Event Subscription and Non-cooperative Network Formation

CHIU KI SO∗ School of Economics and Finance, University of Hong Kong, Hong Kong. E-Mail: [email protected]

VAI-LAM MUI Department of Economics, Monash University, Australia. E-mail: [email protected]

BIRENDRA RAI Department of Economics, Monash University, Australia. E-mail: [email protected]

Abstract. The paper presents a strategic model of the formation of agent-event networks which has erstwhile been modeled as a random process. In existing agent-agent network formation models, agents directly choose to link with other agents. In our model, agents incur costs to ‘subscribe’ to exogenous ‘events’ such as conferences and Facebook groups. The benefit derived by each agent depends on the agent-event network resulting from the subscription decisions of all agents. Agents are motivated by their private payoffs and ignore the externalities generated by their subscription decisions. We analyze the tension between efficient and stable architectures under three frictions – informational decay with distance, imperfect reliability of events, and heterogeneities across agents and events. The tension is strong under heterogeneity but weak under the other frictions. This difference can be explained by the existence or non-existence of a network which is preference-maximal for all agents. The existence of a such a common peak network suffices to ensure the stability of efficient networks.

Keywords: Network formation, Efficiency, Stability, Events, Coordination, Conflict of interest. JEL Codes: D85, C72

∗

Corresponding author

Acknowledgement. We thank Simon Angus, Sanjeev Goyal, Kunal Sengupta, Hugo Sonnenschein, Satoru Takahashi, Christis Tombazos, Liang Choon Wang, Yves Zenou, seminar audiences at Deakin University, Monash University, Purdue University, and the University of Hong Kong for helpful comments. We also thank conference participants at the Australasian Economic Theory Workshop 2014, the Australian Conference of Economists 2013, the Australian PhD Conference in Economics and Business 2013, and the Econometric Society Australasian Meeting 2013. The authors bear the responsibility for any errors and omissions. 1

1

Introduction

The study of social networks is crucial for understanding diverse issues such as job search (Granovetter, 1973), spread of contagious diseases (Bailey, 1975), technology adoption (Bandiera and Rasul, 2006), criminal activities (Ballester et al., 2010), and political protests and civil conflicts (Metternich et al., 2013). Starting with the seminal works of Jackson and Wolinsky (1996) and Bala and Goyal (2000a), an important branch of the literature on networks has studied network formation with a strong emphasis on understanding whether and why efficient networks may diverge from stable networks in the presence of frictions such as informational decay due to distance, imperfect reliability of links, and heterogeneity across agents. The existing models of strategic network formation usually start with a set of agents and assume that each agent can choose to establish a link with any of the remaining agents. We shall refer to this approach as the agent-agent framework. In many contexts, however, agents incur costs to establish or maintain subscriptions to certain events and these subscriptions provide the opportunity for information transmission or emergence of social connections between agents.1 The present paper builds on this observation and studies the formation of agent-event networks. We consider a game form with a finite set of agents and a finite set of events, where an event is conceived as any entity that can serve as a platform for multilateral interactions. Agents simultaneously and independently subscribe to a subset of events. For example, academic conferences can be thought of as events with individual researchers choosing which conferences to attend. In our model, two agents who subscribe to at least one common event are said to be directly connected. A pair of agents who are not directly connected are indirectly connected if there exists some other agent who is directly connected to both of them. In general, subscription decisions of all agents lead to an agent-event network which can be represented as a simple bipartite graph with agents and events as the nodes and subscriptions by agents to events as the links. We analyze whether three prominent frictions that have been identified in the agent-agent framework – informational decay with distance, imperfect reliability, and heterogeneity – lead to a divergence between stable and efficient architectures in the agent-event framework. We focus on these three frictions because they have been shown to have distinct effects on the structure of efficient and stable networks in the agent-agent framework (Bala and Goyal, 2000b; Jackson and Rogers, 2005). As in the agent-agent approach, we model different frictions via different specifications of value functions for agents. A value function specifies the net payoff derived by an agent from each feasible network and incorporates the benefits from connections with other agents net of event subscription costs. Agents are motivated by the private value they derive and do not take into account the externalities generated by their subscription decisions. A stable network is one where the underlying subscription strategies constitute a Nash equilibrium. A network is efficient if no other network generates a strictly greater total value. 1

See Breiger (1974) and Feld (1981), where events are referred to as affiliations and foci, respectively.

2

We first study informational decay in a symmetric setting (where all agents are identical and all events are identical) by assuming that the benefit derived by an agent from another agent decreases with the distance between them. We then study imperfect reliability of events in a model where each event can fail with a strictly positive probability after subscription costs have been incurred but before benefits are derived.2 Finally, heterogeneity is modeled in line with Jackson and Rogers (2005)’s small-worlds model and Galeotti et al. (2006)’s insider-outsider model in the agent-agent framework. Specifically, the cost of subscribing to an event depends upon the identity of the agent and the identity of the event. The key structural features of the efficient architectures and the stable architectures vary with the friction under consideration. For instance, under informational decay, non-empty stable networks are regular, i.e., each agent subscribes to the same number of (possibly distinct) events. In the absence of informational decay, non-empty stable networks are minimal such that each pair of agents is connected by a unique ‘path’. Non-empty stable or efficient networks are necessarily connected under informational decay or imperfect reliability of events, but they can be disconnected under heterogeneity. Super-connected networks, where each pair of agents is connected by multiple paths, can be stable and efficient under imperfect reliability of events but not under heterogeneity. We provide a complete characterization of the efficient architectures under each of the three frictions. This permits us to investigate whether the tension between efficiency and stability manifests in (i) the stability of some inefficient networks or (ii) the instability of some efficient networks. Although some inefficient networks can be stable, every efficient network is stable under informational decay or imperfect reliability of events. Thus, the tension manifests only in the first way under these two frictions despite the fact that both efficient and stable architectures differ across these two frictions. The tension, however, manifests in both ways under heterogeneity. Consequently, the tension is significantly stronger under heterogeneity than under the other two frictions even though agents ignore the externalities generated by their individual decisions in all cases. The tension between efficiency and stability can be attributed to conflict of interest between agents or coordination failure among agents. Roughly, we say that there exists a conflict of interest between agents if no feasible network is the preference-maximal network for all agents. If there exists a network which is preference-maximal for each agent, then such a common peak network will be both efficient and stable. Put differently, if a common peak network exists under a specific friction, then despite its complexity the network formation game is a coordination game with a Pareto dominant equilibrium. The existence of stable but inefficient networks in the absence of conflict of interest reflects coordination failure among agents. Every efficient network is stable under both informational decay and imperfect reliability because a common peak network exists in both cases. However, some inefficient networks can be stable, which reflects coordination failure. Under heterogeneity, a common peak network does not always exist. The tension manifests in both ways and can reflect both conflict of interest and coordination failure. 2

In the agent-agent framework, Bala and Goyal (2000b) examine the effect of link unreliability by assuming that a link between any pair of agents can fail with a strictly positive probability.

3

To place our work in the context of the existing literature, it is useful to note that agentagent network formation was first modeled as a random process (Erd¨os and R´enyi, 1960; Watts and Strogatz, 1998; Barab´ asi and Albert, 1999). The economic literature introduced a strategic perspective to the formation of agent-agent networks. The present paper similarly provides a strategic perspective to the formation of agent-event networks which has erstwhile been modeled as a random process (Zheleva et al., 2009; Lattanzi and Sivakumar, 2009). Events generate a qualitative difference in the sense that the tension between efficiency and stability is often weaker in our models than in comparable agent-agent models (e.g. Jackson and Wolinsky (1996) and Bala and Goyal (2000a)). Our modeling approach is conceptually related to the study on club formation by Page and Wooders (2010) but the two studies have different objectives. While we focus on analyzing the tension between efficiency and stability under different types of frictions, Page and Wooders (2010) identify restrictions on payoff functions which ensure the existence of a pure strategy Nash equilibrium.3 It is also worth noting that it is not immediately obvious whether the agent-event approach analyzed here is best viewed as a variant of the agent-agent approach, or vice-versa. On the one hand, the agent-agent approach may be viewed as a special case of the agent-event approach where each event can accommodate at most two agents and there exists a distinct event for every distinct pair of agents; on the other hand, the agent-event approach can be viewed as a variant of the agent-agent approach where some agents are active and some are passive (i.e., the events), and only links from the active to the passive agents are permitted. Nonetheless, the conceptual origins of the two approaches differ. The literature on the strategic formation of agent-agent networks traces its origins to Myerson (1977) which analyzes coalitional games with an exogenously given bilateral communication structure between pairs of agents. The approach outlined in the present paper relates to Myerson (1980) which generalizes Myerson (1977) to consider coalitional games with an exogenously given multi-lateral communication structure. Section 2 first elaborates on this connection and then outlines our framework. Sections 3, 4, and 5 analyze the structure of stable and efficient networks under infomational decay, imperfect reliability of events, and heterogeneity, respectively. Section 6 formalizes the notions of conflict of interest and common peak networks to provide a unified perspective on the tension between efficiency and stability. Section 7 concludes.

