1 Introduction
The understanding of the effects of collaboration and communication through networks on collective wealth generation dates back to Myerson (1977, 1980). Myerson considered networks to be communication structures that impose constraints on coalition formation in a cooperative game with transferable utilities: A coalition can form if it is connected in the prevailing communication network. This framework—to understand networks as constraints on coalition formation in a cooperative TUgame—is known as a communication situation.
The Shapley Value of the restriction of a cooperative game endowed with a communication network is called the Myerson value, which was seminally introduced by Myerson (1977). In a communication situation, the precise architecture of the communication network is not important; the resulting class of feasible or formable coalitions determines the resulting values. Two different networks inducing the same partition of the player set will yield an identical restricted game and hence an identical Myerson value.
Jackson and Wolinsky (1996) introduced games in which value stems directly from the network rather than a coalition of players. Such a construct is referred to as a network game. In this approach the interaction patterns among players in a network are wealth creating, rather than constraining wealth creation. Allocation rules for network games specify how the value created by the network game is divided among players. Hence, allocation rules for communication situations can be extended to this setting. Thus, allocation rules in the fixed network setting include the Myerson value due to Jackson and Wolinsky (1996) and Position Value due to Slikker (2007).^{1}^{1}1Jackson (2005) argues that the fixed allocation rules are not appropriate in network game settings where the network evolves from players forming and deleting links, and hence came up with the flexible allocation rules.
The (deterministic) Position Value was seminally introduced by Meessen (1988) as an alternative allocation rule for communication situations and subsequently further developed by Borm et al. (1992). In this approach links rather than players are considered as the source of all generated wealth. As such, all generated wealth should therefore be allocated to these links. This transforms a communication situation into a link game in which communication links act as players. The Shapley value of the link game now assigns fair values to all links in the network based on the generated wealth. The Position Value of a communication situation is now the distribution of the assigned Shapley link values to all constituting players of these links. Slikker (2005) characterizes the Position Value for communication situations by component balance and the balanced link contributions property.
Slikker (2007) extended the Position Value to the class of network games and characterized this extension using an appropriate formulation of the balanced link contributions property. Slikker’s characterization of the Position Value makes it fully compatible with his characterization of the Myerson Value on the same class of network games. In subsequent work, this comparative characterization has been pursued for other classes of cooperative wealth generation.
Games with probabilistic networks were first considered by Calvo et al. (1999) in the context of communication situations. In this framework, links between players are formed stochastically independently according to given probabilities, leading to probabilistic constraints on coalition formation. The resulting network formation probabilities are fully determined by the given link formation probabilities. In particular, the network values are formulated as probabilistic formulations, following the multilinear extension first proposed by Owen (1972). The wealth created through a coalition is replaced by the expected value based on networkrestricted games created by all possible networks that might form on that coalition. Calvo et al. (1999) extend the Myerson value to the class of these probabilistic communication situations.
Borkotokey et al. (2021) extend the probabilistic perspective on link formation of Calvo et al. (1999) to the realm of network games. This results in probabilistic network games, where networks are formed stochastically independently based on given link formation probabilities. Borkotokey et al. (2021) extend and characterize the Myerson value as well as the Position Value to this framework of probabilistic network games.
It can be argued that the independence hypothesis of link formation is questionable. In their paper, Borkotokey et al. (2021) give a practical example of probabilistic network games over the airline code sharing networks. Passengers travelling on intercontinental flights often using multiple airlines who have code sharing agreements with one another. The passenger pays an upfront fee which is divided among the relevant code sharing airlines in some fashion. But given the competition among airlines, these links are often unstable with airlines terminating existing agreements and forming new agreements. Further, the independence assumption of link formation seems unrealistic in this setting because airlines are looking at the overall strategic situation rather than considering a bilateral agreement in isolation. So, a more general framework of probabilistic network formation is applicable.
This has been developed by Gómez et al. (2008). In particular, Gomez et al. (2008) consider a generalization of Calvo et al. (1999) where the assumption of independent link formation in communication situations is dispensed with. Instead, one assumes an arbitrary probability distribution on the set of all possible networks on a given player set. Gomez et al. (2008) refer to this as a generalized probabilistic communication situation. They extend and characterize the Myerson value to this more general setting. Ghintran et al. (2012) define and characterize the Position Value in this framework of generalized probabilistic communication situations.
Our framework: Variable network games
The purpose of our paper is to extend the framework of generalized probabilistic communication situations to the realm of network games. We introduce the notion of a variable network game as a combination of a network game and an arbitrary network formation probability distribution. The network game assigns a generated wealth level to every network, while the network formation probability distribution assigns a probability to each network of forming. This framework captures the various frameworks considered in the previous discussion as special cases.
In variable network games, one has consider the expected levels of wealth that are generated among the players through the networks that can form through which they conduct their affairs. We show that certain properties of the underlying network game are retained in the assignment of expected wealth levels created through these probabilistic networks.
