Social networks provide a platform for the flow of information because social connections supply individuals with novel ideas, trade opportunities or job vacancies. This suggests that individuals’ payoffs are highly dependent on their position in the social network.
An important line of work in social network research analyzes this dependence and argues that individuals benefit when they serve as intermediaries or “bridges” between otherwise unconnected groups. The reasons for the success of bridge-individuals have been advanced by Burt’s theory of structural holes: bridge-agents can control the flow of information between different groups in the network, adapting it to specific strategic interests. Moreover, they are well-positioned to synthesize ideas coming from different groups, and this enhances their creative capacity [1
]. The works in [2
] provide empirical evidence that people who bridge structural holes in social networks have significantly higher payoffs. From this perspective, there is a natural strategic aspect when selecting personal and business contacts, especially in settings where brokerage and information access and control incentives can reasonably be expected to dominate behavior with respect to tie formation, as in a firm environment.
The analysis in [1
] constitutes a basic empirical background of the theoretical analysis developed here. Burt’s paper focused on data describing 673 managers who ran the supply chain in 2001 for one of America’s largest electronic companies. The study shows that there are clusters of managers within business units. To make the clusters more apparent, Burt looked at the top 89 senior people to see the core of the supply-chain network, and this provided a stark illustration of the fragile contact across business units. Burt’s results show that compensation, positive performance evaluations, promotions and good ideas are disproportionately in the hands of managers bridging structural holes.
The work in [5
] speculated that this situation could not be sustained in equilibrium if more and more people strategically add and remove ties to gain intermediation rents and to circumvent others who are trying to become intermediary. The work in [6
] formally confirmed that speculation in a setting where agents try to minimize their network constraint, a measure of structural disadvantage developed in [7
]. On the contrary, economic theory has demonstrated that social structures can equilibrate toward highly asymmetric networks with high differential payoffs, even when all agents have complete information and perfectly identical preferences and abilities to strategically modify the network. For example, [8
] or [9
] identify, as a prominent equilibrium network, the “star”, a structure in which a unique agent brokers everyone else. The analysis in [10
] also shows in a symmetric setting that individuals can differentiate and receive very different payoffs in equilibrium. All of these papers provide a game-theoretic rationale to contradict Burt’s speculation about the impossibility to sustain structural advantages in the long run.
In the model presented here, agents are assumed to be exogenously grouped into clusters, replicating the departments of an organization; namely a firm, a university, a family or a criminal group, among others. This connects this paper to [11
], in which a network is endogenously built onto a pre-existing structure, or to [12
], where agents are part of a coalition structure apart from being connected to others through a network. The members of a cluster are assumed to be fully linked among them, forming a clique, a notion widely used in network theory. These cliques have possibly different sizes. Agents face the trade-off between the benefits from connecting to other departments’ members and the cost, in the effort, to maintain those links. These benefits are determined by the payoff function introduced by [9
]: every pair of (directly or indirectly) linked agents creates a unit of surplus that is evenly split across the two involved agents and, possibly, the intermediaries. Thus, individuals form links with other departments’ members to create surplus, to gain intermediation rents and to circumvent others who are trying to become intermediary. Unlike [10
], links can only be created by mutual consent of the two involved agents.
The main results of this paper show that, in this framework, bridge-agents cannot be sustained in equilibrium under the same conditions of [9
]. This is because having groups of completely connected agents multiplies the possibilities of circumventing bridge-agents among different departments. In the setting of the present paper, allowing the same deviation possibilities of [9
] is shown to be incompatible with the existence of bridge-agents in equilibrium. However, if possible deviations are restricted, bridge-agents can be sustained, but only if they connect sufficiently small parts of the population to the rest. In this case, bridge-agents can obtain large payoff differentials in equilibrium as illustrated by the examples of Section 3
. These results formalize Burt’s conjectures about the impossibility to sustain bridge-agents in a firm environment: they cannot connect two big parts of the firm and cannot exist under wide deviation possibilities. Results might also be interpreted as a departure from the sharp results in [9
] because they contribute specific conditions to sustain bridge-managers in the equilibrium network of connections of a population.