2

Motivation and Framework

Consider a coalitional game (N, v) with transferable utilities, where N is the set of agents and the function v specifies the worth of each feasible coalition of agents. Myerson (1977) argues that a coalition T ⊂ N cannot realize its worth v(T ) unless agents in T communicate. For example, the realized worth of the coalition {1, 2} will be v({1, 2}) if agents 1 and 2 can communicate; otherwise, it will be v({1}) + v({2}). 3

Jun and Kim (2009) and Caulier et al. (2013) can be interpreted as modeling the formation of agent-event networks using a strategic and cooperative approach.

4

To formalize this idea, Myerson (1977) supplements the coalition game with an exogenously given communication structure. A communication structure is a set of pairs of agents, which can be represented as an agent-agent network. For example, the communication structure g = {{1, 2}, {1, 3}, {1, 4}} can be represented as a star network where agent 1 is the hub and the remaining agents are the peripherals. The realizable worth of any coalition T is defined by taking into account its potential worth v(T ) and the communication structure g. Myerson (1977) provides an axiomatic characterization of fair allocation rules in this setting under two main assumptions: [A1] the communication structure is exogenous and [A2] connectedness matters but the precise pattern of interconnections does not. Suppose g = {12, 13} such that agents 1 and 2 are directly connected and g0 = {31, 32} where they are indirectly connected. Assumption [A2] implies that the realized worth of the coalition {1, 2} will be the same under the communication structures g and g0 . Myerson (1980) extends the notion of a communication structure by defining it as any set containing sets of at least two agents. For example, the communication structure g00 = {{1, 2}, {2, 3, 4}} is feasible in Myerson (1980) but not in Myerson (1977). In general, one may think of a communication structure g = {g 1 , g 2 , . . . , g K } as representing an agent-event network, where each k ∈ {1, . . . , K} is an event, g k refers to the subset of agents who have subscribed to event k, and all agents who have subscribed to the same event are directly connected with each other. In this terminology, each event is restricted to contain exactly two agents in Myerson (1977) whereas it contains at least 2 agents in Myerson (1980). The literature on strategic formation of networks, as exemplified by Jackson and Wolinsky (1996) and Bala and Goyal (2000a), essentially takes Myerson (1977) as the starting point, endogenizes the communication structure, allows the overall pattern of interconnections to influence the payoffs derived by agents, and explores the impact of different types of frictions on the tension between efficient and stable architectures. The present paper does the same except that our starting point is Myerson (1980).

2.1

Primitives

The two primitive objects of the model are: a finite set of agents N = {1, 2, 3, . . .} with cardinality n ≥ 2 and a finite set of events M = {A, B, C, . . .} with cardinality m ≥ 1. We shall denote agents with lowercase letters and events with uppercase letters. Let e be an n × m matrix where the rows and columns of the matrix respectively represent the agents and the events. Each entry in the matrix is either 0 or 1. We have eJi = 1 if and only if agent i has subscribed to event J. We shall call e an event subscription matrix or an agent-event network.4 Let E denote the finite set of all possible agent-event networks. A typical network in this context is a triple (N, M, e). Unless necessary, we shall denote it simply as e. 4 In the language of graph theory, e is a simple bipartite graph. It may also be referred to as a two-mode network or an affiliation network.

5

Disconnected A c B  B  s 1

B c

B

Bs 2

s 3

s 4

Connected

Min. Connected

A B c c  B@  B @  B @  B  B @  B Bs @s s Bs 1 2 3 4

A B c c @ B@ B  B @  B  B @  B Bs s @s Bs 1 2 3 4

1-compete A B c c b b @ B @b  B @bb  B @ bb Bs @s bs s 1 2 3 4

Figure 1: Terminology. Agents 1 and 4 are not connected in the disconnected network, indirectly connected in the connected network and minimally connected network, and directly connected in the 1complete network. There are two paths between agents 1 and 4 in the connected network: (1, A, 2, B, 4) and (1, A, 3, B, 4). There is only one path between agents 1 and 4 in the minimally connected network and the 1-complete network: (1, A, 3, B, 4) and (1, A, 4), respectively. The distance between agents 1 and 4 is 1 in the disconnected network and the 1-complete network and 2 in the connected network and minimally connected network. Event B is not active in the disconnected network and the 1-complete network. Agents 3 and 4 are inactive in the disconnected network.

2.2

Game form

We consider a game form in which agents simultaneously and independently decide which subset of events to join. Agents do not directly form links with other agents. All agents who subscribe to an event are assumed to be directly connected with each other. Events do not play an active role in the model and serve as exogenous platforms to mediate connections between the agents. A subscription strategy of agent i ∈ N shall be denoted as an m-dimensional vector si = B J (sA i , si , . . .). We will restrict attention to pure strategies such that si ∈ {0, 1} for all i ∈ N and

J ∈ M . Thus, the strategy set Si of any agent i ∈ N is the power set of the set of events. Let Q s = (s1 , s2 , . . . , sn ) denote a representative strategy profile and S = i∈N Si be the set of all strategy profiles. The agent-event network induced by a strategy profile s will be denoted by e(s), where eJi = 1 if and only if sJi = 1. When there is no scope for confusion, we shall suppress the underlying strategy profile and denote a network simply by e. With a slight abuse of notation, we shall denote the network obtained from e upon deleting all subscriptions by agent i as e − i. Similarly, the network obtained from e upon deleting the subscription by agent i to event J will be denoted as e − iJ.

2.3

Path and component

A path between two agents i1 and ir is an alternating sequence of distinct agents and distinct events (i1 , J1 , i2 , J2 , . . . , ir−1 , Jr−1 , ir ) which starts at agent i1 , ends at agent ir , and satisfies the condition that any event along the path contains the agent that immediately precedes it and the agent that immediately succeeds it along the sequence. Two agents are connected to each other if and only if there exists at least one path between them. Otherwise, they are disconnected with each other. If two connected agents are also members of the same event, then they are directly connected; otherwise, they are indirectly connected. The 6

distance `(ik; e) between a pair of agents who are connected in network e is given by the number of events contained in the shortest path between them. In particular, the distance between a pair of directly connected agents is one. The distance between any pair of disconnected agents is assumed to be infinite. Given a graph (N, M, e), a subgraph (N 0 , M 0 , e0 ) describes the subscription decision by agents in N 0 ⊆ N only to the events in M 0 ⊆ M . A subgraph (N 0 , M 0 , e0 ) is regarded as a component of (N, M, e) if there exists at least one path between every pair of agents in N 0 under e0 and no path exists between any i ∈ N 0 and any k ∈ N \N 0 . An agent is said to be active if and only if she subscribes to at least one event. An event is active if it receives subscription by at least one agent. An inactive agent qualifies as a component according to the above definitions but an inactive event does not. The size of a component shall refer to the number of agents in the component.

2.4

Network architectures

The network where no agent subscribes to any event is the empty network. A network is connected if and only if every pair of agents is directly or indirectly connected. If a network is not connected, then it is disconnected. A connected network contains exactly one component and a disconnected network contains at least two components. A subscription in a network is called a bridge if and only if the network obtained by deleting this subscription has strictly larger number of components. A network is minimal if and only if every subscription in the network is a bridge; or, if there is a unique path between any pair of connected agents. A non-empty network is regular if and only if every agent subscribes to the same number of events but the events may differ across agents. A non-empty network is a-complete if and only if all agents subscribe to the same subset of events with cardinality a ∈ {1, . . . , m}. Every a-complete network is regular but the converse is not true.

2.5

Solution concepts

In the following sections, we investigate the impacts of different frictions on the tension between efficiency and stability by making alternative assumptions about the value functions (i.e., payoff functions) of agents. A value function of an agent maps each feasible network into a real number which represents the net payoff derived by the agent. Given any specification of value functions for all the agents, the total value under a network is the sum of the values derived by each agent. A network is efficient if no other feasible network generates a strictly higher total value. A network is stable if the underlying strategy profile constitutes a Nash equilibrium such that no agent can obtain a strictly higher individual value by any unilateral change in her subscription strategy.