This allows us to consider allocation rules on the class of variable network games that are founded on familiar allocation rules on the smaller class of network games. Indeed, using expectations of payoffs of these familiar allocation rules over all networks that can form, we arrive at the allocation of the expected wealth that is created in the given variable network game. This allows us to extend the Myerson Value as well as the Position Value to the class of variable network games as the expected payoff allocation rule. This is referred to as the Expected Myerson Value and the Expected Position Value in the context of our setting.
The Myerson value on the smaller class of network games has been characterized through two main axiomatizations. The first axiomatization is the one based on the properties of component balance—or component efficiency—and equal bargaining power—or “fairness”—as seminally conceived by Myerson (1977). The equal bargaining power property postulates that the change of the allocated payoff is exactly the same for two players if the link between them is removed from the network. In our framework of variable network games this refers to the link being member of a network with zero probability. Jackson and Wolinsky (1996) extended this axiomatization to the setting of network games. In the current paper we extend this axiomatization further to the Expected Myerson Value on the class of variable network games.
Our second axiomatization of the Expected Myerson Value is founded on the component balance property in combination with the balanced contributions property. This axiomatization was seminally formulated for the Myerson value on the class of network games by Slikker (2007). In our setting the balanced contributions property refers to the effects of any player being “removed” in the sense of having zero probability of being a member of a formable network. In particular, the effects on the expected payoffs are equal if other players are “removed” from these probabilistic networks.
Our third axiomatization concerns the extension of Slikker’s axiomatization of the Position Value on the class of network games (Slikker, 2007) to our realm of variable network games. This is founded on the formulation of the balanced link contributions property to the class of variable network games. This property imposes that the accumulated effects of the removal of links in a player’s neighborhood is same for all pairs of players. This axiomatization compares directly with the second axiomatization of the Expected Shapley Value and allows an assessment of these two main values for variable network games.
It is clear that further development of the framework of variable network games is warranted and valuable. It allows for the introduction of more tools to properly represent the features of certain interaction situations. We explore here the case of an intermediated trade, where institutional features of the trade situation are represented by the network formation probability distribution rather than the network payoff function.
Structure of the paper
In Section 2 of the paper we develop the foundations of our approach and introduce the main formal conception of a variable network game. We explore some properties of the expected wealth that is generated in these variable network games. Section 3 introduces allocation rules on the class of variable network games and defines the Expected Myerson Value. Section 4 presents the two main axiomatizations of the Expected Myerson Value and their proofs. Section 5 concludes.
2 Variable network games: Definitions
Throughout we let be a fixed, finite set of players. With a slight abuse of notation, for every set of players , we denote for any player the expanded set and for any player the reduced set . In particular, the set for is the set of all players other than .
Throughout, we denote by the number of elements in a set .
2.1 Network preliminaries
Given the player set , a link between two distinct players and is defined as the binary set , representing an undirected relationship between and .^{2}^{2}2In the following discussion and throughout the remainder of this paper, we use and extend the notation seminally introduced by Jackson and Wolinsky (1996). Clearly, is equivalent to . The set of all possible links on is denoted by .
A network on is any set of links . In particular, is called the complete network, while is the empty network. The class of all possible networks on is given by . Clearly, and .
For every network and every player we denote ’s neighborhood in by and and her link neighborhood by . Furthermore, . In particular, we introduce as the set of all potential links in which player participates.
We also define and let with the convention that if , we let .^{3}^{3}3We emphasize here that if , we have that . Namely, in those cases the network has to consist of at least one link. will be referred to as size of the network .
We say that player is an isolated player in network if . This implies that for isolated player in it holds that . We denote the set of all isolated players in network by .
Connectivity in networks
A path connecting and with in network is an ordered set of distinct players with such that , , and for all . We say that with are connected in if there exists a path between and and disconnected otherwise. A network is connected if all pairs of players with are connected. In particular, for a connected network it holds and .
The network is a component of if is connected and for any and , implies . In other words, a component is a maximally connected subnetwork of . We denote the set of network components of the network by . Note that for any connected network it holds that . In particular, and .
Restrictions of networks
Let and let be some set of players. Then the restriction of to is the network defined as , where and . The restricted network is obviously a subnetwork of .
For a subnetwork , we denote by the network that results after the removal from of all links in the subnetwork . Similarly, for any network we denote by the network that results from after adding all links in to the network . Clearly, for it holds that .
2.2 Network formation probability distributions
Following Gómez et al. (2008) we investigate the formal description of the probabilistic emergence or formation of networks on a given set of players. These probabilistic structures are introduced as additional modelling tools to understand certain phenomena in wealth creation processes observed in the economy. In particular, these probabilistic structures can be used to describe basic link formation failures or fuzziness related to the formation of a network.
Before discussing some motivating examples, we formally introduce the notion of a network formation probability distribution that assigns to every network a probability that it forms.
Definition 2.1
A network formation probability distribution on is a map such that .
The class of all network formation probability distributions on is denoted by
(1) 
Note that .
The notion of a network formation probability distribution was introduced by Gómez et al. (2008), generalising the linkbased network formation approach of Calvo et al. (1999). A network formation probability distribution naturally results in the following conceptions.
Definition 2.2
Let be some network formation probability distribution on .

