Hierarchical Network Formation Games

01/14/2020 ∙ by Pedro Cisneros-Velarde, et al. ∙ The Regents of the University of California 0

We propose a novel network formation game that explains the emergence of various hierarchical structures in groups where self-interested or utility-maximizing individuals decide to establish or severe relationships of authority or collaboration among themselves. We consider two settings: we first consider individuals who do not seek the other party's consent when establishing a relationship and then individuals who do. For both settings, we formally relate the emerged hierarchical structures with the novel inclusion of well-motivated hierarchy promoting terms in the individuals' utility functions. We first analyze the game via a static analysis and characterize all the hierarchical structures that can be formed as its solutions. We then consider the game played dynamically under stochastic interactions among individuals implementing best-response dynamics and analyze the nature of the converged networks.

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1 Introduction

Hierarchies are ubiquitous forms of organization. Not surprisingly, early comparative studies between administrative and economic hierarchies, and other studies of hierarchies in the the context of political organization, date back to the beginning of the twentieth century [26, 27]. Since then, their study has encompassed a large variety of disciplines like, for example, labor and organizational economics [3, 30, 39], moral and evolutionary psychology [24, 29, 8, 37], anthropology [10] and sociology [5, 31]. We remark that in the economics literature, the importance of hierarchies also comes from the study of firms: the structure of firms can be understood as hierarchical structures in production organizations (e.g., superiors having veto power over subordinates to access productive assets) [7, 18]. In general, the empirical study of the emergence of hierarchical structures in social networks has benefited from advancements in the field of sociometry [32], and, theoretically, graph theoretical analysis has proven to be a useful tool for structural analysis of general hierarchical structures [25]. In this paper, we adopt a game theoretical perspective to study the emergence of hierarchical structures through what is known as network formation games. We propose a game and analyze how its play leads to the emergence of hierarchies. The importance of our approach stems from the fact that this game theoretical study allows us to investigate the underlying structures in the individual utilities that promote the emergence of hierarchies from self-interested individuals. Then, the importance of this work is that it defines micro-mechanisms on the individual level that can lead to the emergence of hierarchical structures on the macro-level.

To the best of our knowledge, there is no previous study about the emergence of hierarchical network structures from a network formation perspective. However, we briefly mention that there has been a considerable body of literature on the study of hierarchies from other game theoretical perspectives [34]. For example, there are studies in the literature about allocation rules that take into account explicit hierarchical relationships among the agents (e.g., [33]), and about the analysis of games where agents have veto power over the actions of other players as a result of an underlying hierarchical structure (e.g., [15]).

On the other hand, there is a mature body of work in the network formation game literature related to other applications. Generally speaking, in a network formation game, self-interested or utility-maximizing agents decide to form or severe links (interpreted as some sort of relationship) with other agents so that they can form network structures that maximize some utility or allocation value. The structure of connections resulting from playing the game induces an emergent network whose topology can have different interpretations according to the specific application the game is modeling. The literature on network formation games has a long trajectory which historically started from a cooperative perspective, and has later devised a huge boom from a non-cooperative perspective [19, 34]. These games have been analyzed via a static analysis, that is, the analysis of solution concepts that describes the emergence of particular “stable” networks as a result of the agents playing the game. Additionally, they have also been analyzed via a dynamic analysis, that is, the study of the emergence of a subset (or possibly all) of the networks defined by the static analysis (also known as equilibrium selection) as the result of some defined dynamic rules that describe the evolution of the agents’ actions [36, 38]. For example, [4] uses Nash equilibrium as solution concept, whereas [21] uses the solution concept called pairwise stability (first introduced in [22]); and both works establish a dynamical analysis of convergence towards “stable” networks in their respective solution concept. The two solution concepts mentioned above are the ones that have been typically used in applications where the efficiency of the formed network was the main focus of study; however, in general, solution concepts are proposed (and sometimes renamed) depending on the specific application to study from the network formation perspective (whether there is only a static analysis or a dynamic one too). For example, other solution concepts exist like the generalization of pairwise stability called strong stability also studied in the context of network efficiency [20], or like the solution concept introduced in [23] which is particularly tailored for the problem of routing traffic on networks.