3

A Symmetric Event Subscription Game

We first consider a game where all agents are identical and all events are identical. The value derived by an agent in a given network is assumed to depend on the pattern of her connections

7

with other agents and her number of event subscriptions. The pattern of connections of any agent i in a network e is formalized as the m-dimensional vector ψi = (ψi1 (e), . . . , ψia (e), . . . , ψim (e)), where ψia (e) denotes the number of her connections that lie at a distance of a ∈ {1, . . . , m} in network e. The first entry in this vector, ψi1 (e), gives the number of direct connections agent i has in network e. Let di (e) denote the number of events that agent i has subscribed to in network e. The value derived by agent i under network e is given by Πi (e) = θ(ψi1 (e), . . . , ψia (e), . . . , ψim (e), di (e))

(3.1)

Any value function of the above form implicitly satisfies the following three properties. It is agent anonymous (ANM) in the sense that interchanging the strategies of any pair of agents (and thus their positions in the network) interchanges their values. It is event neutral (ENT) as any relabelling of events has no impact on the values derived by agents. The benefit derived by an agent from being connected with another agent only depends on the distance between them. Since distance is defined as the number of events along the shortest path, agents do not derive any benefit from paths other than the shortest path. The above value function thus exhibits redundancy of longer paths (RLP). The total value under any feasible network e is given by Π(e) =

X

Πi (e)

(3.2)

i∈N

3.1

Efficiency and stability

We consider a class of value functions, denoted by Θ, which is a slight generalization of the ones considered in Bala and Goyal (2000a).5 The value derived by an inactive agent is normalized to zero (hereafter NOR). In addition, every θ ∈ Θ is assumed to satisfy three types of monotonicity restrictions. Formally, for any agent i ∈ N , any a, a0 ∈ {1, . . . , m} with a < a0 , any (feasible) strictly positive integer α, and any θ ∈ Θ, NOR θ(0, . . . , 0, 0) = 0 SCM θ(ψi1 , . . . , ψia + α, . . . , ψim , di ) > θ(ψi1 , . . . , ψia , . . . , ψim , di ) SSM θ(ψi1 , . . . , ψia , . . . , ψim , di + α) < θ(ψi1 , . . . , ψia , . . . , ψim , di ) SDM θ(ψi1 , . . . , ψia + α, . . . , ψia0 − α, . . . ψim , di ) < θ(ψi1 , . . . , ψia , . . . , ψim , di ) Strict connection monotonicity (SCM) requires that a strict increase in the number of connections that an agent has at any finite distance strictly increases the value she derives if her number of 5

Bala and Goyal (2000a) assume separability of benefits and costs. Also note that we suppress the dependence of Θ on the number of agents n and the number of events m for notational simplicity.

8

connections at all other distances and the total number of her event subscriptions remain constant. Strict subscription monotonicity (SSM) stipulates that a strict increase in the number of event subscriptions by an agent strictly decreases the value she derives if the number of her connections at each finite distance remains unchanged. Strict distance monotonicity (SDM) captures the idea of informational decay with increasing distance. It requires the value derived by an agent to strictly increase if she acquires some closer connections at the loss of an equal number of farther connections without any change in her total number of event subscriptions. Proposition 1. Efficient networks are (a) empty if θ(n − 1, 0, . . . , 0, 1) < 0; (b) 1-complete or empty if θ(n − 1, 0, . . . , 0, 1) = 0; and, (c) 1-complete if θ(n − 1, 0, . . . , 0, 1) > 0. Proof. We first show that a non-empty efficient network must have exactly one active event and then show that all agents must subscribe to this event. Let e be an efficient non-empty network. Let e0 be the subgraph obtained from e after removing all the inactive agents and let n0 be the size of e0 . Suppose n0 = 1 and let agent i be the only active agent in e0 such that the value derived by i under e0 is the total value under e0 . By NOR and SSM, Π1 (e0 ) = Π(e0 ) ≤ θ(0, 0, . . . , 0, 1) < θ(0, . . . , 0) = 0, and thus e cannot be efficient. Now suppose n0 > 1. If e0 is disconnected, then the maximum value derived by any agent cannot exceed θ(n0 − 2, 0, . . . , 0, 1). By contrast, consider a ˆ where all the n0 agents subscribe to a single event. Clearly, e cannot be connected network e efficient since by SCM, Π(ˆ e) = n0 θ(n0 − 1, 0, . . . , 0, 1) > n0 θ(n0 − 2, 0, . . . , 0, 1) ≥ Π(e0 ) = Π(e). Next suppose e0 is connected and contains at least two active events. Then, there exists some agent k with Πk (e0 ) ≤ θ(n0 − 1, 0, . . . , 0, 2) < θ(n0 − 1, 0, . . . , 0, 1). However, SSM implies that Π(ˆ e) = n0 θ(n0 − 1, 0, . . . , 0, 1) > (n0 − 1)θ(n0 − 1, 0, . . . , 0, 1) + θ(n0 − 1, 0, . . . , 0, 2) ≥ Π(e0 ) = Π(e) and hence e cannot be efficient. Thus, a non-empty efficient network must contain exactly one active event. Recall that Π(ˆ e) = n0 θ(n0 − 1, 0, . . . , 0, 1). Adding one more agent to the active event ˆ will result in a total value of (n0 +1)θ(n0 , 0, . . . , 0, 1) which will be strictly greater than Π(ˆ in e e) due to SCM. Thus n0 must be equal to n in a non-empty efficient network. Consequently, an efficient network is either empty or 1-complete. The proposition follows from a direct comparison of the total values generated by these networks.  The 1-complete architecture is the only non-empty architecture which can be efficient. A 1complete network, where all agents subscribe to one and the same event, achieves connections between all pairs of agents at the minimum possible distance with minimum number of subscriptions. Each agent derives a value of θ(n − 1, 0, . . . , 0, 1) under a 1-complete network. It is worth noting that SDM does not play any role in the above proof and thus the above result will hold if information is not subject to decay. We begin our analysis of stability by introducing the useful notion of critical agents. An agent k is critical to some other agent i in network e if and only if the removal of k from the network 9

e changes the value derived by agent i from some agent j ∈ N \{i, k}. Since the value derived by i from j in a given network depends on the distance between i and j, agent k can be critical to agent i in network e only if there exists some agent j ∈ N \{i, k} such that `(ij; e) 6= `(ij; e − k). An agent who is not critical to any other agent in a given network is referred to as a non-critical agent in the network. Lemma 1. Given a network e and a pair of agents i, k ∈ N , if `(ik; e) ≥ `(i0 k 0 ; e) for all i0 , k 0 ∈ N , then i and k cannot be critical to each other in e. Proof. Let i and k be a pair of agents at the maximum distance in e. If `(ik; e) = ∞, then clearly they cannot be critical to each other as there is no path between them. Suppose `(ik; e) is finite and k is critical to i. Then k must appear in the shortest path connecting i with some other agent, say i0 . Without loss of generality, assume the path connecting i and i0 via k has the form (i, j1 , . . . , j`(ik;e) , k, j`(ik;e)+1 , . . . , i0 ). Then, the distance between i and k cannot be the maximum distance in the network e. Hence, a contradiction.  The empty network, where no agent subscribes to any event, is always stable. Any unilateral deviation by an agent from the empty network will involve unilaterally subscribing to one or more events. SSM implies any such deviation will make the agent strictly worse off. The following proposition identifies the key structural features of non-empty stable networks. Proposition 2. For any θ ∈ Θ, non-empty stable networks exist if and only if θ(n−1, 0, . . . , 0, 1) ≥ 0. In any such network (a) every pair of agents is directly connected; (b) all agents subscribe to the same number of events (i.e., the network is regular); (c) at most one event receives subscriptions from all agents; and, (d) if more than one event is active, then no event receives more than n − 2 subscriptions. Proof. For any θ ∈ Θ, SCM, SSM, and SDM imply that the maximum value an agent can derive across all feasible networks is either θ(n − 1, 0, . . . , 0, 1) or 0. Hence, if θ(n − 1, 0, . . . , 0, 1) < 0, then a non-empty network cannot be stable since deleting all subscriptions will be a strictly profitable deviation for each agent. Now suppose θ(n−1, 0, . . . , 0, 1) ≥ 0, and e is a non-empty stable network. To prove (a), let i and k be a pair of agents that exhibit the largest distance across all pairs of agents and are thus not critical to each other (see Lemma 1). Further assume that they are not directly connected to each other. Without loss of generality, assume Πi (e) ≥ Πk (e). Let the distance between agents i and k be `(ik; e) > 1. If agent k deviates and mimics the strategy si played ˜ k which will be of the form θ(ψi1 (e) + 1, . . . , ψi`(ik;e) − by agent i under e, then he will obtain Π 1, . . . , ψim (e), di (e)) or θ(ψi1 (e) + 1, . . . , ψi`(ik;e) , . . . , ψim (e), di (e)) depending upon whether i and k are connected or disconnected under e. Such a deviation is strictly profitable for k in both ˜ k > Πi (e) = θ(ψi1 (e), . . . , ψi`(ik;e) (e), . . . , ψim (e), di (e)) ≥ Πk (e) by SDM. Thus, a cases since Π non-empty stable network must involve a direct connection between every pair of agents. 10

A B C c c c " " b b " " b b @  B@  B @b " @b " " " b b  B @  @ B " b " b " @ b  "" B @ bb  " bB " Bs @s" bs @s bBs s 1 2 3 4 5 6 Figure 2: A network satisfying the properties listed in Proposition 2.

