Expected Values for Variable Network Games

A network game assigns a level of collectively generated wealth to every network that can form on a given set of players. A variable network game combines a network game with a network formation probability distribution, describing certain restrictions on network formation. Expected levels of collectively generated wealth and expected individual payoffs can be formulated in this setting. We investigate properties of the resulting expected wealth levels as well as the expected variants of well-established network game values as allocation rules that assign to every variable network game a payoff to the players in a variable network game. We establish two axiomatizations of the Expected Myerson Value, originally formulated and proven on the class of communication situations, based on the well-established component balance, equal bargaining power and balanced contributions properties. Furthermore, we extend an established axiomatization of the Position Value based on the balanced link contribution property to the Expected Position Value.

1 Introduction e understanding of the effects of collaboration and communication through networks on collective wealth generation dates back to Myerson (1977Myerson ( , 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. is framework-to understand networks as constraints on coalition formation in a cooperative TU-game-is known as a communication situation.
e 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 pa erns 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 se ing.us, allocation rules in the fixed network se ing include the Myerson Value due to Jackson and Wolinsky (1996) and Position Value due to Slikker (2007).1 e (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.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 for-mation.e 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).e wealth created through a coalition is replaced by the expected value based on network-restricted games created by all possible networks that might form on that coalition.Calvo et al. (1999)  Our framework: Variable network games e 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.e network game assigns a generated wealth level to every network, while the network formation probability distribution assigns a probability to each network of forming. is framework captures the various frameworks considered in the previous discussion as special cases.
In variable network games, one has to 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.
is 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 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 .Furthermore, we denote by # the number of elements in a set .

Cooperative games and the Shapley value
A cooperative game on a player set is a mapping : 2 → R such that ( ) = 0.A cooperative game assigns to every non-empty coalition of players ⊆ some "worth" ( ), representing a collectively generated wealth.e function is also referred to as a characteristic function.
An allocation rule is a mapping that assigns to every cooperative game some vector ∈ R .(1953) seminally introduced the seminal allocation rule that assigns to every cooperative game an allocation ( ) ∈ R given for every player ∈ by
e Shapley value lies at the foundation of the wealth allocation rules for network games and variable network games considered in this paper.

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 . 3Clearly, is equivalent to .e set of all possible links on is denoted by = { | , ∈ and ≠ }. 0 = is the empty network.e class of all possible networks on is given by G For every network ∈ G and every player ∈ we denote 's neighborhood in by In particular, we introduce = ( ) as the set of all potential links in which player participates.
We say that player ∈ is an isolated player in network ∈ G if ∉ ( ). is implies that for isolated player in it holds that ( ) = .We denote the set of all isolated players in network by 0 ( ) = \ ( ).
and +1 ∈ for all = 1, . . ., − 1.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 ∈ ( ), ∈ implies ∈ ℎ.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 ∈ G it holds that ( ) = { }.In particular, ( ) = { } and Restrictions of networks Let ∈ G and let ⊆ be some set of players.e set of all links within the coalition can be represented as = { | , ∈ and ≠ }.Now, the restriction of to is the network defined as | is obviously a subnetwork of .
For a subnetwork ℎ ⊆ , we denote by − ℎ = \ ℎ the network that results a er the removal from of all links in the subnetwork ℎ.Similarly, for any network ℎ ⊆ − = \ we denote by +ℎ = ∪ℎ the network that results from a er adding all links in ℎ to the network . 6Clearly, for ℎ ⊆ it holds that ( − ℎ) + ℎ = .

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.ese 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.
e class of all network formation probability distributions on is denoted by (2) e notion of a network formation probability distribution was introduced by Gómez et al. ( 2008), generalising the link-based network formation approach of Calvo et al. (1999).A network formation probability distribution naturally results in the following conceptions.
Definition 2.2 Let ∈ P be some network formation probability distribution on .
, which can also be denoted as the support of .
(ii) e extent of the distribution is the network ( ) ∈ G defined as the union of all formable networks, namely, and a player ∈ is isolated in if ∈ 0 ( ( )) is an isolated player in the extent ( ).
Formable networks are those that are assigned a positive formation probability.e 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.erefore, 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.
e 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 de- Example 2.3 (Independent link formation) Consider a trading situation with one seller and one buyer who can generate mutual gains from trade by the transfer of an object.e total wealth that is created is set to one (1).e trade can either be executed directly between and or through the intermediation of an intermediary .
e player set can be identified as = { , , }.
In the standard approach to network games, the links between these three players either exist or not.Myerson (1977Myerson ( , 1980) ) seminally investigated these situations.Under the Myerson hypotheses, the full wealth of 1 can be realised in networks ∈ G 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 G .
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 ∈ [0, 1] and that the critical link is formed with probability ∈ [0, 1], 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. is implies that a network forms with a probability that is determined as a product of link formation probabilities or their non-formation.We can now determine the generated network formation probability distribution as shown in Table 1.
e expected generated wealth can now be computed as Note that E( ) = 1 if = 1 and/or = 1.Calvo et al. (1999) only considered the case of = 1 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