A network is formable under if . The class of formable networks under is given by , which can also be denoted as the support of .

The extent of the distribution is the union of the class of formable networks given by .

A network is a component of if is a component of the extent and a player is isolated in if is an isolated player in the extent .
Formable networks are those that 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.
The extent of a network formation probability distribution is recognised as the network that consists of all links that form with positive probabilities, extending the definition of the support developed in Borkotokey et al. (2021) for probabilistic networks to the class . The components and isolated players in the extent of a network formation probability distribution critically determine the main features of those networks that can emerge under .
A motivating example: Intermediated bilateral trade
The following examples provide some motivation for the study of these structures. We argue that these mathematical devices can represent institutional aspects of network formation into the framework of network games. These examples are mainly motivated by the discussion in Calvo et al. (1999, Section 2.1) of a simple bilateral trading case with potential intermediation. There the case is developed from a linkbased probabilistic perspective. We investigate that here, prior to generalizing to an implementation of networkbased probabilities representing an institutional constraint on network formation.
Example 2.3
(Independent link formation)
Consider a trading situation with one seller and one buyer who can generated mutual gains from trade by the transfer of an object. The total wealth that is created is set to one (1). The trade can either be executed directly between and or through the intermediation of an intermediary . The player set can be identified as .
In the standard approach to network games, the links between these three players either exist or not. Myerson (1977, 1980) seminally investigated these situations. Under the Myerson hypotheses, the full wealth of 1 can be realised in networks such that and/or . In all other networks the generated wealth is zero, since no trade can be accomplished. We can describe this by an appropriately constructed network formation probability distribution on .
Calvo
et al. (1999) introduced the instrument of probabilistic links to describe that relationships are formed subject to certain conditions. Hence, probabilities on the links in are introduced, representing the fundamental uncertainty that links can be formed. Assuming that both and are formed with equal probability and that the critical link is formed with probability , we arrive at a graphical depiction of the communication situation as depicted in Figure 1.
Next we introduce the hypothesis that all links are formed independently. This implies that a network forms with a probability that is determined as a product of link formation probabilities or their nonformation. We can now determine the generated network formation probability distribution as
Network  Probability 