To prove (b), suppose e is an irregular stable network where all agents are directly connected such that no agent is critical to any other agent. Irregularity implies there must exist a pair of agents, say, i and k, such that dk (e) > di (e). Consequently, agent i derives a strictly greater value than does agent k under e. SSM implies that agent k will strictly gain by mimicking i’s strategy. Hence, e cannot be stable and thus every stable network must be regular. To prove (c), suppose e is a regular stable network where all agents are directly connected and at least two events, say, J and J 0 receive subscriptions from all the n agents. SSM implies that an agent can strictly gain by unsubscribing from either event J or J 0 . To prove (d), let us first suppose that event J contains all n agents. SSM implies that any agent who has subscribed to any other event J 0 will strictly gain by deleting the subscription to event J 0 . Now consider the remaining case where some active event J contains n − 1 agents. Any other active event J 0 must contain at least two agents, say i and k. Either both i and k or only one, say i, must have subscribed to event J. In both cases, SSM implies that agent k can gain strictly gain if he subscribes only to event J.  A 1-complete network is a straightforward example of a non-empty network which will be stable if θ(n − 1, 0, . . . , 0, 1) ≥ 0: every pair of agents is directly connected, all agents subscribe to the same single event, and the one and only active event receives subscriptions from all agents. Figure 2 illustrates another network which satisfies the four properties mentioned in Proposition 2. It will be stable for any θ ∈ Θ such that θ(5, 0, . . . , 0, 2) ≥ max{θ(3, 2, 0, . . . , 0, 1), 0}. No agent can gain by unilaterally adding more subscriptions because of SSM. Unilaterally deleting any one or both subscriptions cannot be strictly profitable if θ(5, 0, . . . , 0, 2) ≥ θ(3, 2, 0, . . . , 0, 1) and θ(5, 0, , . . . , 0, 2) ≥ 0, respectively. Note that each agent derives a value of θ(5, 0, . . . , 0, 2) under the network illustrated in Figure 2. Consequently, if it is stable, then it cannot be efficient because SSM implies that it will be dominated by a 1-complete network where each agent derives a value of θ(5, 0, . . . , 0, 1). In a network formation model, the tension between stability and efficiency can manifest in two ways: stable networks can be inefficient and efficient networks can be unstable. Here, the tension manifests only in the first way. Corollary 1. Stable networks may be inefficient but every efficient network is stable. 11

Proof. Consider the network in Figure 2 which, according to Proposition 2, is stable if θ(5, 0, . . . , 0, 2) ≥ max{θ(3, 2, 0, . . . , 0, 1), 0}. Proposition 1 implies that an efficient network is either empty or 1complete. Since the network in Figure 2 is neither empty nor 1-complete, it is inefficient. This establishes that a stable network can be inefficient. Let e be an efficient network. If it is empty, then it is stable because empty network is always stable. Now suppose e is 1-complete. Since each agent in the 1-complete network is using the same strategy, and the set of possible deviations is identical across all agents, we can focus on any agent i. If i makes any additional subscriptions, then SSM implies that the value she derives will strictly decrease since subscription costs will increase but no additional benefit will be obtained. If i unsubscribes from the only active event, then she will obtain a value of zero as she will lose all her connections. Efficiency of the 1-complete network implies θ(n − 1, 0, . . . , 0, 1) ≥ 0. This condition also guarantees that the value derived by each agent in a 1-complete network is non-negative. Consequently, every efficient network is stable.  Two remarks are in order. First, the corollary suggests that although there is a tension between efficiency and stability it is quite weak since efficient networks are stable. Second, the reason behind the stability of some inefficient networks is coordination failure among the agents. Note that for any θ ∈ Θ with θ(n − 1, 0, . . . , 0, 1) > 0, the value derived by each agent in a 1-complete network is strictly greater than that under any other network. Put differently, each agent would prefer the efficient and stable 1-complete networks over all other networks including stable but inefficient networks such as the one illustrated in Figure 2. Nonetheless, this inefficient network can be stable as no individual can strictly gain by any unilateral deviation. We say that the stability of such an inefficient network is due to coordination failure among agents because each agent would strictly gain if all agents could deviate in a coordinated fashion to a 1-complete network.

3.2

The role of informational decay

To better understand the role of informational decay with distance, suppose information is not subject to decay with distance between agents. Formally, let the value derived by an agent under any feasible network e be now defined by θ(ψi (e), di (e)), where ψi (e) and di (e) are the agent’s total number of (direct and indirect) connections and total number of subscriptions under e. Consider the class of value functions Θ such that for any θ ∈ Θ, θ(0, 0) = 0, and for any feasible positive integer α, θ(ψi (e) + α, di (e)) > θ(ψi (e), di (e)) and θ(ψi (e), di (e)) > θ(ψi (e), di (e) + α). Note that these restrictions are simply the suitable adaptations of normalization, connection monotonicity, and subscription monotonicity. In the absence of informational decay, stable networks turn out to be either empty or minimally connected. Non-empty stable networks exhibit minimality and connectedness since being connected still matters but distance becomes irrelevant when the quality of connections does not decay with distance. Informational decay induces agents to from direct connections with other agents as

12

highlighted by Proposition 2. The modified connection and subscription monotonicity properties imply that each agent derives the highest value in a 1-complete network and/or the empty network depending on the comparison between θ(n − 1, 1) and 0. Hence, efficient networks are empty or 1-complete both in the presence and in the absence of informational decay. In summary, informational decay with distance affects the structure of stable networks but has no impact on the structure of efficient networks.6

3.3

Alternative notions of stability

The above discussion suggests that nonempty stable networks are regular directly connected or minimally connected depending on whether or not information decays with distance. We now briefly comment on how an alternative solution concept – strict stability – affects the results. A network is strictly stable if the underlying strategy profile constitutes a strict Nash equilibrium such that any unilateral deviation by any agent leads to a strict decrease in the value she derives. Strictly stable networks are thus stable, while stable networks are not necessarily strictly stable. Consider the minimally connected network containing two active events as illustrated in Figure 1). Agents who subscribe to exactly one event – agents 1, 2 and 4 – are indifferent between subscribing to any of the active events. This indifference implies that such a network cannot be strictly stable as agents who subscribe to only one event will have a weakly profitable deviation. Similarly, the network illustrated in Figure 2 is also not strictly stable because each agent is indifferent between subscribing to events A and B, or A and C, or B and C.7 In fact, it is not difficult to show that, non-empty strictly stable networks are 1-complete regardless of whether information is subject to decay or not. This result implies that the sets of efficient and strictly stable networks are the same. Tension between strict stability and efficiency arises only when θ(n − 1, 0, . . . , 0, 1) = 0 because the 1-complete network architecture is efficient but not strictly stable.8 In summary, the tension between stability and efficiency almost vanishes under the stronger solution concept of strict stability. When strict Nash is used as the solution concept for stability, then the set of stable networks coincides with the set of efficient networks. It is worth pointing out that the absence of tension between strict stability and efficiency is specific to the symmetric 6

This observation can also be seen from the fact that SDM plays no role in the proof of Proposition 1.

7

In general, suppose e is a non-empty strictly stable minimally connected network where at least two events are active. By Lemma 2.1.3 of West (1996), every minimally connected network with at least two nodes has at least two end-nodes. An end-node is a node which has exactly one link. In our context, an end-node can be an event with only one member or an agent with only one subscription. Since e is minimal, it contains at least two end-nodes which are either events or agents. However, because e is strictly stable, each event in e must contain at least two agents. Therefore, an event cannot be an end-node and hence every end-node in e must be an agent. However, any agent who is an end-node is indifferent between playing his strategy or mimicking the strategy of any other agent who is an end-node. The resulting contradiction establishes that a minimally connected network with at least two active events cannot be strictly stable. 8

As the empty network is always strictly stable, it will be strictly stable but inefficient if θ(n − 1, 0, . . . , 0, 1) > 0.

13

event subscription game analyzed in this section. We conduct the analysis of event reliability and heterogeneity using the weaker notion of stability as most of our results remain unchanged under the stronger notion of stability.9

4

Imperfect Reliability of Events and Superconnected Networks

Information transmission in networks need not be perfectly reliable. In the agent-agent network formation models, imperfect reliability has been studied by assuming that a link may fail (due to exogenous unmodeled reasons) after agents have incurred the cost to establish the link but before they derive any benefits (Jackson and Wolinsky, 1996; Bala and Goyal, 2000b). In the agent-event framework, imperfect reliability might arise due to the potential failure of events or subscriptions. We restrict attention to the first case and assume that each event succeeds with an exogenous probability p ∈ (0, 1) which does not depend on the identity of the event and is independent of the success of other events. The network that would result from subscription decisions of all the agents if no event fails will be referred to as the initial network. Given any initial network e, let L(e) denote the set of all possible networks that can be realized ex-post. To isolate the impact of imperfect reliability, we draw upon Bala and Goyal (2000b) to specify the value function of agents. Specifically, conditional on any realized network, the benefit derived by an agent is assumed to be the total number of (direct or indirect) connections he has. Each subscription is assumed to cost c > 0. Consequently, the ex-ante (expected) value derived by agent i under any initial network e is given by Πi (e) =

X

λ(e0 |e)ψi (e0 ) − di (e)c,

(4.1)

e0 ∈L(e) 0

0

where λ(e0 |e) = pγ(e ) (1−p)γ(e)−γ(e ) is the probability that network e0 ∈ L(e) is realized conditional on the initial network e, ψi (e0 ) is the total number of direct and indirect connections that agent i has in network e0 , and di (e) is the total number of subscriptions by agent i in the initial network e. An initial network is stable if the underlying strategies constitute a Nash equilibrium. We say an initial network is ex-ante efficient if no other feasible network generates a strictly greater total ex-ante value. The above value function violates redundancy of longer path (RLP), satisfies agent anonymity (ANM) and event neutrality (ENT). In addition, there is no decay with distance, additional connections (subscriptions) without additional subscriptions (connections) strictly increase (decrease) the value derived by an agent, and an inactive agent derives zero value. 9

The interested reader may refer to So (2014) for a detailed discussion of adopting strict Nash and weak Nash as the solution concept for stability.