Broadly, the literature on network formation games can be classified according to the nature of the agents that play the game and/or the nature of the connections they can establish. We discuss first the possible nature of the agents. Traditionally, a first distinction is whether the agents are

non-consensual or consensual. A non-consensual agent can establish links unilaterally with agent without ’s consent, i.e., without having to agree to accept such link (see, e.g., [4]); and a consensual is the one who needs ’s consent or approval to establish a link (see, e.g., [21, 16, 2]). Another second major distinction regarding the agents is whether they all have the same structure in their utility or payoff function, i.e., the rewards or costs of making/maintaining/severing a link is the same for any agent, which is know as the homogeneous case; or whether they have different structures in their utility function, which is the heterogeneous case [13].

Regarding the distinction on the nature of the connections that agents can establish, network formation games can be classified depending on whether the formed network is undirected (i.e., the edge or link between agents and has no direction) or directed (i.e., every edge or link has an intrinsic direction, so that an edge can be directed from to ). The interpretation is that an undirected edge between and gives a two-way benefit of flows, i.e., the utility generated by this edge benefits both and . On the contrary, a directed edge from to has a one-way flow of benefits since the utility generated by this edge benefits but may or may not benefit  [4, 13]. Examples of undirected relationships are friendships, collaborations (e.g., co-authorship), mutual insurance, etc. Examples of directed relationships arise naturally in communication networks (e.g., someone send an email to another person), loan relationships, subordinate-authority relationships, buyer-seller relationships, etc. We refer to [14, Chapter 5] for more discussion on the use of undirected vs directed graphs in game theoretical modeling.

We stress that not all network formation games in the literature have been studied from a dynamic analysis (indeed, all the classifications presented above only have to do with the static analysis aspect). A first classification from a dynamic analysis is to consider whether the agents, at any time-step of the (discrete) evolution of the game, are myopic (e.g., [4, 21, 2]), i.e., they want to only maximize their current utility at any time step, or, otherwise, foresighted (e.g., [11, 9, 35]), i.e., they also want to maximize their (expected cumulative) future utility. Myopic agents are a more reasonable assumption in contexts where agents live in a big and/or complex network, where it is virtually impossible for a single agent to completely predict how her actions would influence over the rest of the whole network and thus how current actions will affect future utility.

1.1 Contributions

Our first main contribution is to propose, to the best of our knowledge, a first study on the emergence of hierarchical structures from the perspective of network formation games. We propose a game which allows us to study and understand how self-interested or utility-maximizing individuals lead to the emergence of networks with hierarchical structures. In particular, our work shows this through the judicious inclusion of well-motivated hierarchical promoting terms in the utility functions of the agents. We assume all agents are homogeneous, i.e., that we have an egalitarian society, and we also assume that all individuals can be of two types: non-consensual or consensual. For each type, we perform a static analysis of the emergent network structures that satisfy the solution concept proposed for our game and which is closely related to other notions in the classic literature of network formation games. Then, we present a dynamic analysis where we analyze the dynamic formation of these networks as a result of a stochastic process of pairwise interactions among agents playing best-response dynamics.

Our second contribution is a complete static analysis where we show that network structures with different levels of hierarchy emerge as equilibrium networks, i.e., as solutions of the game, for any type of agents. In particular, individuals within the same level or rank of hierarchy in these structures can only have cooperative relationships. Moreover, we show the relationship between the structure or topology of the formed network and the different parameters in the utility function of the agents. For example, we study how the total number of individuals with a specific rank in the hierarchy is related to the utility parameters. Intuitively, the formation of hierarchies within homogeneous agents has the interesting interpretation that even in egalitarian societies, hierarchies seem to be a natural outcome of socio-economic relationships. Finally, we show that all possible equilibrium networks that can be formed by non-consensual agents in any game are a strict subset of the ones that are possible to be formed in the same game by consensual agents.