2. The Model
The basic setting is borrowed from [9
], except for the group structure of the population. Let
be the finite set of individuals. These individuals are exogenously distributed across different groups or departments. Let
be the finite set of departments of the population, where
. Each individual is located at exactly one department. Let
be the department of individual i
, and let
denote the size of this department. Let
Agents are connected by a graph or network g, a collection of direct links that represent pairwise relations between the respective two individuals, with the particularity that all members of the same department are fixedly connected to each other, forming a clique. The set of all of these possible networks in N is G. The subset of N containing two individuals i and j is denoted by and is referred to as the link . Individuals i and j are connected if and only if . Thus, if , then , by assumption. When i and j belong to different departments, then link is said to be external. A path in g connecting and is a set of distinct nodes such that . All individuals with whom i has a path constitute the component of i in g, which is denoted by . Two departments and are connected if and only if there is a pair of individuals and such that .
Individuals play a network-formation game where the strategy of a player consists of making an announcement of intended external links. Let be the strategy vector of player i, which has elements. Let be a particular element of this vector, where means that player intends to form a link with player (where ), while means the opposite. A link between two individuals is undirected, can be severed by one of them unilaterally, but can only be created by mutual consent of the two implied individuals. Formally, a link between i and j is formed if and only if . Notice that a strategy profile induces a unique network .
As in [9
], the payoff function is such that any pair of connected players (i
) generates one unit of surplus. The distribution of this unit depends on the intermediaries between i
and on the nature of competition between intermediaries. It is assumed that any two paths between any two players fully compete away the entire surplus (à la Bertrand competition). Therefore, an intermediary between i
) can retain part of the surplus generated by i
if and only if this intermediary lies on all paths connecting i
. In such a case, k
is said to be an essential agent for i
. For example, in a star network where a unique agent i
is connected to any other agent, whereas others do not hold any additional link, agent i
is essential since no pair of players can ever avoid her/him on any path connecting them. Throughout the paper, essential players will also be called bridge-agents. Let c
be the cost of an external link. Let
be the set of essential agents in g
, and let
. Then, for every strategy profile s
, net payoffs to player i
are given by:
is an indicator function specifying whether i
is essential for j
denotes the number of external links of i
. The first term represents i
’s access payoffs while the second term represents her/his intermediation payoffs. A network
is efficient if
Definition 1. Agent is central if and only if:
In other words, x is central if no member of receives more access payoffs than x. For example, in a star network, the agent connected to all others is the unique central agent of the network. On the other hand, in a cycle network where all agents are connected to two other agents forming a circle, all agents are central.
At this point, I present some graph-theoretic notions that will be used repeatedly throughout the paper. A link is said to be critical if it defines the unique path between the two players involved and whose deletion increases the number of components. If all individuals belong to the same component, the population is said to be connected. An isolated department does not have external links. A network without external links is said to be empty. If a component contains an essential player i, then the rest of the members of can be distributed among two or more i-groups; are members of different i-groups if i is essential for connecting them. A department is essential if there is a pair of departments and such that every path that links some member of to some member of contains a member of (not necessarily the same). Notice that essential agents can be members of both essential and non-essential departments. A non-essential department can be extreme or not. Department is extreme if it is only connected to another department. Otherwise, is said to be non-extreme. If a component contains an essential department , then the rest of the departments of can be distributed among two or more -groups: are members of different -groups if all paths connecting j and k include some member of . Finally, a group of p departments constitutes a cycle if they can be ordered in a list such that is connected to and is connected to for and there are no other external links.
Given that link creation requires mutual consent of the two players involved and that agents can announce any combination of links they wish (multidimensional strategy space), a coordination problem arises. As such, the game displays a multiplicity of Nash equilibria where mutually-beneficial links can be left aside.1
This is solved if players are allowed to coordinate bilaterally. For this reason, refinements on the Nash equilibrium that allow for coalitional moves are usually applied to this kind of network formation game. One of the most widely-used refinements is the pairwise-Nash equilibrium, created by [13
], which is defined as follows:
Definition 2. A strategy profile is a Pairwise-Nash Equilibrium (PNE) if the following conditions hold:
for any and every ,
for any pair of players and every strategy pair in which ,
Networks generated by a PNE strategy profile are robust to deviations of unilateral multilink severance (that is, the usual Nash Equilibrium requirement) and to deviations of bilateral commonly-agreed one-link creation. That is, a PNE network is a Nash equilibrium network where, in addition, no mutually-beneficial link can be formed.