14

4.1

Efficiency and stability

The key feature that distinguishes this model from the ones studied in the previous section is that multiplicity of paths between agents can be valuable. In general, given any initial network, if a pair of agents has multiple paths connecting them, then each of these paths will be the unique path connecting them under some ex-post realizable network. Intuitively, unless subscription costs are too high, one would expect stable and efficient networks to involve multiple subscriptions by agents that can insure against potential failure of events and the resulting loss of connections. This motivates the definition of a superconnected network: a connected network where every pair of agents remains connected ex-post upon the failure of any single event. For example, an a-complete network is superconnected if and only if a ≥ 2 as agents will remain ex-post directly connected with each other even if one or more but not all the a events fail. The following proposition characterizes the efficient networks and confirms that efficient networks are superconnected if subscription costs are below a certain threshold. Proposition 3. Ex-ante efficient networks are (a) empty if c ≥ p(n − 1); (b) m-complete if c ≤ p(1 − p)m (n − 1); and, (c) b-complete if p(1−p)m (n−1) < c ≤ p(n−1), where b ∈ {1, 2, . . . , m−1} is the integer satisfying p(1 − p)b (n − 1) < c ≤ p(1 − p)b−1 (n − 1). Proof. Suppose some agent i subscribes to the subset Mi ⊆ M containing d ≥ 0 number of events. Let E(Mi ) denote the set of all feasible networks wherein agent i subscribes to each event in Mi . Given any network e ∈ E(Mi ), the maximum benefit i can derive from all possible networks that can be realized conditional on e is n − 1. The probability that at least one of the events that agent i has subscribed to is realized is 1 − (1 − p)d . The maximum ex-ante value i can derive will either ¯ i (d; p, n, c) = [1 − (1 − p)d ](n − 1) − dc. Agent i will obtain Π ¯ i (d; p, n, c) if she remains be 0 or Π connected with all the remaining (n − 1) agents in all ex-post realizable networks where at least one event succeeds. For any vector of parameters ω = (p, n, m, c), let d∗ (ω) ∈ {0, 1, . . . , m} be the number of subscrip¯ i (d; p, n, c). The marginal change in Π ¯ i (d; p, n, c) from the dth subscription tions that maximize Π ¯ ¯ ¯ − 1) = p(1 − p)d−1 (n − 1) − c. As ∆Π(d) ¯ will be ∆Π(d) = Π(d) − Π(d is decreasing in d, the dth subscription will have a non-negative marginal contribution as long as c ≤ p(1−p)d−1 (n−1). Thus, if (1−p)1−1 p(n−1)−c = (n−1)p−c < 0, we have d∗ = 0. Similarly, if n(n−1)p(1−p)m−1 −nc ≥ 0, then we have d∗ = m. Finally, for all c between (n − 1)p(1 − p)m and (n − 1)p, d∗ ∈ {1, . . . , m − 1} ∗

∗ −1

will be such that (n − 1)p(1 − p)d < c ≤ (n − 1)p(1 − p)d

. ¯ ∗ (ω)) if (i) she subscribes to d∗ (ω) For a given ω, any agent can obtain the ex-ante value of Π(d

number of events and (ii) each event that she subscribes to contains all the remaining n − 1 agents. The d∗ -complete networks, with d∗ ∈ {0, 1, . . . , m}, are the only architectures that fulfil these conditions as no other network can simultaneously satisfy (i) and (ii) from the perspective of all agents. 

15

The above result tells us that efficient networks are regular since for any subscription cost the efficient network is either empty or a-complete with a ∈ {1, . . . , m}. Second, efficient networks are superconnected if and only if c ≤ p(1 − p)(n − 1) since for any such subscription cost efficient networks are a-complete with a ≥ 2. We next turn to stability. Proposition 4. Non-empty stable networks exist if and only if c ≤ p(n − 1). Every such network is connected. Further, super-connected stable networks exist if c ≤ p(1 − p)(n − 1). Proof. The empty network is always stable. The maximum ex-ante value that an agent can derive from a single subscription is p(n − 1) − c. If c > p(n − 1), no agent will have the incentive to subscribe to even one event. Hence, if c > p(n − 1), the empty network is the unique stable network. If c ≤ p(n − 1), then 1-complete networks are stable since no agent can strictly gain from any unilateral deviation. Hence, non-empty stable networks exist if and only if c ≤ p(n − 1). Note that 1-complete networks are connected. In the remainder of the proof we show that connectedness is a feature common to all non-empty stable networks. Let e be a non-empty stable network which is disconnected. Let i and k be two active agents in e who belong to different components in e, and are thus not critical to each other. Stability of e implies Πi (e) ≥ 0. Suppose agent k maintains all his subscriptions under e and also subscribes to all the events that agent i has subscribed to under e. Since agents i and k are disconnected under e, the additional value that k can derive from such a deviation will be at least Πi (e). Moreover, agent k will gain an extra connection with agent i whenever at least one of the events that both i and k are subscribed to is realized. Hence, such a deviation is strictly profitable for agent k. Consequently, a non-empty disconnected network cannot be stable. Proposition 3 suggests that 2-complete networks, which are super-connected, are efficient if c ≤ p(1 − p)(n − 1). The proof of Proposition 3 highlights that, given any subscription cost c, each agent obtains the same value under all efficient networks for that c, and this value is strictly greater than that under any inefficient network. Consequently, for any c, an efficient network will be stable. Hence, superconnected stable networks exist if c ≤ p(1 − p)(n − 1).  Proposition 4 suggests that non-empty stable networks can be potentially quite complex, as Proposition 4 only establishes that they must be connected. Non-empty stable networks can be minimally connected such that there is a unique path between each pair of agents, or they may be non-minimally connected with a unique path between some pairs and multiple paths between other pairs of agents, or they may be superconnected with more than one path between all pairs of agents. The analytical difficulty in obtaining a sharp characterization of stable networks under imperfect reliability arises because understanding the payoff implications of a unilateral deviation by an agent requires accounting for its impact on all possible networks that can be realized following the deviation. Similar difficulties are encountered in the agent-agent framework (see Bala and Goyal (2000b) for details). The following proposition is therefore useful since it helps identify some connected networks that cannot be stable. 16

Proposition 5. A non-empty connected network e cannot be stable if it contains a pair of agents who (i) play different strategies and (ii) are not critical to each other in any realizable network e0 ∈ L(e). Proof. Let e be a stable network containing a pair of agents i and k who satisfy (i) and (ii). At least one of them must be active under e otherwise (i) will be violated. Let us first consider the case where both i and k are active in e. Without loss of generality, suppose Πi (e) ≥ Πk (e). The set of all possible realizations conditional on e, L(e), can be partitioned into four subsets: the sets L0 (e), Li (e), Lk (e), and Lik (e) of all the realizable networks where both i and k are inactive, only i is active, only k is active, and both i and k are active, respectively. Suppose k deviates and mimics the strategy played by agent i in e. Then, the value derived by k under the resulting network will be at least ˜ k = P 0 0 λ(e0 |e)[0] + P 0 i λ(e0 |e)[ψi (e0 ) + 1] + P 0 k λ(e0 |e)[0] + P 0 ik λ(e0 |e)[ψi (e0 )] − Π e ∈L e ∈L e ∈L e ∈L P P 0 i di (e)c = Πi (e) + e0 ∈Li λ(e |e)[1]. Since L (e) cannot be empty, e0 ∈Li λ(e0 |e)[1] is strictly posi˜ k = Πi (e) + P 0 i λ(e0 |e)[1] > Πi (e) ≥ Πk (e), which contradicts the stability tive. Thus we have Π e ∈L

of e. Now suppose e is a stable network containing a pair of agents i and k who satisfy (i) and (ii) where i is active but k is inactive. Consequently, Πi (e) ≥ Πk (e) = 0. Mimicking the strategy being played by i in e is a strictly profitable deviation for k as she will then obtain a connection with i while i does not have the connection with k under e.  The above proposition helps clarify how unreliability of events impacts on the structure of stable networks. Recall that the regularly directly connected network illustrated in Figure 2 can be stable in the symmetric model analyzed in Section 3 when information is subject to decay with distance and all events are perfectly reliable. This network, however, can never be stable according to Proposition 5 if events are not perfectly reliable since there exist pairs of agents who play different strategies and no agent is critical to any other agent in any ex-post realizable network. The instability of this network is neither due to regularity nor due to direct connections between all pairs of agents. Figure 3 provides two examples of regular networks – one involves direct connections between all agents while the other does not – which can be stable under imperfect reliability. Also note that minimally connected networks with multiple active events can be stable if information does not decay with distance and events are perfectly reliable (Subsection 3.2). However, when events are not perfectly reliable, every minimal network with at least two active events is unstable. Every minimally connected network with at least two active events contains at least two agents who subscribe to only one event and the events they subscribe to are different. Such a network cannot be stable since the two agents would be playing different strategies, and will not be critical to each other in any ex-post realizable network since they subscribe to only one event. Consequently, no minimally connected network with at least two active events is stable when events are not perfectly reliable. The above discussion reveals that the structure of efficient and stable networks differs markedly in the presence versus the absence of perfect reliability. Nonetheless, the following corollary highlights