Our third contribution is to analyze our proposed network formation game when it is played dynamically starting from any fixed initial graph. In such dynamic setting, agents are stochastically chosen to perform an action and they play their best-response dynamics. We show that the networks, being formed dynamically, eventually converge to some equilibrium network in finite time. Finally, we show that for any type of agent, the topology of the converged hierarchical network, besides depending on the parameters of the utility functions, also depends on the particular realization of the underlying stochastic process (i.e., the solution path).

1.2 Notation and preliminary modeling

Let be the finite set of agents, and, throughout the paper, consider . Let . Any element of is a node of the directed graph (digraph) , and

is thus defined by a set of ordered pairs between different elements of

111Although, technically, a graph is defined by the tuple , i.e., the node and the edge sets, we simply refer to the graph by with the node set being understood by the context. An element is called an edge from to . Let and . We define and , with the in-degree of being . Given and , if () then is a root (sink).

Whenever there exists a sequence of edges that connects with with no edge appearing more than once and no node appearing more than in two edges, we say that there exists a path of length between and . We denote by the condensation graph of , and by the connected component of . We refer to Section A of the Appendix for basic graph theoretical terminology and concepts that will be used throughout the paper.

New notation and definitions

Given a directed acyclic graph and a node of , the level of , denoted by , is the largest number of arcs or edges from any root node of to . Also, for digraph , the level of denoted by , is the largest number of arcs in from any root of to the strongly connected component containing . See Figure 3(a).

We also define and .

Consider a graph and any . Given a node , we define and . We use the term undirected connected component whenever all edges of the connected component are undirected, and we call it complete connected component when additionally it is a complete subgraph.

Given an undirected connected component that contains nodes and , we say the undirected edge is critical if .

Finally, throughout the paper, we use the terms agent, node and individual interchangeably, as well as the terms network and graph, and the terms edge and link.

(a)
(b)
Figure 3: In (a), we number the levels of all nodes in a directed acyclic graph. In (b), given two agents and , we represent relationships of subordination in the upper figure (here, is ’s subordinate), and collaboration in the lower figure.

2 Proposed model and game

In our model, any single edge is a pairwise relationship that describes an unambiguous hierarchical or authority relationship where agent has a higher hierarchy or exercises some authority over ; so we say is a supervisor of , and, respectively, that agent is a subordinate of

. This simple concept has been used extensively in the game theory literature 

[14]. On the other hand, whenever and have an undirected edge, we understand that they have a collaborative or cooperative relationship, and thus, they must share the same hierarchical status and no agent is the subordinate of the other one. See Figure 3(b).

2.1 Utility definition

We first specify the costs and rewards that any agent incurs according to our proposed model.

  1. Subordinate-collaborative reward: From an organizational perspective, this represents the positive reward an individual obtains by being the subordinate or collaborator of another competent agent. For example, consider agent wants to pursue her own goals (e.g., as an employee), then she will seek direction and help from perhaps an equally or better skilled person (e.g., a competent boss or co-worker). This reward can also be interpreted as an exogenous reward (e.g., a salary) that an agent receives by working for or with other people inside an organization. We represent the subordinate-collaborative reward per link by the symbol .

  2. Hierarchical reward: Individuals who have a higher hierarchical status or position receive an inherent reward for being in a higher position of power. For example, from an organizational perspective, individuals with higher management or directive positions have skill sets that are highly specialized to maintain such high hierarchical positions, and, as a consequence, they are (mostly) better paid in the job market. Moreover, in political organizations and public institutions, individuals with higher hierarchical positions can exert more influence than those in lower positions (e.g., leaders or directives have more influence over ordinary individuals in the society or over their political base). Then, the reward for these individuals comes from the strength of their influence and the reach of the consequences of their decisions. Mathematically, it is natural that if and both appear in some cycle, then agent has no authority or any hierarchical power over , and so must have the same hierarchical reward as . Therefore, two agents in the same strongly connected component have the same reward. In summary, we model the hierarchical reward for agent in the graph as a (strictly) monotonic function which takes as an argument the level of . Formally, the hierarchical reward for an agent situated in the network is

    (1)

    We call the reward function, and we assume it is non-negative and (strictly) monotonic. Intuitively, the monotonic property of means that individuals with higher positions of status are better rewarded.

