {\mathbb{P}}^N\times {\mathbb{V}}^N$, $M_i^P\left(\mathcal{P}(\rho ,v)\right)={\Psi }_i^m(\rho ,v)$ holds.

Proof. 

First, I will prove that a probabilistic player potential function exists. First, define $\mathcal{P}(\rho ,v)={\displaystyle {\displaystyle \sum }_{g\subseteq g\left(\rho \right)}}\rho \left(g\right)·$$P(g,v)$, where $P:{\mathbb{G}}^N\times {\mathbb{V}}^N\to \mathbb{R}$ is the player potential function of network games.

Now,

$$\begin{array}{cc}& {\displaystyle \sum }_{i\in N}{M}_i^P\left(\mathcal{P}(\rho ,v)\right)\hfill \\ & ={\displaystyle \sum }_{i\in N}\left[\mathcal{P}\left(\rho ,v\right)-\mathcal{P}\left({\rho }^{-i},v\right)\right]\hfill \\ & ={\displaystyle \sum }_{i\in N}\left[{\displaystyle \sum }_{g\subseteq g\left(\rho \right)}\rho \left(g\right)·P(g,v)-{\displaystyle \sum }_{g\subseteq g\left({\rho }^{-i}\right)}{\rho }^{-i}\left(g\right)·P(g,v)\right]\hfill \end{array}$$

$$\begin{array}{cc}& ={\displaystyle \sum }_{i\in N}\left[{\displaystyle \sum }_{g\subseteq g\left(\rho \right):L_i\left(g\right)\ne \varnothing }\rho \left(g\right)·P(g,v)+{\displaystyle \sum }_{g\subseteq g\left({\rho }^{-i}\right)}\rho \left(g\right)·P(g,v)-{\displaystyle \sum }_{g\subseteq g\left({\rho }^{-i}\right)}{\rho }^{-i}\left(g\right)·P(g,v)\right]\hfill \\ & ={\displaystyle \sum }_{i\in N}\left[{\displaystyle \sum }_{g\subseteq g\left(\rho \right):L_i\left(g\right)\ne \varnothing }\rho \left(g\right)·P(g,v)+{\displaystyle \sum }_{g\subseteq g\left({\rho }^{-i}\right)}P(g,v)\left(\rho \left(g\right)-{\rho }^{-i}\left(g\right)\right)\right].\hfill \end{array}$$

since by Proposition 1, $g\left({\rho }^{-i}\right)=g\left(\rho \right)-L_i\left(g\left(\rho \right)\right)$, and hence, $g\subseteq g\left({\rho }^{-i}\right)=g\subseteq g\left(\rho \right):L_i\left(g\left(\rho \right)\right)=\varnothing$. By Proposition 1, ${\rho }^{-i}\left(g\right)=\rho \left(g\right)+{\displaystyle \sum _{\varnothing \ne h\subseteq L_i\left(g\left(\rho \right)\right)}}\rho (g\cup h)$ for all $g\subset g\left({\rho }^{-i}\right)$, and hence, $\rho \left(g\right)-{\rho }^{-i}\left(g\right)=-{\displaystyle \sum _{\varnothing \ne h\subseteq L_i\left(g\left(\rho \right)\right)}}\rho (g\cup h)$.

Hence,

$$\begin{array}{cc}& {\displaystyle \sum }_{i\in N}{M}_i^P\left(\mathcal{P}(\rho ,v)\right)\hfill \\ & ={\displaystyle \sum }_{i\in N}\left[{\displaystyle \sum }_{g\subseteq g\left(\rho \right):L_i\left(g\right)\ne \varnothing }\rho \left(g\right)·P(g,v)-{\displaystyle \sum }_{g\subseteq g\left({\rho }^{-i}\right)}\left(\sum _{\varnothing \ne h\subset L_i\left(g\left(\rho \right)\right)}\rho (g\cup h)\right)·P(g,v)\right].\hfill \end{array}$$

Now, the set of networks $g\subseteq g\left(\rho \right):L_i\left(g\right)\ne \varnothing$ is the same as the set of networks $g\cup h:g\subseteq g\left({\rho }^{-i}\right)$& $\varnothing \ne h\subseteq L_i\left(g\left(\rho \right)\right)$.

Hence,

$$\begin{array}{cc}& {\displaystyle \sum }_{i\in N}{M}_i^P\left(\mathcal{P}(\rho ,v)\right)\hfill \\ & ={\displaystyle \sum }_{i\in N}\left[{\displaystyle \sum }_{g\subseteq g\left(\rho \right):L_i\left(g\right)\ne \varnothing }\rho \left(g\right)·\left(P(g,v)-P(g-L_i\left(g\right),v)\right)\right].\hfill \end{array}$$

By Lemma 1, $P(g,v)-P(g-L_i\left(g\right),v)=Y_i^m(g,v)$. Hence,

$$\begin{array}{cc}& {\displaystyle \sum }_{i\in N}{M}_i^P\left(\mathcal{P}(\rho ,v)\right)\hfill \\ & ={\displaystyle \sum }_{i\in N}\left[{\displaystyle \sum }_{g\subseteq g\left(\rho \right):L_i\left(g\right)\ne \varnothing }\rho \left(g\right)·Y_i^m(g,v)\right].\hfill \end{array}$$

Given that $Y_i^m(g,v)=0$ for all i such that $L_i\left(g\right)=\varnothing$, I get

$$\begin{array}{cc}& {\displaystyle \sum }_{i\in N}{M}_i^P\left(\mathcal{P}(\rho ,v)\right)\hfill \\ & ={\displaystyle \sum }_{i\in N}\left[{\displaystyle \sum }_{g\subseteq g\left(\rho \right)}\rho \left(g\right)·Y_i^m(g,v)\right]\hfill \\ & ={\displaystyle \sum }_{i\in N}{\Psi }_i^m(\rho ,v)\hfill \\ & =\mathbb{E}(\rho ,v)\hfill \end{array}$$

given that the Expected Myerson value is balanced by of Proposition 2. It is also clear from the above steps that $M_i^P\left(\mathcal{P}(\rho ,v)\right)={\Psi }_i^m(\rho ,v)$ for all $i\in N$.