17

Ac

1 s

B c @

Ac

@ Dc @cB s 3 @ @ @ @ @c @s C 2

@ @s2

4 s @ @ @c D

s 3

1 s @

c C

Figure 3: The regular network on the left is a circle with four agents and four events where some pairs of agents are not directly connected. It is unstable under informational decay√but stable under imperfect reliability if p2 (1 − p)2 + 4p3 (1 − p) ≥ c ≥ 5p2 (1 − p)2 + 2p(1 − p)3 and p ≥ (1 + 13)/6. Every pair of agents is directly connected in the regular network on the right. It is unstable under informational decay but stable under imperfect reliability if 2p2 (1 − p)2 ≥ c ≥ 2p(1 − p)3 and p ≥ 1/2.

that the nature of the tension between efficiency and stability is not affected by imperfect reliability of events. Corollary 2. Stable networks may be inefficient but every efficient network is stable. Proof. If c > p(n−1), then the empty network is the unique efficient and stable network. Given any c ≤ p(n−1), non-empty efficient networks are a-complete for some a ∈ {1, . . . , m}. As demonstrated in Proposition 3, the efficient a-complete networks will be stable as well since every agent derives the maximum feasible value in an efficient network. Consequently, every efficient network is stable. If c is such that efficient networks are b-complete with b ≥ 2, then any a-complete network with 1 ≤ a < b will be inefficient. We now show that any such a-complete network will be stable because any additional subscription or any deletion of existing subscriptions will strictly reduce the value derived by an agent. Let e be an a-complete network where every agent obtains [1 − (1 − p)a ](n − 1) − ac. No agent in e has an incentive to subscribe to an additional event since all the other events are inactive. Suppose agent i deviates and unsubscribes from an event J. His value under the resulting network will be [1 − (1 − p)a−1 ](n − 1) − (a − 1)c. The deviation is profitable if Πi (e − iJ) − Πi (e) = c − p(1 − p)a−1 (n − 1) ≥ 0. Since p(1 − p)a−1 (n − 1) > p(1 − p)b−1 (n − 1) ≥ c for all a < b, the deviation will be strictly unprofitable if a ∈ {1, . . . , b − 1}. As unsubscribing from more than one event will only increase the loss, c ≤ (n − 1)p(1 − p)b−1 is sufficient to deter any unsubscription from any a-complete network with a < b.  As in the models considered in the previous section, although there is a tension between efficiency and stability it is quite weak since efficient networks are stable. Further, the stability of some inefficient networks, once again, purely reflects coordination failure among the agents. As shown in the proof of Proposition 3, given any c, each agent derives the maximum feasible value in the efficient networks for that c. Consequently, given any stable but inefficient network, each agent would strictly gain if all agents could deviate in a coordinated fashion to an efficient network.

18

5

Heterogeneity

In the models analyzed so far, all agents are identical and all event are identical from the perspective of every agent. In this section, we allow the cost of subscribing to a given event to differ across agents. More precisely, we assume that each event is tailored to serve a specific subset of agents, such that any given event serves as the internal event for that subset of agents and also serves as an external event for all the remaining agents. The structure of the model closely parallels Jackson and Rogers (2005)’s small world model and Galeotti et al. (2006)’s insider-outsider model in the agent-agent framework. Formally, we consider an economy with n agents who can be partitioned into m ≥ 2 distinct groups. Let NJ be the set of agents belonging to group J. We assume each group has n ¯ ≥ 2 agents such that the total number of agents in the economy is m¯ n = n. Further suppose the set of events M contains m distinct events. Event J ∈ M is tailored for agents in group J in the sense that the cost of subscribing to event J is cL > 0 for agents in group J and cH > cL for agents not in group J. In general, subscribing to one’s internal event is cheaper than subscribing to any external event, and all external events have the same subscription cost. Such a value function will violate both agent anonymity (ANM) and event neutrality (ENT) since the cost of subscription depends both on the identity of the agent and the identity of the event. Depending upon the context, such a cost structure could be attributed to different characteristics such as language, location, race, gender, and profession, among other things. To isolate the impact of heterogeneity, we follow Galeotti et al. (2006) and assume the value derived by an agent in any network is given by her total number of connections net of her subscription costs. Formally, Πi (e) = ψi (e) −

X

eJi cJi ,

(5.1)

J∈M

where ψi (e) is the total number of direct and indirect connections that agent i has in network e, eJi is either 1 or 0 depending upon whether or not agent i has subscribed to event J in the network e, and cJi is the cost to agent i in subscribing to event J (which will be cL or cH depending upon whether or not agent i belongs to group J).

5.1

Stability and efficiency

The above value function implies there is no decay with distance, longer paths are redundant, and additional connections (subscriptions) without additional subscriptions (connections) strictly increase (decrease) the value derived by an agent. In addition, an inactive agent derives a zero value. These properties have two immediate implications. First, as the benefit derived by an agent depends only on his total number of (direct or indirect) connections, we would expect non-empty stable and efficient networks to be minimal such that there is a unique path between any pair of connected agents. The reason being that when the pattern of connections is irrelevant, having more than one path between any pair of agents will impose both private and social costs but provide no 19

1 s s 2

A 4 c N @ @ s @N 3 5

B 4 N 6

Figure 4: A minimally connected network which is not internally complete. This network is unstable because agent 4 can obtain a strictly greater value by unsubscribing from the external event A and subscribing to the internal event B. The network is inefficient because agent 4 strictly gains from such a deviation and the value derived by the other agents remains the same.

additional private or social benefits in return. Second, if the internal subscription to any event J is worthwhile for even one agent in group J, then it is worthwhile for all agents in group J (see Figure 4 for an illustration). Consequently, an active event in a stable or efficient networks would be expected to receive subscriptions from all internal agents. Formally, an active event J ∈ M is internally complete if all agents belonging to group J subscribe to their internal event J. An internally complete event may or may not receive subscriptions from external agents. The following proposition formalizes the preceding discussion. Proposition 6. (a) Non-empty stable or efficient networks are minimal. (b) Every active event in any stable or efficient network is internally complete. (c) If a stable or efficient network is disconnected, then it cannot contain any external subscription. Proof. In the Appendix. Parts (a) and (b) imply that non-empty stable networks will be minimal and internally complete regardless of whether they are connected or disconnected. Part (c) highlights a further restriction on the structure of stable or efficient networks that are disconnected. The three conditions together imply that a non-empty stable or efficient network must belong to one of the following two architectures. a-ICMC A network is a-internally complete and minimally connected if and only if (i) a ≥ 1 out of the m events are active, (ii) each active event is internally complete, and (iii) the network is minimally connected. a-ICA A network is an a-internally complete autarky if (i) a ≥ 1 out of the m events are active, (ii) each active event is internally complete, and (iii) there are no external subscriptions in the network such that all the agents whose internal events are inactive are themselves inactive. The a-ICMC networks are the only connected networks that satisfy minimality and internal completeness. The a-ICA networks are the only disconnected non-empty networks which satisfy

20

2-ICA

1-ICA

1 s

A c

4 N

B 4

1 s

s 2

s 3

N 5

N 6

s 2

1 s

A c

4 N

B 4

1 s

s 2

s 3

N 5

N 6

2-ICMC

1-ICMC

s 2

A 4 c N @ @ s @N 3 5

B 4 N 6

A 4 B c 4 HH N @ H @ HH s @N HN 3 5 6

Figure 5: Examples of a-ICMC and a-ICA networks. Agents 1 to 3 belong to group A and agents 4 to 6 belong to group B.

minimality and internal completeness and involve no external subscriptions. Note that the a-ICMC and a-ICA architectures have been defined above in a way such that the empty network is not a special case of any of these networks. A 1-ICMC network is equivalent to what we have referred to as a 1-complete network in the preceding sections. Figure 5 illustrates these architectures for the case with two events and six agents. The next result characterizes the stable architectures as a function of internal and external subscription costs. Proposition 7. Efficient networks are (a) empty if cL ≥ n ¯ − 1, cH ≥

n(n−1) m−1

−

n m−1 cL

and cH ≥

m(n−1) m−1

−

1 m−1 cL ;

(b) m-ICA if cL ≤ n ¯ − 1 and cH ≥ n¯ n; (c) a-ICA if cL = n ¯ − 1 and cH ≥ n¯ n; (e) a-ICMC if (f ) 1-ICMC if

n(n−1) m−1

n m−1 cL and cH n ¯ cL ≤ m(¯ n − 1) and cH = n¯ −1 cL ; and m(n−1) n ¯ 1 cH ≤ n¯ −1 cL and cH ≤ m−1 − m−1 cL .