The expected generated wealth can now be computed as
Note that if and/or . Calvo et al. (1999) only considered the case of in their motivating discussion.
Example 2.3 discusses the special case of network formation being based on the independent formation of the individual links that make up the network—further explored in Calvo et al. (1999) and Borkotokey et al. (2021). In particular, let with be a link formation probability distribution. Under independence of link formation, the probability that a network forms is then given by the multilinear form
(2) 
It holds that for every link formation probability distribution with .
The next example discusses a case in which the network formation probability distribution is not based on a link formation probability distribution, but rather expresses an institutional feature of network formation.
Example 2.4
(Institutional network formation)
Consider again the intermediated bilateral trade situation discussed in Example 2.3. We amend this case by assuming that this trade occurs in an institutional framework of a certain implementation of contract law and that a relationship can only be effectuated if notarised. We assume further that the intermediary is a notary and that trade can, therefore, only occur in a network to which is connected.^{4}^{4}4Therefore, we de facto assume that either or or both of these players need to know the intermediary to have access to her notary services. Hence, trade can only be executed in networks such that and can trade as well as .
Furthermore, we assume that the costs related to the formal notarisation of the contract by is negligible in relation to the wealth created through the trade between and .
We apply again the assumption that network formation is founded on the link formation probabilities of of and forming and of of forming. However, now networks can only be formed if is connected. This results in conditional probabilities of network formation in relation to the probabilities reported in Example 2.3. These conditional probabilities are given in the table below.
Network  Probability 

The expected generated wealth in this institutional network formation setting can be computed as
Note that as before if and/or .
Moreover, we easily compute that if and only if if and only if as well as . This implies that this simple application shows that institutional embedding increases the expected generated wealth in this simple bilateral trade situation.
Restrictions of network formation probability distributions
Next we discuss the notion of restricting a network formation probability distribution to a certain given (deterministic) network. This implies that its extent is limited to the given network. The following definition formalises this by transferring network formation probabilities of networks extending beyond the imposed restriction to the networks that form within the imposed rerstricted extent. The applied conception is due to Gómez et al. (2008, page 543).
Definition 2.5
Let be some network formation probability distribution and let be some given network on . Then the restriction of to is the modified network formation probability distribution defined by
(3) 
In this definition the formation probabilities of subnetworks of are transferred to subnetworks of itself. It should be clear that the extent of the restriction of to some network is determined as .
As a special cases of this notion of a restriction of a network formation probability distribution to a given network, we introduce devices for removing individual links and players. For the computation of these restrictions we can state the following proposition.
Proposition 2.6
Let and let with .

