Game theoretic foundations of the Gately power measure for directed networks

We introduce a new network centrality measure founded on the Gately value for cooperative games with transferable utilities. A directed network is interpreted as representing control or authority relations between players--constituting a hierarchical network. The power distribution of a hierarchical network can be represented through a TU-game. We investigate the properties of this TU-representation and investigate the Gately value of the TU-representation resulting in the Gately power measure. We establish when the Gately measure is a Core power gauge, investigate the relationship of the Gately with the $\beta$-measure, and construct an axiomatisation of the Gately measure.

1 Introduction e concept of network centrality has emerged from sociology, social network analysis and network science (Newman, 2010;Barabási, 2016) into the field of cooperative game theory, giving rise to game theoretic methods to measure the most important and dominant nodes in a hierarchical social network (Redhead and Power, 2022).
e underlying method is to construct a cooperative game theoretic representation of characterising features of a network and to apply cooperative game theoretic analysis to create centrality measures for these networks. 1  We limit ourselves to directed networks as representations of collections of hierarchical or control relationships between the constituting players in a network.Such hierarchical relationships can be found in employment dynamics between managers and subordinates, the interaction between a professor and her students, rivalry between different sports teams based on past performance between them, or the connections between a government agent and the individuals they oversee.
We refer to these relationships as "hierarchical" since it implies that the predecessor node has some level of control or authority over the successor node. 2 Taking this interpretation central, we refer to these directed networks as hierarchical.
A hierarchical relationship is between a predecessor and a successor, where the predecessor exercises some form of control or authority over the successor.e most natural representation is that through a TU-game that assigns to every group of players their "total number of successors", which can be interpreted in various ways.We consider the two standard ways: Simply counting the successors of all members of the group, i.e., the number of players that have at least one predecessor that is member of the group; or, counting the number of players for which all predecessors are member of that group. 3We show that these two TU-representations are dual games.We remark that van den Brink and Borm (2002) already characterised the main "strong" successor representation as a convex game (Shapley, 1971), implying that its dual "weak" successor representation is a concave game.
A power gauge for a network is now introduced as a vector of weights that are assigned to players in the hierarchical network that represent or measure each player's authority in that network.
A power measure is now introduced as a map that assigns to every hierarchical network a single power gauge.In this paper we investigate some power measures that assign such gauges founded on game theoretic principles related to the two TU-representations of hierarchical networks considered here.In particular, each simple hierarchical network-in which each player has at most one predecessor-has a natural power gauge in the form of the outdegree of each node in the network, representing the number of successors of a player in that network.
Application of the Shapley value (Shapley, 1953) to the successor representations results in 1 We refer to Tarkowski et al. (2018) for an overview of the literature on cooperative game theoretic constructions of centrality measures in directed networks.
2 is authority can also be psychological and be influential on reputational features in the relationship.An example might be the relationship between two chess players.One of these players can have a psychological advantage over the other based on outcomes of past games between the two players and/or the Elo rating differential between the two players.
3 As Tarkowski et al. ( 2018) point out, the successor representations are only one type of representation of a hierarchical network.More advanced TU-representations have been pursued by Gavilán et al. (2023).
the -power measure (van den Brink and Gilles, 1994). is is the centre of the set of Core power gauges for each network.In the -measure the weight of a player is equally divided among its predecessors.As such it has a purely individualistic foundation to measuring power.
Applying the Gately value of the successor representations results in a fundamentally different conception of a power measure.Here, the set of dominated nodes is treated as a collective resource that is distributed according to a chosen principle.In the Gately measure this is the proportional distribution.4 is stands in contrast to the individualistic perspective of the -measure.
Since, the Gately measure is founded on such different principles, it is not a surprise that the assigned Gately power gauges are not necessarily Core power gauges.We identify conditions under which the assigned Gately power gauge is a Core power gauge in eorem 3.3.In particular, we show that for the class of (weakly) regular hierarchical networks, the Gately power measure assigns a Core power gauge for that network.