Network
Probability Borkotokey et al. (2021).In particular, let : → [0, 1] be some assignment of link formation probabilities on .Under independence of link formation, the probability that a network ∈ G forms is then given by the multilinear form It holds that ∈ P for every link formation probability assignment : e next example discusses a case in which the network formation probability distribution is based on a link formation probability distribution representing an institutional feature in the network formation process.
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.7Hence, trade can only be executed in networks ∈ G 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 probabili-  e expected generated wealth in this institutional network formation se ing can be computed as Note that as before E( ′ ) = 1 if = 1 and/or = 1.
Moreover, we easily compute that E( ′ ) > E( ) if and only if ( − 1) ( − 1) 2 < 0 if and only if < 1 as well as < 1. is 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. is implies that its extent is limited to the given network.
e following definition formalises this by transferring network formation probabilities of networks extending beyond the imposed restriction to the networks that form within the imposed restricted extent.e applied conception is due to Gómez et al. (2008, page 543).
Definition 2.5 Let ∈ P be some network formation probability distribution and let ∈ G be some given network on .en the restriction of to is the modified network formation probability distribution ∈ P defined by 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 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.
(a) Let − = − be the restriction of to − .en for every network ∈ G : Furthermore, it holds that − = ( ) − and and for any − : − ( ) = 0. Furthermore, it holds that − = ( ) \ ( ) and Proof.It is clear that (5) in assertion (a) follows immediately from the definition of − as introduced in Definition 2.5.
To show (7) in assertion (b), let ∈ P and ∈ .From Definition 2.5 it is obvious that for every Furthermore, from the definitions it is clear that all networks with links outside the extent ( ) have zero formation probability under .Hence, is shows (7).

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.is leads to the introduction of a function that assigns a wealth level to every network that can be formed on .
Formally, a network game on is a function : G → R such that ( 0 ) = 0. is leads to the class of all network games on to be defined as e class V has been the subject of study of numerous contributions to game theoretic network We recall that a network game ∈ V is component additive if ( ) = ℎ∈ ( ) (ℎ) for all ∈ G . is additional property will be used frequently throughout the next sections on variable network games.
Allocation rules on V × G Following the notation introduced by Jackson and Wolinsky, an allocation rule on the class of network games is a mapping : V × G → R that for every network game ∈ V assigns to every player ∈ in a network ∈ G an allocated value ( , ) such that ( , ) = 0 for every isolated player ∈ 0 ( ).8 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 V × G is balanced if for every network game ∈ V and every network ∈ G : • An allocation rule on V × G is component balanced if for every component additive network game ∈ V , every network ∈ G and all of its components ℎ ∈ ( ) : Component balance implies balance for component additive network games.
• An allocation rule on V × G satisfies the balanced contributions property if for every network game ∈ V and every network ∈ G it holds for all players , ∈ that ( , ) − ( , \ ) = ( , ) − ( , \ ).
• An allocation rule on V × G satisfies the balanced link contributions property if for every network game ∈ V and every network ∈ G it holds for all players , ∈ with ≠  (1988) to this extended framework.Formally, the Position Value on the class of network games is the