Let be the restriction of to . Then for every network
(4) Furthermore, it holds that and
(5) 
Let be the restriction of to . Then for every network it holds that
(6) and for any .
Furthermore, and(7)
Proof. It is clear that (4) in assertion (a) follows immediately from the definition of as introduced in Definition 2.5.
Next, note that follows immediately from (4).
To prove (5), take . Then by (4) it follows that .
Next, take any with . Then by (4), if and only if and/or if and only if .
Finally, take any with . Then , implying that .
This show that (5) indeed holds.
To show (6) in assertion (b), let and . From Definition 2.5 it is obvious that for every with .
Next let with . Then it holds by definition that
Furthermore, from the definitions it is clear that all networks with links outside the extent have zero formation probability under . Hence,
This shows (6).
To show (7), take . Now if , if and only if for some (including ) if and only if .
Moreover, if , it follows immediately that and, therefore, .
Finally, from (7) we immediately see that . This shows assertion (b) of the proposition.
2.3 Variable network games
Jackson and Wolinsky (1996) seminally introduced the notion of a network game on the player set . 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 .
Thus, a network game on is a function such that . This leads to the class of all network games on to be defined as
(8) 
The class has been the subject of study of numerous contribution to game theoretic network analysis. It is clear that is a dimensional Euclidean space.
In this paper we extend the class of network games to the larger class of network wealth creation situations in which network formation processes are assumed to be probabilistic. This is represented by a combination of a network game and a network formation probability distribution.
Definition 2.7
A variable network game is a pair consisting of a network game describing the potential wealth levels created through the formed networks and a network formation probability distribution describing the probabilities with which networks form.
The expected wealth that is created in the variable network game is defined as
(9) 
There are some properties satisfied by the expected wealth that is created in a variable network game. First, we consider the case that a network game is component additive. Component additivity reflects explicitly that all collective wealth is generated through the network through which the players interact. Therefore, disconnected components of a network do not interact and the total generated wealth is simply the sum of the wealth generated in the constituting components of the network.
Formally, a network game is component additive if for every network it holds that .
Proposition 2.8
Let be such that is component additive. Then it holds that
(10) 
Proof. From the component additivity of it immediately follows that
by definition of the restriction of to any . This shows the assertion.
Second, we consider the marginal contributions of individual links and players to the expected wealth that is created by a variable network game. Recall that in this context the deletion of a link or a player from a network formation probability distribution is expressed through the restrictions and , respectively. The next definitions formalise the contributions made by links and players in regular network games.
Let be a network game and let be a given network. The marginal contribution of a link to the network game at network is given by .
Similarly, the marginal contribution of a player to the network game at network is given by .
The next proposition collects properties that describe the marginal contributions of links and players to a variable network game.
Proposition 2.9
Let be a variable network game.

For every link it holds that
(11) 
For every player it holds that
(12)
Proof. To show assertion (a) we use property (a) of Proposition 2.6 that for every and . Hence,
To show assertion (b), we use property (b) stated in Proposition 2.6 to derive that
This shows the assertion.
The properties stated in Proposition 2.9 show that if links and players are contributing, their removal from a variable network game reduces the expected wealth that is generated. Furthermore, the removal or addition of socalled null players—as defined throughout the literature as players such that for all —do not affect the expected wealth generated in variable network games.
3 Allocation rules on variable network games
The main objective of this paper is to investigate the allocation of the expected generated wealth in variable network games and the properties of the associated allocation rules. We do this in two stages. First, we recall the notion of allocation rules on the class of network games and, subsequently, we consider natural probabilistic extensions of allocation rules from the class of network games to the class of variable network games. We focus hereby on the particular allocation rule of the Myerson Value based on the seminal formulation by Myerson (1977) for communication situations as well as an extension of the Position Value seminally developed by Meessen (1988) and Borm et al. (1992) for communication situations.
Allocation rules on
Following the notation introduced by Jackson and Wolinsky, an allocation rule on the class of network games is a mapping that for every network game assigns to every player in a network an allocated value such that for every isolated player .^{5}^{5}5The imposed property that isolated players are assigned a zero value is a required hypothesis, since under standard properties such as balance and component balance one can only show that . For a further discussion we also refer to Borkotokey et al. (2021).
Allocation rules on network games can satisfy a number of standard properties that have been introduced and investigated in the literature. We list the most relevant of these properties below.

An allocation rule on is balanced if for every network game and every network .

An allocation rule on is component balanced if for every component additive network game ,^{6}^{6}6We recall that a network game is component additive if for all . every network and all of its components .
Component balancedness implies balancedness for component additive network games. 
An allocation rule on satisfies equal bargaining power^{7}^{7}7The equal bargaining power property was referred to as the “fairness” property by Myerson (1977). if for every network game and every network it holds for all pairs with that

An allocation rule on satisfies the balanced contributions property if for every network game and every network it holds for all players that