We are able to devise an axiomatic characterisation of the Gately value as the unique power measure that satisfies three properties.First, it is normalised to the number of nodes that have predecessors, which is satisfied by many other power measures as well.Second, it satisfies "normality" which imposes that a power measure assigns the full weight of controlling successors with no other predecessors and the power measure of the reduced network with only those nodes that have multiple predecessors.Finally, it satisfies a proportionality property in the sense that the power measure assigned is proportional to how many successors a node has.
Finally, we address the question under which conditions the -and Gately power measures are equivalent.We show that for the class of weakly regular hierarchical networks this equivalence holds.
is is exactly the class of networks for which the Gately measure assigns a Core power gauge.is insight cannot be reversed, since there are non-regular networks for which the Gately and -measures are equivalent.
Relationship to the literature e study of centrality in networks has evolved to be a significant part of network science (Newman, 2010;Barabási, 2016).In economics and the social sciences there has been a focus on Bonacich centrality in social networks.is centrality measure is founded on the eigenvector of the adjacency matrix that represents the network (Bonacich, 1987).In economics this has been linked to performance indicators of network representations of economic interactions such as production networks (Ballester et al., 2006;Huremovic and Vega-Redondo, 2016;Allouch et al., 2021).e nature of these networks is that they are undirected and, therefore, fundamentally different from the hierarchical networks considered here.
Traditionally, the investigation of directed networks focussed on degree centrality-measuring direct dominance relationships (van den Brink and Gilles, 2003;van den Brink and Rusinowska, 2022)and on betweenness centrality, which considers the position of nodes in relation to membership of (critical) pathways in the directed network (Bavelas, 1948;White and Borga i, 1994;Newman, 2005;Arrigo et al., 2018).
Authority and control in networks has only more recently been investigated from different perspectives.Yang-Yu et al. (2012) considers an innovative perspective founded on control theory.
More prevalent is the study of centrality in hierarchical networks through the -measure and its close relatives.van den Brink and Gilles (1994)  Finally, with regard to the Gately value as a solution concept for TU-games, this conception was seminally introduced for some specific 3-player cost games by Gately (1974).generalised the scope of the Gately value and identified exact conditions under which this value is well-defined.is has further been developed by Gilles and Mallozzi (2023), which showed that the Gately value is always a Core imputation for 3-player games, devised an axiomatisation for the Gately value for arbitrary TU-games, and introduced a generalised Gately value founded on weighted propensities to disrupt.
Structure of the paper Section 2 discusses the foundations of the game theoretic approach that is pursued in this paper.It defines the successor representations and presents their main properties.
Furthermore, the standard solution concepts of the Core and the Shapley value are applied to these successor representations.Section 3 introduces the Gately measure, which represents a different philosophy of measuring the exercise of control and power in a network.We investigate when the Gately measure assigns a Core power gauge to a network and we devise an axiomatisation of the Gately measure.e paper concludes with a comprehensive comparison of the Gately and -measures, identifying exact conditions under which these two measures are equivalent.
2 Game theoretic representations of hierarchical networks In our study, we focus on networks with directed links, where each link has specifically the interpretation of being the representation of a hierarchical relationship.In a directed network, the direction of a link indicates that one node is positioned as a predecessor while the other node is considered a successor in that particular relationship.Here we interpret this explicitly as a control or authority relationship.erefore, we denote these networks as hierarchical throughout this paper.
In hierarchical networks, predecessors exercise some form of authority over its successors, allowing the assignment of that control to that particular node. is results in a natural game theoretic representation.We explore these game theoretic representations in this section and investigate the properties of these games.
Notation: Representing hierarchical networks Let = {1, . . ., } be a finite set of nodes, where ∈ N is the number of nodes considered.Usually we assume that 3. A hierarchical network on is a map : → 2 that assigns to every node ∈ a set of successors ( ) ⊆ \{ }.
We explicitly exclude that a node succeeds itself, i.e., ∉ ( ). e class of all directed networks on node set is denoted as Inversely, in a directed network ∈ D , for every node ∈ , the subset −1 ( ) = { ∈ | ∈ ( )} denotes the set of its predecessors in .Due to the general nature of the networks considered here, we remark that it might be the case that ( ) ∩ −1 ( ) ≠ , i.e., some nodes can be successors as well as predecessors of a node.