Variable network games
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.is is represented by a combination of a network game and a network formation probability distribution.
Definition 3.1 A variable network game is a pair ( , ) ∈ V × P 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.
e expected wealth that is created in the variable network game ( , ) ∈ V × P is defined as ere 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 ∈ V is component additive.Component additivity reflects explicitly that all collective wealth is generated in the component through which the players interact.erefore, there are no externalities across disconnected components of a network and the total generated wealth is simply the sum of the wealth generated in the constituting components of the network.
Formally, a network game ∈ V is component additive if for every network ∈ G it holds that ( ) = ℎ∈ ( ) (ℎ). roughout the remainder of the paper, we assume that all network games considered are component additive.
Proposition 3.2 Let ( , ) ∈ V × P be such that is component additive.en it holds that Proof.From the component additivity of it immediately follows that by definition of the restriction ℎ of to any ℎ ⊆ ( ). is 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 ∈ P is expressed through the restrictions − and − , respectively.e next definitions formalise the contributions made by links and players in regular network games.
Let ∈ V be a network game and let ∈ G be a given network.e marginal contribution of a link ∉ to the network game at network is given by Δ ( , ) = ( + ) − ( ).
Similarly, the marginal contribution of a player ∈ ( ) to the network game at network is given by Δ ( , ) = ( ) − ( \ ).
e next proposition collects properties that describe the marginal contributions of links and players to a variable network game.
Proposition 3.3 Let ( , ) ∈ V × P be a variable network game such that is component additive.
(a) For every link ∈ it holds that (b) For every player ∈ it holds that Proof.To show assertion (a) we use Proposition 2.6(a) that for every ∈ G it holds that − ( ) = To show assertion (b), we use Proposition 2.6(b) to derive that is shows the assertion.
e properties stated in Proposition 3.3 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 so-called null players do not affect the expected wealth generated in variable network games, i.e., Δ ( , ) = 0 for some ∈ G and ∈ ( ) implies that E( , − ) = E( , ).
Similarly, superfluous links do not affect wealth generation either in the sense that Δ ( , ) = 0 for some ∈ G and ∉ implies that E( , − ) = E( , ).

Allocation rules on variable network games
e 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.In particular, 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 Myerson Value and the Position Value for network games.We show that the standard axiomatizations for both of these allocation rules extend to our framework of variable network games.
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.e allocation of the generated wealth is, therefore, conditioned on the particular network 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.is implies that allocation rules should account for the stochastic nature of network formation processes and, consequently, the probabilistic relationships between the players.is is formalised in the next two definitions.
Definition 3.4 An allocation rule on the class of variable network games is a mapping Ψ : V × P → R such that for every variable network game ( , ) ∈ V × P it holds that Ψ ( , ) = 0 for every isolated player ∈ 0 ( ( )).
e definition of an allocation rule is clearly a straightforward conceptual extension of the definition of allocation rules on the class of network games to the class of variable network games V ×P .In Component balance implies that all wealth that can be a ributed 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 3.7 Let : V × G → R be a component balanced allocation rule on the class of network games.en its standard extension Ψ : V × P → R is component balanced.
Proof.Let ∈ V be component additive and take any ∈ P .
Next, take a component ℎ ∈ ( ( )) and consider any network ⊆ ( ).Let ℎ ′ ∈ ( ) be a component of with (ℎ) ∩ (ℎ ′ ) ≠ .en, clearly, ℎ ′ ⊆ ℎ and, therefore, from the component balance of it then follows that Hence, by component additivity of , it follows that erefore, we can conclude that is completes the proof of the assertion.

e Expected Myerson and Position Values
As discussed in Section 2.4, there are traditionally two principal allocation rules on the class of network games, namely the Myerson Value (10) and the Position Value (11).Both of these allocation rules 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.
First, we explore extending the Myerson Value to the class of variable network games using the method explored above.Referring to Table 1, we denote by 1 the network formation probability distribution representing network formation under the hypothesis of independence of link formation with probabilities > 0 of and forming and > 0 of forming.
For each resulting network we can thus compute the corresponding Myerson and Position Values.
e following table collects the information for this case: Furthermore, the Expected Position Value under independent link formation is Next consider as in Example 2.4 that the intermediary has a position of formalised leadership in the institutional se ing and that a network is formable only if ∈ ( ). is is described by the probability distribution 2 given in Table 2. is results in different Expected Myerson and Position Values, computed from Table 4.
From these computations, we conclude that for all values of , ∈ [0, 1] it holds that Ψ ( , 1 ) Ψ ( , 2 ) for every player ∈ { , , }.Hence, the improved wealth generation situation due to institutional constraints on network formation translates to increased Expected Myerson payoffs to all constituting players in this intermediated trade situation.To elaborate on this, we compare the Expected Position Values of the buyer and seller under two network formation rules.In particular, Ψ ( , only if (4 − 3) ( − 1) 2 > 0, which is the case if and only if 3 4 < 1 and 0 < < 1.