An allocation rule satisfies the balanced link contributions property if for every network game and every network it holds for all players with that
The properties listed above have been used to characterise the most common allocation rule on the class of network games, namely the Myerson Value (Jackson and Wolinsky, 1996) and the Position Value (Slikker, 2007).
Allocation rules on
An allocation rule on the class of (regular) network games is assigned to a network game as well as a certain given deterministic network, representing the interaction between the players. The allocation of the generated wealth is, therefore, conditioned on the particular relationships between the players in the game.
On the other hand, variable network games are introduced as combinations of a network game and a network formation probability distribution. This implies that allocation rules should account for the stochastic nature of network formation processes and, consequently, the relationships between the players. This is formalised in the next two definitions.
Definition 3.1
An allocation rule on the class of variable network games is a mapping such that for every variable network game it holds that for every isolated player .
The definition of an allocation rule is clearly a straightforward conceptual extension from the definition of allocation rules on the class of network games to the class of variable network games. In particular, all allocation rules on can be extended to the larger class of variable network games by taking its expected payoffs. This is formalised as follows.
Definition 3.2
Let be an allocation rule on the class of network games. Then its standard extension to the class of variable network games is the allocation rule defined by
(13) 
for every variable network game .
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.
3.1 Balancedness properties of allocation rules
The properties listed above for allocation rules on the class of network games can easily be extended to allocation rules on the class of variable network games. This is explored next for the properties introduced for network game allocation rules.
Balanced allocation rules
An allocation rule satisfies balancedness—or is balanced—if for every variable network game it holds that , i.e., the allocation rule exactly covers the expected wealth that is created in the variable network game.
Proposition 3.3
Let be a balanced allocation rule on the class of network games. Then its standard extension is balanced.
Proof. The proof of the assertion rests on the following:
This indeed shows the assertion.
Component Balancedness
An allocation rule satisfies component balancedness—or is component balanced—if for every variable network game such that is component additive, it holds that for every component
(14) 
Component balancedness implies that all wealth that can be attributed to a certain component in 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 game.
Proposition 3.4
Let be a component balanced allocation rule on the class of network games. Then its standard extension is component balanced.
Proof. Let be component additive and take any .
Next, take a component and consider any network . By definition there are no connections between players in and in . Hence, any component of containing players from is a subset of .
From the component balance of and the component additivity of it then follows that
Therefore, we can conclude that
This completes the proof of the assertion.
3.2 The Expected Myerson and Position Values
As discussed in the introduction to this paper, there are traditionally two principal allocation rules on the class of network games, namely the Myerson Value and the Position Value. Both of these values can be extended to the class of variable network games through the consideration of their standard extension, reflecting the expected allocation of the generated wealth under these two values. The definition of these two allocation rules is discussed here, while in the next sections we consider the axiomatization of these two allocation rules on the class of variable network games.
Jackson and Wolinsky (1996) formulated the Myerson Value as an allocation rule on the class of network games as an extension of the allocation rule for communication situations seminally introduced by Myerson (1977). Formally, the Myerson Value on the class of network games is the allocation rule defined by
(15) 
Next we explore extending the Myerson Value to the class of variable network games using the method explored above.
Definition 3.5
The Expected Myerson Value is the allocation rule defined as the standard extension of the Myerson Value on the class of network games to the class of variable network games.
Slikker (2007) introduced the Position Value as an allocation rule on the class of network games by extending the earlier definition to the new framework. Formally, the Position Value on the class of network games is the allocation rule defined by
(16) 
As applied for the Myerson Value, we can also base an allocation rule on the class of variable network games on the formulated Position Value. This introduces the Expected Position Value.
Definition 3.6
The Expected Position Value is the allocation rule defined as the standard extension of the Position Value on the class of network games to the class of variable network games.
Exploring expected values in the intermediated trade situation
The Expected Myerson Value is simply the expectation of the Myerson payoff to a player that arises in every possible network that can form under the imposed network formation probability distribution. Similarly, the Expected Position Value is the expectation of the player’s Position Value in the possible networks. We illustrate the computation of these two Expected Values by returning to the case of intermediated trade discussed in Examples 2.3 and 2.4.
Example 3.7
Again consider the intermediated trade situation with player set , described in Figure 1. The generated wealth in this bilateral trade situation can be formulated as a network game given by if and only if and/or , and otherwise.
We recall that the network formation probability distribution represents network formation under the hypothesis of independence of link formation with probabilities of and forming and of forming.
For each resulting network we can thus compute the corresponding Myerson and Position Values. The following table collects the information for this case:
0  
0  
0  
1  
1  
1 
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