We introduce the following additional notation to count the number of successors and predecessors of a node in a network ∈ D : In particular, { , , } forms a partitioning of the node set .We introduce counters = # , = # and = # , leading to the conclusion that = + .
e constructed partitioning informs the analysis of the game theoretic representation of the hierarchical authority structure imposed by on the node set .Our analysis will show that for certain centrality measures, the class of nodes that have multiple predecessors plays a critical role in the determination of the assignment of a power index to these predecessors.
e partitioning of the node set based on the structure imposed by ∈ D allows further notation to be introduced for every node ∈ : , resulting in the conclusion that ( ) = ( ) + ( ).
• From the definitions above we conclude immediately that Classes of hierarchical networks e next definition introduces some normality properties on hierarchical networks that will be used for certain theorems.
Definition 2.1 Let ∈ D be some hierarchical network on node set .In a regular network, each node has either no predecessors, or a given fixed number of predecessors.
Hence, all nodes with predecessors have exactly the same number of predecessors.In a weakly regular network, each node has either no predecessors, or exactly one predecessors, or a given fixed number 2 of predecessors.
e notion of a simple network further strengthens the requirement of a regular network.It imposes that all nodes either have no predecessors, or exactly one predecessor.
Furthermore, van den Brink and Borm (2002) introduced the notion of a simple subnetwork of a given network ∈ D on the node set .We elaborate here on that definition.
Definition 2.2 Let ∈ D be a given hierarchical network on .
A network ∈ D is a simple subnetwork of if it satisfies the following two properties: (i) For every node ∈ : ( ) ⊆ ( ), and (ii) For every node ∈ : ( ) = 1.
e collection of a simple subnetworks of is denoted by S ( ).
e collection of simple subnetworks of a given network can be used to analyse the Core of the game theoretic representations of hierarchical games as shown below.It is easy to establish that a hierarchical network is simple if and only if S ( ) = { }.

Game theoretic representations of hierarchical networks
Using the notation introduced above, we are able to device cooperative game theoretic representations of hierarchical networks.We recall that a cooperative game with transferable utilities-or a TU-game-on the node set is a map : 2 → R such that ( ) = 0.A TU-game assigns to every group of nodes ⊆ a certain "worth" ( ) ∈ R. A group of nodes ⊆ is also denoted as a coalition of nodes, to use a more familiar terminology from cooperative game theory.
To embody the control or authority represented by a hierarchical network ∈ D on the node set as a cooperative game, we introduce some additional notation.For every group of nodes ⊆ we denote as the (weak) successors of coalition in .A node is a (weak) successor of a node group if at least one of its predecessors is a member of that group.

Similarly, we introduce
as the strong successors of coalition in .A node is a strong successor of a node group if all predecessors of that node are members of that group.Clearly, strong successors of a node group are completely controlled by the nodes in that particular group and full control can be exercised.
is compares to regular or weak successors of a node group over which the nodes in that group only exercise partial control.
e next definition introduces the two main cooperative game theoretic embodiments of this control over other nodes in a network.
Definition 2.3 Let ∈ D be some hierarchical network on node set .e successor representation is also known as the "successor game" in the literature and the strong successor representation as the "conservative successor game" on (Gilles, 2010).It is clear that the four TU-games introduced in Definition 2.3 embody different aspects of the control exercised over nodes in a given hierarchical network.In particular, these TU-games count the number of successors that are under the control of nodes in a selected coalition.
Properties of successor representations e next list collects some simple properties of these four games introduced here.
Proposition 2.4 Let ∈ D be some hierarchical network on node set .en the following properties hold regarding the successor representations , , and : (i) For every node ∈ : ( { } ) = ( ) and the worth of the whole node set is determined as (iii) For every coalition ⊆ : ( ) = ∈ ( ), implying that the partial successor representation is an additive game.