  3. Management cost: An individual who is in charge of a group of people incurs on a management cost directly proportional to the number of her subordinates. There are two possible reasons for this. First, leading or directing more subordinates requires more managerial or logistic work, thus demanding more energy than doing it with a smaller group, since has to respond to the demands of more people. Naturally, people who collaborate with (i.e., any such that ) do not incur a management cost to . Second, from an organizational perspective, a manager is responsible for the group she leads, and she also needs to report to her immediate superiors (i.e., the people, if any, to whom she is a subordinate) about how the people she is responsible for are performing. This reporting task, obviously, requires more work when is directing larger groups. Observe that management costs must be reduced if is part of a team of people who also manage or direct her same subordinates since this team will collaborate with her and thus reduce the workload on . Formally, we model the management cost as (observe that this quantity is well-defined) for some positive constant .

We conclude that the utility of an individual is the sum of her subordinate-collaborative and hierarchical rewards minus her management cost.

Definition 2.1 (Utility function).

The utility that an individual incurs given a graph is

(2)

Clearly, if has no connections and no other agent is her subordinate in , then .

Example 1: We present a couple of graphs for which we compute the utilities of some agents to better illustrate the calculation using equation (2). Consider Figure 6. If is as in (a), then

If is as in (b), then

(a)
(b)
Figure 6: Network examples.

2.2 Game formulation and solution concept

Having defined the utility that individuals obtain from their membership in the network , we now complete the definition of our proposed network formation game in strategic (or normal) form [12]. An agent can only perform two types of actions over the network : establishing an edge so that , or severing an existing edge so that . Regarding the agent’s type, we say is consensual whenever requires ’s agreement to establish the link ; otherwise, is non-consensual whenever she can perform such action irrespective of ’s agreement.

Definition 2.2 (Hierarchical network formation game).

A hierarchical network formation game is defined by the set of homogeneous agents , with , the utility function for any agent as defined in (2). Let be the action space of , so that any is an action (or pure strategy) for agent . Given a graph , we say agent has taken action if ; and, otherwise, has taken action if . Let be the utility parameters that define the payoff of any agent as in Definition 2.1, and be the action (or pure strategy) profile which takes values from the set . Then, we represent the hierarchical network formation game by the tuple .

As it is usual in the literature on network formation games, we only consider pure strategies throughout the paper. We also consider that any agent is self-interested or utility-maximizing and so she wants to chose the action profile that maximizes her utility (2).

Remark 2.1.

If , then cannot sever the link , i.e., there exists no action that could take such that . In other words, agent cannot “stop” having an already established subordinate or collaborator. Intuitively, this induces some sense of responsibility on agent : accepts the connection attempts to establish (whether the case is non-consensual or consensual) knowing that she will not be able to severe it if she “later regrets” having such connection.

We propose and use the following solution concept for hierarchical network formation games:

Definition 2.3 (Equilibrium network).

A network is an equilibrium network when, for any ,

  1. if then and;

  2. if , then

    • for non-consensual agents: ,

    • for consensual agents: either , or, if then .

Remark 2.2 (Connection to other solution concepts).

The concept of an equilibrium network becomes the concept of strict Nash equilibrium whenever the agents are non-consensual, and it becomes the concept of pairwise stability when the agents are consensual (with the difference that our concept employ strict inequalities in contrast to the definitions of pairwise stability found in other works, e.g., [22]).

Remark 2.3.

When agents are consensual, any agent who wants to establish a link with , needs ’s consent; however, can severe any relationship she has with unilaterally.

If there exists some edge with that can be added (or severed) so that (or ), i.e., so that some condition on the utilities of as described in Definition 2.3 is violated, then we say that has an incentive or intention to establish (or sever) the link respectively. Obviously, when any agent has an incentive to severe some link, then is not an equilibrium network for any type of agents. However, if agent has an incentive to establish a link, then is not an equilibrium network for non-consensual agents, but may or may not be one for consensual agents (recall that for consensual agents, the addition of the edge also depends on ).