Alternative equilibrium notions include unilateral stability concepts as in [6
] or [10
] and concepts that allow for other coalitional moves.2
The Bilateral Equilibrium (BE) concept used in [9
] deserves special attention. The bilateral equilibrium concept is defined as follows:
Definition 3. A strategy profile is a Bilateral Equilibrium (BE) if the following conditions hold:
for any and every ,
for any pair of players and every strategy pair ,
A BE network must be robust to bilateral commonly-agreed one-link creation, to unilateral multilink severance and to deviations consisting of a simultaneous combination of the previous two deviations by any given pair of individuals. In particular, Condition (ii) implies that a BE network should be robust to any possible coalitional deviation that involves two agents. Thus, under the BE concept, agents can use more complex deviations than under the PNE concept. Proposition 1 shows that, unlike [9
], bridge-agents cannot be sustained in a BE network in the present setting. However, if the deviation possibilities are restricted, so that either only one individual considers deleting her/his links or two individuals consider creating a link between them at a time, equilibrium networks can display bridge-agents enjoying large payoff differentials.
First, the set of efficient networks is characterized. It is trivial to see that a network is efficient if and only if it is empty or minimally connected, i.e., all departments are grouped into the same component, and there are external links. As a consequence, notice that in the present setting, all agents with external links in an efficient network are essential. Therefore, efficient networks other than the empty network display multiple essential agents.
Now, the study turns to the equilibrium analysis. The following two results narrow the set of PNE networks and, since BE is stricter than PNE, also the set of BE networks. All proofs are relegated to the Appendix A
Lemma 1. A PNE network cannot include more than one multi-department component . Moreover, an isolated department can coexist with only if:
non-essential agents do not have external links and
where i is a central member of .
The proof shows that the marginal access and intermediation payoffs for those who create a critical link between two multi-department components always exceed the payoffs of (at least) one member of these components, so if this member does not have incentives to cut all of his/her links off, then creating a critical link should be a profitable deviation. Similarly, the proof of Part (i) shows that the marginal gross intermediation and access payoffs obtained by a pair of agents who create a critical link between them exceed the actual gross payoff of a non-essential agent with some external link, so if the linking cost is sufficiently low to sustain this non-essential agent in equilibrium, then a pair of members of different components will have incentives to create a link between them. Finally, Part (ii) states that in order to sustain two separate components in equilibrium, the linking cost should exceed the gross marginal payoff of the most profitable creation of a critical link between them. This result implies that there can be three different equilibria: (a) a connected network; (b) an empty network or (c) a network with a unique multi-department component and sufficiently small isolated departments.3
The last equilibrium is inefficient, whereas the last two equilibria reflect a coordination problem and can only be sustained for sufficiently high values of c
In order to analyze the conditions under which bridge-agents can be sustained in equilibrium, the analysis should focus on the unique multi-department component.
Lemma 2. Given a sufficiently large multi-department component,5 a PNE network cannot display more than one essential agent for any c.
If two agents create a link circumventing (at least) one essential agent, then each of them will eliminate (at least) one intermediary to access a specific part of the population. Thus, only when the size of some of these two parts is lower than an upper bound (that depends on c), this link will not be formed. The proof shows that if there are two or more essential agents in the same component, then this upper bound will be exceeded for some pair of agents; so, there always exist two individuals who can profitably circumvent some essential agent. Furthermore, a unique essential agent can be sustained in a large multi-department component and only if he/she connects sufficiently small parts of the component to the rest.
Since a multi-department component with (at most) one essential agent cannot be minimally connected, the previous result implies that if the population is large, then PNE (and also BE) networks will not be efficient.6
Moreover, the previous lemma implies that some non-essential agent will hold external links in this multi-department component. Thus, from Lemma 1, the next directly follows:
Given a sufficiently large multi-department component, a PNE network should be connected for any c.
As commented above, this list of necessary conditions stated by Lemmas 1 and 2 restricts both the set of potential PNE and BE networks. Based on these results, the next proposition follows:
Given a sufficiently large multi-department component, a BE network cannot include any essential agent.
From Lemma 2, a PNE network cannot have more than one essential agent. This will also be true for BE networks. The proof focuses on this unique essential agent to show that, under the BE concept, agents have the deviation possibilities they need to circumvent it. In particular, a pair of deviators is allowed to coordinate the replacement of a set of their links with a new link between them. In our setting, this allows deviators to circumvent essential agents without any extra cost. Consequently, bridge-agents cannot be sustained in BE networks.
This result contrasts with [9
] where the unique (strict) BE network displays one essential agent. The existence of multi-personal nodes, as the departments of our setting, has a double effect: on the one hand, this potentially increases the benefits from holding links to other nodes, but on the other hand, this may also smooth payoff differentials because there are more possibilities to circumvent bridge-agents. For this reason, only when these possibilities are restricted with respect to [9
], by using a weaker equilibrium concept as is done next, large payoff differentials can be sustained in equilibrium.