(vi) For every node ∈ : ( { } ) = ( ) and the worth of the whole node set is determined as ese properties follow straightforwardly from the definitions, therefore a proof is omi ed.
e next theorem collects some properties of the successor representations that have not been remarked explicitly in the literature on cooperative game theoretic approaches to representations of hierarchical networks.6eorem 2.5 Let ∈ D be some hierarchical network on node set .en the following properties hold for the successor representations and : (i) e strong successor representation is the dual of the successor representation in the sense that e strong successor representation is decomposable into unanimity games with (iii) e strong successor representation is a convex TU-game (Shapley, 1971) in the sense that (iv) e successor representation is concave in the sense that ( ) + ( ) ( ∪ ) + ( ∩ ) for all , ⊆ Proof.Let ∈ D be some hierarchical network on node set and let the TU-games and be as defined in Definition 2.3.
To show assertion (i), let ⊆ , then it holds that is shows that is indeed the dual game of .
Assertion (ii) is Lemma 2.2 in van den Brink and Borm (2002) and assertion (iii) follows immediately from (ii).Finally, assertion (iv) is implied by the fact that is the dual game of -following from assertion (i)-and is convex.
e duality between the successor representation and the strong successor representation implies that some cooperative game theoretic solution concepts result in exactly the same outcomes for both games.In particular, we refer to the Core, the Weber set, the Shapley value, and the Gately value of these successor representations as explored below.
2.2 Some standard solutions of the successor representations e cooperative game theoretic approach to measuring power or hierarchical centrality is based on the assignment of a quantified control gauge to every individual node in a given hierarchical network.A power or hierarchical centrality measure now refers to a rule or procedure that assigns to every node in any hierarchical network such a gauge.In this section we set out the foundations for this approach.
Definition 2.6 Let ∈ D be some hierarchical network.A power gauge for is a vector ∈ R + such that ∈ = .A power measure on D is a function e normalisation of a power gauge for a network ∈ D to the allocation of the total number of nodes in is a yardstick that is adopted in the literature, which we use here as well. is nor-malisation is in some sense arbitrary, but it allows a straightforward application of the cooperative game theoretic methodology as advocated here.
e game theoretic approach adopted here allows us to apply basic solution concepts to impose well-accepted properties on power gauges and power measures.e well-known notion of the Core of a TU-game imposes lower bounds on the power gauges in a given hierarchical network. is leads to the following notion.
Definition 2.7 A Core power gauge for a given hierarchical network ∈ D is a power gauge ∈ R + which satisfies that for every group of nodes ⊆ : e set of Core power gauges for is denoted by C ( ) ⊂ R + .
e Core requirements on a power gauge impose that every group of nodes is collectively assigned at least the number of nodes that it fully controls.is seems a rather natural requirement.e following insight investigates the structure of the set of Core power gauges for a hierarchical network.
Proposition 2.8 Let ∈ D be some hierarchical network on node set .en the following hold: (i) If is a simple hierarchical network, then there exists a unique Core power gauge, C ( ) = , where = ( ) for every node ∈ .
(ii) More generally, C ( ) is equal to the Weber set of , which is the convex hull of the unique Core power gauges of all simple subnetworks of given by C ( ) = Conv | ∈ S ( ) ≠ .
Furthermore, suppose that ∈ C ( ). en from ( ) = for every node ∈ and ∈ it immediately follows that = for all ∈ . is shows that is the unique Core power gauge for the simple hierarchical network , showing assertion (i).
Assertion (ii) follows immediately from eorem 4.2 in van den Brink and Borm (2002) in combination with assertion (i).
e -measure A well-established power measure for hierarchical networks was first introduced by van den Brink and Gilles (1994) and further developed in van den Brink and Gilles (2000) and van den Brink et al. (2008).is -measure is for every node ∈ defined by e following proposition collects the main insights from the literature on the -measure.
Proposition 2.9 Let ∈ D be a hierarchical network.en the following properties hold: (i) ( ) ∈ C ( ) is the geometric centre of the set of Core power gauges of .
ples than the -measure or other power measures.Now, the Gately power measure is member of a family of values that considers the control exercise over the nodes in to be a collective resource in any hierarchical network ∈ D .e control is then allocated according to some well-chosen principle.In particular, e Gately measure allocates the control over proportionally to the predecessor of the nodes in .Assuming ≠ , this proportional allocator is for every node ∈ with ( ) ∩ ≠ defined as where the Gately measure is now given by ( ) = ( ) + ( ) • . is compares, for example, to the allocation principle based on the egalitarian allocator of the power over the nodes in ≠ given by and the resulting Restricted Egalitarian power measure given by ( ) = ( ) + ( ) .We emphasise that the Gately and Restricted Egalitarian power measures are members of the same family of power measures for hierarchical networks, which have a collective allocative perspective on the control over the nodes in .