We conclude by providing some further interpretations and intuition on the agents’ types. A non-consensual agent can be thought of being “greedy” because accepts any agent that intends to be her subordinate or collaborator. Intuitively, we can think of greediness from two perspectives: as a bounded rationality aspect, since can accept more subordinates even though this would mean a negative impact on her utility; or perhaps as a foresightedness feature of , since may accept any subordinate with the hope of having a larger future hierarchical reward. On the other hand, we can think of a consensual agent as being “non-greedy” since she does not want to acquire more subordinates than what she can really handle, i.e., if acquiring more subordinates will affect her utility negatively.

3 Static Analysis

3.1 Hierarchical structures

We introduce a useful class of networks that abstracts and represents the concept of hierarchies. Intuitively, agents with a higher level have higher positions of power, and formally, this is true, since agents with a higher level have a greater hierarchical reward (see equation (1)). When different connected components have the same level , we say there are components per level . If the maximum level that any agent in the network has is , then we say the network has levels.

Definition 3.1 (Hierarchical structures).

A (weakly) connected digraph is a hierarchical structure if each connected component is an undirected subgraph, and it is additionally a sequential hierarchy if there is only one connected component per any level, and, if there is more than one level, then any node has a single edge towards any other node of a higher level.

A hierarchical structure has the particularity that any of its connected components are composed of agents that form strong collaborative units or teams, i.e., all agents in a connected component are collaborators. Thus, is naturally a hierarchical representation of the organization, power and/or influences among teams of people (with the understanding that a team may be composed of a single individual). Now, a sequential hierarchy is a hierarchical representation in which there is only one team at each rank or level of the hierarchy. Observe that a particular trivial case of a hierarchical structure which is also a sequential hierarchy is when the network is complete, i.e., all agents form a unique team. See Figure 9 for examples of hierarchical structures.

Observe that a pyramidal structure is formed by a hierarchical structure with two or more levels, generally with one connected component per any level, and such that the number of agents that have a lower level is greater than the ones that have a higher one. This type or class of graphs is easily found in real life hierarchies; for example, military organizations (there are less generals than colonels, but less colonels than lieutenants, etc.) and clerical hierarchies in the Catholic church [1] (there is only one Pope, and more bishops, but there are less bishops than priests) have this type of hierarchies. Moreover, it is known that many companies naturally adopt a pyramidal structure for its management organizations [28], with the CEO or president at the top, followed by a small executive leadership or vice-presidents, followed by tiers of middle managers, and all the way down to the lowest level employees.

(a)
(b)
Figure 9: Consider each subfigure, (a) and (b), as an example of a digraph which has its condensation graph represented by the graph with white-colored nodes on the left, and which has a subgraph corresponding to the nodes and of represented by a graph with gray-colored nodes on the right. In (a) we have a particular example of a hierarchical structure which is not a sequential hierarchy, whereas in (b) we have a hierarchical structure which is a sequential hierarchy. In both cases, all agents corresponding to node in have level , whereas all agents corresponding to node in have level .

3.2 Solution analysis

We first introduce a couple of useful technical results.

Lemma 3.1.

For both non-consensual and consensual agents, an equilibrium network cannot have directed cycles, and thus, only has undirected connected components. In particular, for the case of non-consensual agents, all connected components are complete subgraphs.

Proof.

By contradiction, consider an equilibrium network and a single edge such that a directed cycle involving the node exists with the edge belonging to such cycle. We analyze whether has an incentive to start a collaboration with . Observe that , for any with . Then,

Thus, has an incentive to establish the edge . For non-consensual agents, this would contradict the fact that is an equilibrium network, and so cannot have single edges in any connected component, which implies that all connected components are undirected subgraphs. Now, assume that agents are consensual and continue the analysis to see if has an incentive in accepting a link from . We can easily show that

since for any . Thus, will accept collaborating with , and this immediately implies that must only have undirected connected components for consensual agents.