Axiomatizing the Expected Myerson Value
e Myerson Value on the class of network games can be fully characterized using standard axioms, in particular component balance and one other property, usually a fairness property-also denoted as the equal bargaining power property (Jackson and Wolinsky, 1996)-and the balanced contributions property (Slikker, 2007).In this section we discuss similar characterizations of the Expected Myerson Value on the class of variable network games.
Equal Bargaining Power One of the main properties investigated in the literature since its inception by Myerson (1977) is that of "fairness" in the allocation of wealth generated in a network.
is refers to the idea that the removal of a single link from a network would affect both of its constituting players in equal measure. is has been referred to by Jackson and Wolinsky (1996) as "equal bargaining power", which terminology we adopt here as well.
Definition 4.1 Let Ψ : V × P → R be an allocation rule on the class of variable network games.
en Ψ satisfies the equal bargaining power property if for every variable network game ( , ) ∈ V × P and every link ∈ ( ) in 's extent it holds that e equal bargaining power property states that the removal of a single link affects the expected payoff to each of its constituting players equally.Note that the removal of a link can affect a player's payoff in a negative or a positive fashion.e formalisation of this property on the class of variable network games provides us with our first axiomatization of the Expected Myerson Value.
Axiomatization I e Expected Myerson Value Ψ is the unique allocation rule on the class of component additive variable network games that satisfies component balance as well as the equal bargaining power property.
For a proof of this axiomatization we refer to Section 4.1.

Balanced Contributions Slikker (2007) considered a different axiomatization of the Myerson
Value on the class of network games.His approach is founded on the consideration of the effects of the removal of players from a network on the allocated values.Slikker proved an axiomatization based on the property that the effects of the removal of players are equalised.is is formalised on the class of variable network games as follows.
Definition 4.2 Let Ψ : V × P → R be an allocation rule on the class of variable network games.
en Ψ satisfies the balanced contributions property if for every variable network game ( , ) ∈ V × P and all players , ∈ with ≠ it holds that Next we extend Slikker's axiomatization to the class of component additive variable network games founded on this definition of the balanced contributions property.is is stated as our second axiomatization of the Expected Myerson Value.
Axiomatization II e Expected Myerson Value Ψ is the unique allocation rule on the class of component additive variable network games that satisfies component balance as well as the balanced contributions property.

An axiomatization of the Expected Position Value
e balanced link contributions property implies that the sum of changes to 's payoff when 's links are removed one at a time probabilistically from the extent of is the same as the sum of changes to 's payoffs when 's links are removed probabilistically one at a time.
Next we extend Slikker's axiomatization (Slikker, 2007) to the class of variable network games founded on this definition of the balanced link contributions property.
Axiomatization III e Expected Position Value Ψ is the unique allocation rule on the class of component additive variable network games that satisfies component balance as well as the balanced link contributions property.

Proof of Axiomatization III
We proceed with a proof of Axiomatization III by first showing that the Expected Position Value indeed satisfies the properties of component balance and balanced link contributions property.Second, we show that it is actually the unique allocation rule that satisfies these two properties.

Ψ satisfies component balance
is transforms a communication situation into a link game in which communication links act as players.e Shapley value of the link game now assigns fair values to all links in the network based on the generated wealth.e 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.
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. is 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 o en using multiple airlines who have code sharing agreements with one another.e passenger pays an up-front fee which is divided among the relevant code sharing airlines in some fashion.But given the competition among airlines, these links are o en unstable with airlines terminating existing agreements and forming new agreements.Further, the independence assumption of link formation seems unrealistic in this se ing 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. is 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.ey extend and characterize the Myerson Value to this more general se ing.Ghintran et al. (2012) define and characterize the Position Value in this framework of generalized probabilistic communication situations.
Figure 1: Network of an intermediated buyer-seller situation − ( , − ) .e properties listed above have been used to characterise the most common allocation rules on the class of network games.e Myerson Value for network games Jackson and Wolinsky (1996) formulated the Myerson Value as an allocation rule on the class of network games that is 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 : V × G → R defined by ( (1996) show-as an extension of the main result of Myerson (1977) for communication situations-that the Myerson Value is the unique allocation rule on the class of network games V × G that satisfies component balance and the equal bargaining power property.Furthermore, Slikker (2007) shows that the Myerson Value is the unique allocation rule on the class of network games V × G that satisfies component balance and the balanced contributions property.Jackson and Wolinsky (1996) also showed that the Myerson Value on the class of network games V × G is the Shapley Value of an associated cooperative game ˆ ( , ) : 2 → R defined for every network game ∈ V and network ∈ G defined by ˆ ( , ) ( ) = ( | ) for any coalition ⊆ .Hence, ( , ) = ( ˆ ( , ) ). e Position Value for network games Slikker (2007) introduced the Position Value as an allocation rule on the class of network games by extending the earlier definition of Meessen 11) Slikker (2007) shows that the Position Value is the unique allocation rule on the class of network games V × G that satisfies component balance and the balanced link contributions property.is insight allows a complete comparison between the Myerson and Position Values for network games.