Our proof of uniqueness closely follows Ghintran et al. (2012). From (2) and (4), I get

$${\displaystyle \sum }_{i\in N}\left[\mathcal{P}\left(\rho ,v\right)-\mathcal{P}\left({\rho }^{-i},v\right)\right]=\mathbb{E}(\rho ,v)$$

$$\begin{array}{cc}& \Rightarrow {\displaystyle \sum }_{i\in N}\mathcal{P}\left(\rho ,v\right)-{\displaystyle \sum }_{i\in N}\mathcal{P}\left({\rho }^{-i},v\right)=\mathbb{E}(\rho ,v)\hfill \\ & \Rightarrow n·\mathcal{P}\left(\rho ,v\right)-{\displaystyle \sum }_{i\in N}\mathcal{P}\left({\rho }^{-i},v\right)=\mathbb{E}(\rho ,v)\hfill \\ & \Rightarrow \mathcal{P}\left(\rho ,v\right)={\displaystyle \frac{1}{n}}\left[\mathbb{E}(\rho ,v)+{\displaystyle \sum }_{i\in N}\mathcal{P}\left({\rho }^{-i},v\right)\right].\hfill \end{array}$$

(6)

Starting with $\mathcal{P}\left({\rho }_0,v\right)=0$, for every $({\rho }_0,v)\in {\mathbb{P}}^N\times {\mathbb{V}}^N$, using (6), I can recursively define $\mathcal{P}\left(\rho ,v\right)$ in a unique way. □

Before we conclude, we provide an example illustrating the calculation of these potentials for a uniform probability distribution.

Example 1.

Let $n=3$ and $\rho \left(g\right)={\displaystyle \frac{1}{8}}$ for all $g\in {\mathbb{G}}^N$. Further, let all one link networks yield a value of 1, all 2 link networks yield a value of 2 and the complete network yield a value of 3. From Slikker (2005), the link potential function $P^L$ follows:

$$P^L(g,v)={\displaystyle \frac{1}{2(\#g)}}\left[v\left(g\right)+2{\displaystyle \sum }_{ij\in g}{P}^L(g-ij,v)\right].$$

Hence, we get

$$\begin{array}{ll}g\hfill & P^L(g,v)\hfill \\ \varnothing \hfill & 0\\ \left\{\right\{12\left\}\right\}\hfill & {\displaystyle \frac{1}{2}}\\ \left\{\right\{13\left\}\right\}\hfill & {\displaystyle \frac{1}{2}}\\ \left\{\right\{23\left\}\right\}\hfill & {\displaystyle \frac{1}{2}}\\ \left\{\right\{12\},\{13\left\}\right\}\hfill & {\displaystyle \frac{3}{4}}\\ \left\{\right\{12\},\{23\left\}\right\}\hfill & {\displaystyle \frac{3}{4}}\\ \left\{\right\{13\},\{23\left\}\right\}\hfill & {\displaystyle \frac{3}{4}}\\ g_N\hfill & {\displaystyle \frac{3}{4}}\end{array}$$

and hence, the probabilistic link potential function is given by ${\mathcal{P}}^L(\rho ,v)={\displaystyle {\displaystyle \sum }_{g\subseteq g\left(\rho \right)}}\rho \left(g\right)·$$P^L(g,v)={\displaystyle \frac{9}{16}}.$

From Slikker (2005), the player potential function $P^P$ follows

$$P^P(g,v)={\displaystyle \frac{1}{n}}\left[v\left(g\right)+{\displaystyle \sum }_{i\in N}{P}^P(g-L_i\left(g\right),v)\right].$$

Hence, we get

$$\begin{array}{ll}g\hfill & P^P(g,v)\hfill \\ \varnothing \hfill & 0\\ \left\{\right\{12\left\}\right\}\hfill & {\displaystyle \frac{1}{3}}\\ \left\{\right\{13\left\}\right\}\hfill & {\displaystyle \frac{1}{3}}\\ \left\{\right\{23\left\}\right\}\hfill & {\displaystyle \frac{1}{3}}\\ \left\{\right\{12\},\{13\left\}\right\}\hfill & {\displaystyle \frac{8}{9}}\\ \left\{\right\{12\},\{23\left\}\right\}\hfill & {\displaystyle \frac{8}{9}}\\ \left\{\right\{13\},\{23\left\}\right\}\hfill & {\displaystyle \frac{8}{9}}\\ g_N\hfill & {\displaystyle \frac{4}{3}}\end{array}$$

and hence, the probabilistic player potential function is given by ${\mathcal{P}}^P(\rho ,v)={\displaystyle {\displaystyle \sum }_{g\subseteq g\left(\rho \right)}}\rho \left(g\right)·$$P^P(g,v)={\displaystyle \frac{5}{8}}.$

4. Conclusions

I conclude by pointing out some directions in which this body of research can be extended. My results can be extended to potentials related to other fixed network allocation rules such as the network Banzhaf value (see Gogoi et al., 2026) in the framework of both deterministic and variable network situations. Under variable network situations, the network value function and the probability distribution of network formation are completely independent. However, these could be related.