Now, assume we have any with belonging to the same undirected connected component of and that this connected component is not a complete subgraph. Then, it is easy to show that , and so has an incentive to form the connection . For non-consensual agents, this contradicts being an equilibrium network, from which we immediately conclude that all connected components must be complete subgraphs. Now, consider consensual agents. Then, it is easy to show that , and so has no incentive in accepting a single edge from . Thus, the connected components of are not necessarily complete for consensual agents. ∎

Corollary 3.1.

Consider any graph without directed cycles. Then, no agent has an incentive to sever any single edge.

Proof.

Given the graph , consider any such that and ; and observe that . ∎

Now, we introduce the main results for this section.

Theorem 3.2 (Non-consensual agents).

Consider non-consensual agents. A network is an equilibrium network for some game if and only if it is a sequential hierarchy that satisfies the following conditions:

  1. If the network has levels, then the following condition holds:

    (3)

    for any such that and with .

  2. If there exists some critical edge , then

    (4)

Moreover, there exist sequential hierarchies that are not equilibrium networks for any hierarchical network formation game.

Proof.

Throughout the proof, let . We denote by the fact that there exists a path between and , and, with a slight abuse of notation, we will also use it to denote the set of edges that compose some path between and , so that clearly 222Note that there might be multiple paths between and and so there is some ambiguity as to which specific path refers to, but this will not be a problem throughout the paper.

Part I: We first prove that any given sequential hierarchy as stipulated in items (i) and (ii) of the theorem statement, is the equilibrium network for some game . Consider a given sequential hierarchy . Now, we choose any and fix their values. Then, the remaining problem is to find a reward function so that is an equilibrium network for the game . Then, for to be an equilibrium network, we need to ensure that no agent is able to establish new links or severe existing ones as pointed out in Definition 2.3. Let be the number of levels of .

If , then the network is complete with and for any ; which gives since . Thus, there is no incentive for any agent of the network to severe a link, irrespective of the function . Thus, any non-negative monotonic function can be chosen as the reward function.

Let us assume now that we have .

Case 1: Assume has level and that she considers establishing a link with some agent that has level (recall that ), which we call an immediate backward connection. Then, and , and thus

Having this previous expression being less than zero would suffice for a non-consensual agent to not have an incentive to establish immediate backward links, which is the same as choosing a function that satisfies condition (i) of the theorem statement (observe that must be, indeed, strictly increasing since the sequence is also strictly increasing for ).

Case 2: Assume has level and that she considers establishing a connection with an agent that has level for (assuming, obviously, that such exists). Let be the number of agents that have rank in the graph . Then, and ; and thus

Now, if (3) holds, then for any ; from which it follows that

and so has no incentive to establish such new link. In conclusion, if condition (i) from the theorem statement holds, then no agent has in incentive to establish a link with any agent that has a lower level.

Case 3: Assume has any level and that she considers severing any of her existing edges. From Corollary 3.1, we know that she has no incentive to sever any single edge. Now, consider any agent such that belongs to some undirected edge. If the undirected edge is a critical edge, i.e., , then it is easy to show that . Having this previous expression being less than zero would suffice for to not have an incentive to severe the edge , which is the same as choosing a function that satisfies condition (ii) from the theorem statement. On the other hand, if the undirected edge is not a critical edge, so that , then it is easy to show that ; and thus, has no incentive in severing the edge .

In conclusion, from all the previous analyzed cases, if is a sequential hierarchy with more than one level, then we only need to define the non-negative monotonic function as stipulated by the conditions (i) and (ii) of the theorem statement in order to finally obtain the sought game such that is an equilibrium network for .