In spite of the list of necessary conditions stated by Lemmas 1 and 2, the set on PNE networks is large. However, the next result shows that under certain network structures, bridge-agents must necessarily exist in equilibrium.
Given a sufficiently large multi-department component with an essential department and an -group containing t agents, there exists a such that, for any , a PNE network must display a unique essential agent who connects those t agents to the rest of the component.
In other words, if an essential department has an -group relatively small with respect to c in a sufficiently large component, then a unique agent j will be essential for connecting the members of this -group to the rest of the component. The proof shows that j must have two external links to this -group, say and , and that there is no other way to connect this -group to . Intuitively, there are several factors that explain this result. First, agent k (or h) does not have incentives to cut (or ) off because this would involve an additional essential agent to access the rest of the component, which is sufficiently large to make this deviation nonprofitable. Similarly, agent j does not have incentives to cut some of her/his external links off because this would imply a loss of intermediation payoffs that exceeds the cost saving. Finally, no other agent has incentives to circumvent agent j to access this -group because this group is too small to compensate the cost of a new link. Thus, the smaller the -group, the smaller will be , so the essential department will have to include one essential agent for a wider range of c.
In general, a multi-department component without essential agents can present multiple structures in equilibrium. However, the previous result narrows the possibilities: a multi-department component without essential agents can be sustained when (1) it does not include essential departments, i.e., this component is a cycle or a group of cycles; or (2) all -groups are sufficiently large, for any essential department in the component.
The next result follows from Proposition 2 and Lemma 2.
Corollary 2. Given a sufficiently large multi-department component, a PNE network can only include one essential department with some -group sufficiently small.7 Moreover, if there are multiple -groups sufficiently small, then they must be connected to through the same essential agent, .
Moreover, a sufficiently small -group cannot include an essential department itself because, by Proposition 2, this would imply a new essential agent, which contradicts Lemma 2.
The following example illustrates a PNE network with one essential department.
Example 1. Consider a connected network with a unique essential player i who has two external links to every other department. Moreover, there are no additional links. Let m be the number of departments. For simplicity, it is assumed that all departments have the same size s. I claim that such a network is a PNE if m is sufficiently large and:
so that c is sufficiently high to avoid an additional link and sufficiently low to avoid the deletion of an external link.
In this case, player i’s payoffs are positive and equal to:
Player i’s marginal payoff from cutting one link off is:
which is negative for a sufficiently high m. Likewise, it can be concluded that i’s marginal payoff for cutting two links to a department is also negative. On the other hand, if a player in a peripheral department deletes one external link, then she/he obtains a marginal payoff equal to:which is negative for sufficiently large m. The creation of an additional link generates the following marginal payoff for one of the deviators:which is negative given the conditions stated in this example. Therefore, it can be concluded that this network is PNE under those conditions of m and c.
Moreover, by Proposition 2, this is the unique form of connecting all of the peripheral departments to the essential one when .
Notice that for moderate values of c, agent i will enjoy large payoff differentials with respect to others because of the large intermediation payoffs she/he receives.
The previous example illustrates that bridge-agents not only can be sustained in PNE networks, but also that they must exist, under certain configurations of the network. Moreover, the example shows that these agents can enjoy much larger payoffs than others in equilibrium.
Essential agents can also be members of non-essential departments if the following necessary conditions hold.
An extreme department can contain an essential agent only if . A PNE network can display an essential agent in a non-essential department only if .
Therefore, an essential agent can monopolize the intermediation between a group of agents and the rest of the component only if this group is sufficiently small. The previous result establishes specific size upper-bounds for non-essential departments. Below these upper-bounds, no other agent has incentives to circumvent the essential agent to access such a small group because the additional access payoffs do not compensate the cost of the new link. Nevertheless, the essential agent enjoys large payoff differentials with respect to the rest of the agents, as the next example illustrates.
Consider a network consisting of a cycle of m departments. Only one of them, say , has a single player i with external links. All other departments have at least two players with external links. In that case, agent i is essential for connecting the rest of the members of her/his department to the rest of the component. This network is a PNE if m is sufficiently large and .
For any given c, it is easy to see that the marginal payoff for deleting one external link will be negative for a sufficiently large m. On the other hand, the most profitable possibility for creating a new link, i.e., adding a link circumventing the essential player i, generates a marginal payoff to one of the deviators equal to:which is negative under the initial conditions stated above. Thus, the network is a PNE. Notice that the payoff of i is:
whereas the payoff of any agent not in without external links, say j, is:
Therefore, for moderate values of c, the payoff differential between these two players is high.