Definition 3. 8 e
Expected Myerson Value is the allocation rule Ψ : V × P → R defined as the standard extension Ψ = Ψ of the Myerson Value on the class of network games to the class of variable network games.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. is introduces the Expected Position Value.Definition 3.9 e Expected Position Value is the allocation rule Ψ : V × P → R defined as the standard extension Ψ = Ψ 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 e 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 3.10 Again consider the intermediated trade situation with player set = { , , }, described in Example 2.3 and Figure 1. e generated wealth in this bilateral trade situation can be formulated as a network game : G → R given by ( ) = 1 if and only if ∈ and/or { , } ⊆ , and ( ) = 0 otherwise.

Furthermore
, regarding the Expected Position Value we compute Ψ ( , 2 ) = Ψ ( , 2 ) = Contrary to the Expected Myerson Value, for the Expected Position Value there is no unequivocal improvement of the expected payoffs under the introduced institutional restrictions on network formation.Indeed, the intermediary 's Expected Position Value increases, but whether the Expected Position Value increases of and , depends on the exact probabilities and .
Next we consider axiomatizing the Expected Position Value along the lines of the axiomatization developed by Slikker (2007) for network games.With regard to axiomatizations of the Position Value for probabilistic se ings, we note that Ghintran et al. (2012) has done this for probabilistic communication situations and that Borkotokey et al. (2021) develop an axiomatization for probabilistic network games.Here we pursue an extension of these axiomatizations to the class of variable network games.Balanced Link Contributions We first extend the balanced link contributions property to the class of variable network games and, subsequently, extend Slikker's axiomatization founded on this property to variable network games.We can quite straightforwardly extend the balanced link contributions property from the class of network games to the class of variable network games.Definition 5.1 Let Ψ : V × P → R be an allocation rule on the class of variable network games.en Ψ satisfies the balanced link contributions property if for every variable network game ( , ) ∈ V × P and all players , ∈ with ≠ , it holds that ∈ ( ( )) Ψ ( , ) − Ψ , − = ∈ ( ( )) Ψ ( , ) − Ψ , − .
We refer toSlikker (2007, eorem 3.1)  for the fact that the Position Value satisfies component balance on the class of network games V .Hence, by Proposition 3.7, it immediately follows that the Expected Position Value Ψ = Ψ is component balanced on the expanded class of variable network games V × P .Ψ satisfies the balanced link contributions property Let ( , ) ∈ V × P be a variable network game.Furthermore, let , ∈ with ≠ .en that ( − ) = ( ) − .Furthermore, by Proposition 2.6(a), if ∈ , then − ( ) = 0. On the other hand, if ∉ , it holds that − ( ) = ( ) + ( + ).Hence, ( ) − − ( = − ( + ).erefore, Note that { ⊆ ( )| ∈ } = { + | ⊆ ( ) and ∉ }. is allows us to conclude that allocation of the expected wealth that is created in the given variable network game. is 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. is is referred to as the Expected Myerson Value and the Expected Position Value in the context of our se ing.).e 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.{1, 2, . . ., } be a fixed, finite set of players.With a slight abuse of notation, and forming and of ∈ [0, 1] of forming.However, now networks can only be formed if is connected, which probability is given by (2 − ). is results in conditional probabilities of network formation in relation to the probabilities reported in Example 2.3.eseconditionalprobabilities are given in Table2below.

Table 2 :
Network formation probabilities in Example 2.4.

Table 3 :
e Myerson and Position Values for Example 2.3.From this we compute the Expected Myerson Value under independent link formation as