Part II: Now, we need to prove that any equilibrium network is a sequential hierarchy as stipulated in items (i) and (ii) of the theorem statement. Consider the given equilibrium network . From Lemma 3.1 we immediately conclude that is a hierarchical structure with complete connected components. Note that this lemma does not specify that must be connected. However, from the proof of the same lemma, if we assume is not connected, then there is always an incentive for some agent to form a single edge with an agent from another complete connected component of the same level, which by contradiction implies that there is only one connected component per any level in . Now, we claim that any agent in must have single edges towards any other agent with a higher rank than her. Assume this is not the case and take any such that and , then it follows easily that has an incentive to form links with since such action would increase her utility by , giving a contradiction. Thus, in observance of Definition 3.1, we have proved that is a sequential hierarchy. It should be noted that, from our previous analysis in Part I, the number of agents in each complete connected component of (when there is more than one level) is determined by the utility parameters satisfying the conditions (i) and (ii) of the theorem statement. This concludes the proof that any equilibrium network is a sequential hierarchy as stipulated in items (i) and (ii) of the theorem statement.

Finally, assume we have a sequential hierarchy such that there exists some level such that: 1) for any that has level , we have ; and 2) for any that has level , we have . Then, we claim that cannot be an equilibrium network for any game. To see this, observe condition 1) implies , which immediately implies both (3) and (4) cannot hold simultaneously because of condition 2), for any utility parameters . ∎

Remark 3.3.

The importance of equations (3) and (4) is that they relate the agents’ utility parameters with the specific topology of the sequential hierarchy resulting from the hierarchical network formation game. For example, whenever has a number of levels greater than one, specify a range of allowable values for the number of agents at each level of the hierarchical structure. In particular, from inequalities (3) and (4), we observe the importance of the increments on the hierarchical reward function in defining the number of agents in each level and may, for example, define if a specific equilibrium network has a pyramidal structure.

Theorem 3.4 (Consensual agents).

Consider consensual agents. A network is an equilibrium network for some game if and only if is composed by one or multiple hierarchical structures and satisfy the following conditions:

  1. If a hierarchical structure in has levels, then

    (5)

    for any belonging to this hierarchical structure and such that , with and for any appropriate .

  2. If there exists such that and for any appropriate , and such that there exists no path from to (whether and belong to the same hierarchical structure or not), then

    (6)
  3. If there exists a critical edge such that and , then

    (7)

Moreover, there exist hierarchical structures, which are possibly sequential hierarchies, that are not equilibrium networks for any hierarchical network formation game.

Proof.

Throughout the proof, let . We adopt the same notation introduced at the beginning of the proof of Theorem 3.2, and we additionally introduce the following notation: given nodes , let denote the fact that there does not exist any path between and in , and let denote that and are (weakly) connected, i.e., or .

Part I: We first analyze the conditions under which any given hierarchical structure, as in items (i)(ii) and (iii) of the theorem statement, is the equilibrium network for some game . Consider a given network that has one or more hierarchical structures. The remaining problem is to find the utility parameters so that is an equilibrium network for the game . Now, we analyze the conditions that ensure no agent has an incentive and/or is able to form a new link or severe any existing link, so that can be properly defined as an equilibrium network of .

Case 1: Assume any agent that has level , with being the number of levels in the hierarchical structure for which belongs to, is considering establishing a link with any agent that has rank for any appropriate , i.e., establish a backward connection. First, assume . Then, following a similar procedure as for Case 1 of Part I in the proof of Theorem 3.2, we obtain with for any , so that does not have an incentive to establish a link with if the inequality (5) from condition (i) of the theorem statement is satisfied. Assume has an incentive to establish this link with , then since agents are consensual, we need to analyze if would consent to such action from . Then, it is easy to show that since for any , and thus would accept such proposition. Then, the sufficient and necessary condition for not incentivizing a backward connection from is the satisfaction of condition (i) from the theorem statement. Second, assume and . Then, it is easy to see that . Assume the worst case, i.e., has an incentive in making such connection. Then, we find that , and so will not accept a new link from . Third, assume and (notice that and may or may not belong to the same hierarchical structure). Then, it is easy to show that has an incentive in establishing such link, and that for we have , and so will not accept such connection if and only if

(8)

Case 2: Assume any agent that has level , with being the number of levels in the hierarchical structure for which belongs to, considers making a forward connection with any agent that has rank for any appropriate . We already know from Part I in the proof of Theorem 3.2 that has an incentive to establish a link with . If and , then it follows that . If and , or (whether or not), then . Then, we conclude that has no incentive in accepting a connection from .