Properties of the Gately measure
We investigate the properties of the Gately measure from the cooperative game theoretic perspective developed in this paper.We first investigate whether the Gately measure assigns a Core power gauge as is the case for the -measure.Second, we consider some characterisations of the Gately measure.In particular, we derive an axiomatisation as well as investigate some interesting properties of the Gately measure on some special subclasses of hierarchical networks.
e Gately measure is not necessarily a Core power gauge We first establish that contrary to the property that the -measure is the geometric centre of the set of Core power gauges of a given hierarchical network, its Gately power gauge does not necessarily have to satisfy the Core constraints.
e next example provides a hierarchical network on a node set of 8 nodes which Gately measure is not a Core power gauge.
Clearly, we have established that in this case ( ) ≠ ( ) even though the network satisfies the condition of eorem 3.3(i), implying that the Gately measure is a Core power gauge.
A question remaining is whether the assertion of eorem 3.5 can be reversed, i.e., if ( ) = ( ) then has to be weakly regular.e answer to that is negative as the following example shows.
introduced the -measure as a natural measure of influence and considered some non-game theoretic characterisations.e -measure is closely related to the PageRank measure introduced by Brin and Page (1998) and considered throughout the literature on social network centrality measurement.e -measure has been linked to game theoretic measurement of centrality in directed networks by van den Brink and Gilles (2000) and van den Brink and Borm (2002).e -measure was identified as the Shapley value of the standard successor representations as TU-representations of domination and control in directed networks.van den Brink et al. (2008) develop this further through additional characterisations.Gavilán et al. (2023) introduce other, more advanced TUrepresentations of directed networks and study their Shapley values.ey consider a family of centrality measures resulting from this methodology.Gómez et al. (2003), del Pozo et al. (2011) and Skibski et al. (2018) introduce and explore a game theoretic methodology for measuring network power that is fundamentally different from the methodology used in this paper and the literature reviewed above.ese authors consider a well-chosen TU-game on a networked population of players and subsequently compare the allocated payoffs based on the Shapley value in the unrestricted game with the Shapley value of the network-restricted TU-game.e normalisation of the generated differences now exactly measure the network-positional effects on the players, which can be interpreted as a centrality measure.
is contribution inspired a further development of the underlying conception of "propensity to disrupt" byLi lechild and Vaidya (1976)  andCharnes et al. (1978), including the definition of several related solution concepts.Li lechild and Vaidya (1976) also developed an example of a 4-player TU-game in which the Gately value is not a Core imputation.More recently,Staudacher and Anwander (2019) → N counts the number of successors of a node defined by ( ) = # ( ) the number of predecessors of a node defined by ( ) = # −1 ( ) for ∈ ; e previous analysis leads to a natural partitioning of the node set into different classes based on the number of predecessors of the nodes in a given network ∈ D : weakly regular if for all nodes , ∈ : ( ) = ( ). e collection of weakly regular hierarchical networks is denoted by D ⊂ D .(b) e network is regular if for all nodes , ∈ : ( ) = ( ). e collection of regular hierarchical networks is denoted by D ⊂ D .(c) e network is simple if for every node ∈ : ( ) = 1.e collection of simple hierarchical networks is denoted by D ⊂ D .
of is the TU-game : 2 → N for every coalition ⊆ given by ( ) = # ( ), the number of successors of the coalition in the network .(b) We additionally introduce two partial successor representations as the two TU-games , : 2 → N, which for every coalition ⊆ are given by ( ) = # ( ) ∩ and ( ) = # ( ) ∩ .(c) e strong successor representation of is the TU-game : 2 → N for every coalition ⊆ given by ( ) = # * ( ), the number of strong successors of the coalition in the network .
Figure 1: e hierarchical network considered in Example 3.2.