(d) m-ICMC if cH ≤ n¯ n, cH ≤

−

≥

n ¯ n ¯ −1 cL ;

Proof. In the Appendix. The first three parts refer to cases where the efficient architecture is disconnected while the last three refer to cases where the efficient architecture is connected. The empty network is efficient when both internal and external subscription costs are sufficiently high. Efficient networks are autarkic when external subscription costs are sufficiently high and internal costs are sufficiently low. Minimally connected networks are efficient when both internal and external subscription costs are sufficiently low. We next turn to stability. Proposition 8. Stable networks are either empty, a-ICA or a-ICMC. Further, (a) the empty network is always stable; and, it is the unique stable network if cH > n − 1 and 21

cL > n ¯ − 1. (b) Non-empty stable networks are a-ICMC if and only if cH ≤ n − 1. (c) Non-empty stable networks are a-ICA if and only if cH ≥ n ¯ and cL ≤ n ¯ − 1. Proof. In the Appendix. The effect of internal and external subscription costs on the structure of non-empty stable architectures is quite intuitive. Part (a) states that the empty network is the unique stable network when both internal and external costs are sufficiently high.10 Part (b) states that when the costs of both internal and external subscriptions are sufficiently low, then ICMC networks involving both internal and external subscriptions can be stable. Part (c) suggests that the ICA networks involving only internal subscriptions are stable when the cost of internal subscription is sufficiently low but the cost of external subscription is sufficiently high. It also highlights the main impact of heterogeneity by showing that, in contrast to the previous models, non-empty stable networks can be disconnected.11 The leftmost and the rightmost panels in Figure 6 illustrate the efficient and the stable architectures, respectively. The line in the middle highlights that the sets of efficient and stable networks have an empty intersection if cL < n ¯ − 1 and cH ∈ (n − 1, n¯ n). The following corollary follows immediately from this observation. Corollary 3. Stable networks can be inefficient. Efficient networks can be unstable. The first part of the corollary holds in all the previous models. The second part highlights the impact of heterogeneity. While every efficient network is stable in the models considered in the preceding sections, this is no longer true with heterogeneity. As an example, consider the 2-ICA network in Figure 5. Proposition 8 implies it is the unique stable architecture if cL < n ¯−1 = 2 and cH > n − 1 = 5. Proposition 7 implies that, given cL < 2, the 2-ICMC network architecture is the unique efficient architecture if cH ∈ (3, 18). Thus, for all cH ∈ (5, 18), every stable network is inefficient and every efficient network is unstable.12 10 Specifically, the conditions are cL > n ¯ − 1 and cH > n − 1. The former says that an internal subscriptions costs more than the benefit derived from connections with the remaining agents belonging to one’s own group. The latter says that an external subscription costs more than the benefit from being connected with all agents in the economy. 11

This result also relates to the notion of homophily which broadly refers to the idea that individuals are often observed to be friends with agents of their own ‘type.’ Homophily may be driven by subjective preferences and/or objective constraints ( Feld (1982), McPherson (1983), McPherson et al. (2001)). In the context of our model, suppose co-membership in events provides the opportunity for friendships to develop. The ICA networks would lead to high levels of homophily as each agent will have the opportunity to only befriend agents belonging to one’s own group. The a-ICMC networks will lead to increasing levels of homophily as a increases. For example, in a 1-ICMC network each agent will have the opportunity to befriend the entire population but the opportunity to do so will be severely limited in an m-ICMC network. 12

The efficient 2-ICMC network is unstable because establishing the link 5A, though socially desirable, is not in the private interest of agent 5. The marginal social benefit of link 5A is 18, the private marginal benefit to agent 5 is 3, and the (private and social) cost is cH > 3.

22

Stable networks

Efficient networks

cH

cH

6

45o

n¯ n

Empty

B B

450

6

B B B n m-ICMC BPP

P i PP

P 1-ICMC I @

@ @

a-ICMC



n ¯ -1 m(¯ n-1)

a-ICA

n-1

n-1 n ¯

-

cL

Empty only

a-ICA

m-ICMC

n¯ n

m-ICA 6  9 y X X a-ICA

cH

c
a-ICA I @ P i PP @ P  a-ICMC     ) 

n ¯ -1

-

cL

Figure 6: Summary of Proposition 7 and Proposition 8

In summary, heterogeneity has two main impacts. First, non-empty disconnected networks can be efficient or stable. A non-empty disconnected efficient network does not contain inactive agents but a disconnected stable network may contain inactive agents. Second, the tension between efficiency and stability is quite severe. In particular, there exists a range of parameters such that all the stable networks are inefficient and all efficient networks are unstable. In all models considered in this paper, agents do not take into account the externalities generated by their private subscription decisions . Yet, the tension is very weak in some cases but quite acute in the presence of heterogeneity. The following section provides a unified perspective on why this is the case.

6

Efficiency versus Stability

Consider any strategic network formation model and let E denote the finite set of networks that are feasible. For any agent i ∈ N , let %i (ω) ⊆ E × E be a binary preference relation defined over E which may depend on ω – the ‘parameters’ of the underlying model. What qualifies as the ‘parameters’ will depend on the specific model. For example, a specific ω corresponds to a tuple of (n, m, θ) in the symmetric agent-event model analyzed in Section 3. A specific ω corresponds to a tuple of (n, c, δ, π) in the symmetric connections model of Jackson and Wolinsky (1996) where n is the finite number of agents, c is the cost of forming a link, δ captures informational decay with distance, and the function π specifies the value derived by an agent under each feasible network. In a strategic and cooperative context as in Jackson and Wolinsky (1996), a specific ω is a tuple of (n, v, Y ), where n is the number of agents, v : E → R is a network value function and Y : E × V → Rn is an allocation rule.

23

6.1

Common peak and conflict of interest

Given any parameter ω in a finite network formation model, each agent has weakly ordered preferences over the set of feasible networks E. Let Ei∗ (ω) ⊆ E be the set of preference-maximal networks for agent i at ω such that for all e ∈ Ei∗ (ω), e %i (ω) e0 for all e0 ∈ E. An element of Ei∗ (ω) will be referred to as a peak of %i at ω. Let E ∗ (ω) be the set of networks which are preference-maximal T ∗ for every agent at ω. This set of common peak networks at ω is thus E ∗ (ω) = i∈N Ei (ω). Each network in E ∗ (ω) will be termed a common peak at ω. Note that Ei∗ (ω) will be non-empty at every ω but E ∗ (ω) may be empty at none, some, or all feasible ω. We say that conflict of interest between agents exists at ω if no feasible network is simultaneously preference-maximal for all agents at ω, i.e., if E ∗ (ω) = ∅. By contrast, there is no conflict of interest between agents at ω if a common peak exists at ω, i.e., if E ∗ (ω) is non-empty. If a common peak exists at ω, then every network in E ∗ (ω) will be efficient and stable at ω.13 Consequently, if E ∗ (ω) is non-empty at every ω in a network formation model, then the tension between efficiency and stability will be quite weak. In particular, if there is no conflict of interest between agents at any ω, then every efficient network will be stable.

6.2

Common peak in network formation models

In the following, we use the notion of common peak and conflict of interest to discuss the tension between stability and efficiency. We first note that a common peak exists at all parameter values in some prominent models in the agent-agent framework. For instance, in the one-way flow model without decay of Bala and Goyal (2000a), the preference-maximal network for every agent is a wheel or the empty network. In other agent-agent models, a common peak exists at some but not all parameter values. For example, in the symmetric connections model of Jackson and Wolinsky (1996), every agent prefers the complete network (empty network) over all the other networks when the link cost is lower (higher) than a threshold; however, over an intermediate range of link costs, a common peak does not exist since every agent prefers being a peripheral rather than the hub in a star network. Similar situation also occurs in the two-way model with decay of Bala and Goyal (2000b). Consider the symmetric model analyzed in Section 3. The maximum value any agent can derive at any ω = (n, m, θ) across all feasible networks is max{θ(n − 1, 0, . . . , 0, 1), 0}. Each agent obtains θ(n − 1, 0, . . . , 0, 1) in every 1-complete network, and zero in the empty network. The set of common peak networks, E ∗ , contains only 1-complete networks, only the empty network, or both depending upon whether θ(n − 1, 0, . . . , 0, 1) is strictly greater than, strictly less than, or equal to zero, respectively. Consequently, there is no conflict of interest between agents at any ω and thus every efficient network is stable. The same holds in the model with unreliable events analyzed in Section 4. Given any ω, the efficient network delivers the maximum feasible private value to each 13

Here, stability may be defined with respect to unilateral, pairwise, or coalitional deviations depending on the specific model. Similarly, efficiency may refer to either utilitarian or Pareto efficiency.