Case 3: Assume any agent that has level , with being the number of levels in the hierarchical structure for which belongs to, considers making a connection with any agent of rank (so that ). Then, we already know has an incentive for making such connection. If , then we have and thus will not accept such link from . If (whether belongs or not to the same hierarchical structure as ), then we obtain and so has no incentive in accepting such link from if and only if

(9)

Case 4: Assume any agent that has any rank considers severing any of her existing edges. From Corollary 3.1, we know that she has no incentive to sever any single edge. Now, consider any agent such that belongs to some undirected edge. If the undirected edge is a critical edge and can increase its level by severing the edge , then it is easy to show that . Then, the condition would suffice for to not have an incentive to sever the edge , which is expressed in equation (7) from condition (iii) of the theorem statement. On the other hand, if the undirected edge is not a critical edge, then and it is easy to show that since for any ; and thus, has no incentive in severing the edge .

Notice that enforcing inequalities (8) and (9) amounts to the condition (ii) of the theorem statement. Therefore, if there exist utility parameters that satisfy the conditions (i)(ii) and (iii) from the theorem statement, we conclude that the given sequential hierarchy is an equilibrium network.

Part II: Now, consider a given equilibrium network . From Lemma 3.1 we immediately conclude that is a hierarchical structure or is composed of isolated hierarchical structures (since this lemma does not specify whether is connected or not). Then, from our previous analysis in Part I, it follows that the utility parameters must be such that the conditions (i)(ii) and (iii) in the theorem statement are satisfied. This concludes the proof that any equilibrium network is composed by one or multiple hierarchical structures.

Finally, we prove the existence of hierarchical structures that are not equilibrium networks of any possible hierarchical network formation game. For the case in which the hierarchical structures are sequential hierarchies, we can use the same construction given in the proof of Theorem 3.2. Now, we focus on the case in which the hierarchical structures are not sequential hierarchies. Consider a hierarchical structure with more than one component per any level and with complete connected components, and such that the only single edges in the graph are as follows: for any with , it holds (and so ) for any , . Moreover, also assume for any , i.e., there are no critical edges in . We immediately notice from the topology of and our discussion above that the expression in equation (5) from condition (i) of the theorem statement becomes

(10)

for any with and , . Assume has more than one hierarchical structure, and that one of them has levels with denoting any agent of such hierarchical structure such that . Consider any level . Now, observe that condition (i) (using (10)) and condition (ii) imply

(11)
(12)

respectively. We claim that if there exists some level such that , then there are no utility parameters that can define a game such that is its equilibrium network. To see this, observe that this condition implies for any , which makes it impossible to choose a reward function such that conditions (11) and (12) are satisfied, which in turn violates both conditions (i) and (ii) from the theorem. ∎

It is easy to construct games with consensual agents in which one or multiple sequential hierarchies (including complete networks) are equilibrium networks. The next proposition gives other non-trivial examples of equilibrium networks for a game with consensual agents.

Proposition 3.5 (Examples of equilibrium networks for consensual agents).

Consider any hierarchical structure with more than one level, one complete connected component per any level whose size is greater than two, and such that the only existing edges in the graph are as follows: for any with and , for any , . Then,

  1. , or

  2. any network composed by two or more (isolated) hierarchical structures with the same properties as and the additional property that for any with .

is an an equilibrium network for some game with consensual agents.

Proof.

We analyze the conditions under which any given network as in (i) or (ii) is the equilibrium network for some game with consensual agents. Given the network, the remaining problem is to find the appropriate utility parameters so that it becomes the equilibrium network for the game .

Consider to be a given hierarchical structure as in statement (i) with levels. First, observe that since all connected components are complete subgraphs with more than two nodes, then there are no critical edges in , so that we do not need to analyze the satisfaction of condition (iii) of Theorem 3.4. We immediately notice from the structure of and the proof of Theorem 3.4 that equation (6) of condition (ii) from Theorem 3.4 becomes

(13)

with and any with