24

agent and ensures there is no conflict of interest between agents at any ω. In the absence of conflict of interest, the existence of stable but inefficient networks reflects purely the coordination failure among agents. Under similar assumptions – symmetry, informational decay, connection monotonicity and link monotonicity – conflict of interest exists in models of agent-agent networks (Jackson and Wolinsky, 1996; Bala and Goyal, 2000a) but not in our model of agent-event networks. Thus it seems that events can reduce the conflict of interest between agents and thereby weaken the tension between efficiency and stability by serving as a coordinating device14 .

6.3

The common peak condition and asymmetry

Consider an asymmetric variant of the model analyzed in Section 3. Suppose the value functions can differ across agents but the value function for each agent satisfies the four properties NOR, SCM, SSM, and SDM. At any given ω = (n, m, θ1 , θ2 , . . . , θn ), each agent will derive the maximum private value in a 1-complete network and/or the empty network. If there exists an ω ˆ at which some agents obtain a strictly higher private value under a 1-complete than under the empty network while the reverse holds for some other agents, then E ∗ (ˆ ω ) will be empty. Consequently, the efficient network at ω ˆ can be unstable. Consider the insider-outsider model analyzed in Section 5. When cL < n ¯ −1 and cH ∈ (n−1, n¯ n), the m-ICMC networks are efficient but unstable. Efficiency demands that some agents take up the role of connectors. However, each agent prefers a network where some other agent(s) serves as the connector over a network in which he is the connector. This conflict of interest between agents precludes the existence of a common peak network and undermines the stability of the efficient network. While heterogeneity is likely to intensify the tension between efficiency and stability, it is useful to note that symmetric value functions are neither necessary nor sufficient for overcoming the tension between efficiency and stability. Since value functions simply serve as a device to represent the preference ordering of subjects over the set of feasible networks, if agents have different value functions but a common peak exists at all parameter values then efficient networks will be stable. Jackson and Wolinsky (1996) show that rules which allocate the network value across agents in a fixed proportion can eliminate the tension between efficiency and stability (under some additional mild restrictions). Given any network value function and a proportional allocation rule, the preference ordering of all agents over the set of feasible networks will be identical and coincide with the weak ordering of networks in terms of the total value they generate. Such rules imply the existence 14

Bala and Goyal (2000a) point out that the two-way flow model with decay has a weaker tension between efficiency and stability than the symmetric connections model in Jackson and Wolinsky (1996). In the symmetric connections model, where both agents must incur costs to establish a direct link, the hub bears half of the total cost. In Bala and Goyal (2000a) an agent can form a link unilaterally. Hence, the burden on the hub is significantly lower in star network where the peripherals pay the cost of establishing links with the hub. However, conflict still exists because the peripherals ignore the positive externalities on the hub. Events effectively push this idea further by serving as a non-strategic ‘hub’ for all the strategic agents who are akin to peripherals.

25

of a common peak while the converse need not be true.

6.4

The common peak condition: Sufficiency vs necessity

In the following, we present an example to highlight that the existence of a common peak network is sufficient but not necessary for the stability of efficient networks. Although this can be demonstrated in other ways, we also wish to highlight the effect of relaxing connection monotonicity (SCM) of value functions.15 The basic idea we wish to capture is that the quality of links between agents within an event may deteriorate with an increase in the number of agents within the event. The possibility of congestion implies that an agent may impose positive or negative externalities on other agents. Let φ(ηJ ) be the direct benefit that a subscriber of event J derives from any other subscriber of J, given that the size of J is ηJ . Assume φ(1) = 0 and φ(ηJ ) ∈ (0, 1) for all finite ηJ ≥ 2. To capture congestion, we further assume that φ(ηJ +1) ≤ φ(ηJ ) for all ηJ ∈ {2, . . . , n−1}. In general, each agent subscribed to an event containing ηJ agents will derive a direct benefit of (ηJ − 1)φ(ηJ ). One way to formalize the indirect benefit that an agent can derive from agents belonging to other events is as follows16 . Let τ (ρ) = φ(ηJ1 )φ(ηJ2 ) · · · φ(ηJr ) be the benefit delivered by the path ρ = (i, J1 , i2 , J2 , . . . , ir , Jr , k) connecting agents i and k. Let P (ik; e) be the set of all the paths connecting i and k in network e. Different paths connecting agent i to agent k will provide different benefits to agent i. We assume that two connected agents always communicate via the path that provides the highest benefit. Then, the value derived by agent i under network e will be Πi (e) =

X k∈N

max τ (ρ) − di (e)c

(6.1)

ρ∈P (ik;e)

The main trade-off agents face in this model is between (i) joining few events which contain a low number of agents that provide a small number of high quality direct connections and (ii) joining several events which contain large number of agents that provide a large number of low quality direct connections. To examine this trade-off, let us consider a simple setting with φ(ηJ ) = α(ηJ )β , where α > 0 and β < 0. Further, assume α = 1 and β = −1 such that the benefit derived by an agent from any other agent in the same event is monotonically decreasing in the total number of agents within the event for all ηJ ≥ 2. Suppose there are three agents and three events (i.e., n = m = 3). Clearly, every active event in a stable network must contain at least two agents. Networks with at least two events having the same set of agents are also unstable because subscriptions to at least one of the events will be redundant. This leaves six non-empty architectures that can potentially be stable (see Figure 7). In 15

We have already discussed the impact of distance monotonicity (SDM) in Subsection 3.2. Subscription monotonicity (SSM) seems natural enough in most contexts and thus we do not assess the implications of relaxing SSM. 16

Bloch and Dutta (2009) and Dero¨ıan (2009) use a similar formulation to investigate the tension between efficiency and stability in a network formation game where the strength of a link is endogenously determined by the investments of agents.

26

A c @ @

e1

s 1

e4

B c

C c e2

@

@s

s

2

3

1

A B C c c c HH @ @ @HHH@ @ HH @ s @s H @s 1

2

A B C c c c HH @ H @ HH @ H HHs @s s

3

e5

2

3

A c @ @

B C c c @ @ @  @  @s s  @s

1

e3

2

3

e6

A c @ @

B c @ @

C c

s

@ @s

@ @s

1

2

3

A B C c c c HH @ @HHH @ HH s @s Hs 1

2

3

Figure 7: List of architectures in the model with congestion

fact, we can focus on architectures e2 , e3 , e4 and e5 because networks of the form e1 and e6 cannot be efficient or stable. Network e6 is not stable because the subscription 2A provides no additional benefit to agent 2 but involves a cost. Since this subscription also imposes negative externalities on agents 1 and 3, network e6 cannot be efficient. Network e1 is unstable because an efficient or stable network which contains only one active event must be 1-complete. This is because the value derived by each active agent in any network which contains only one active event increases with the number of active agents. Efficiency strikes a balance between two conflicting goals: reducing congestion versus reducing total subscription costs, while trying to maintain connectedness between agents. As c increases, reducing the number of subscriptions becomes relatively more important for efficiency. Consequently, as c increases, the efficient network changes from a directly connected network (e5 ) to a minimally connected network (e3 ), and then from a 1-complete network (e2 ) to the empty network. Figure 8 reports the efficient networks above the cost line and the stable networks below the cost line. This example allows us to highlight several points. When connection monotonicity does not hold, then both positive and negative externalities contribute to the tension between efficiency and stability. Further, stable and efficient architectures exhibit a rich variety even though events are perfectly reliable and all agents and events are ex-ante identical. In particular, note that efficient networks are symmetric in the models where agents are ex-ante symmetric. Efficient networks are not necessarily symmetric in the insider-outsider model where we assumed exogenous heterogeneities. In the absence of connection monotonicity, efficient networks over some parameter ranges display endogenous heterogeneity. However, the resulting heterogeneity does not necessarily lead to a severe tension between efficiency and stability. In this example, every efficient network

27

e5

1 12

e5

1 6

e5

e3

e3

e2 -

0

1 4

e2

e2

e2

e4

e3

e3

e3

e5

e4

e5

e5

e2

1 3

e2

1 2

e2

c

2 3

e3

e5 Figure 8: Efficient and non-empty stable networks in the model with congestion (n = m = 3). Efficient (stable) networks are listed above (below) the cost line. The empty network is the unique efficient network and the unique stable network for c > 23 .

is stable even though common peak networks do not exist over some range of parameters.17 For instance, when c ∈ ( 14 , 12 ), the minimally connected network is both efficient and stable. However, it is not the preference-maximal network for any agent since every agent prefers to be a non-connector instead of being the connector. Thus, the existence of common peak networks is sufficient but not necessary to ensure the stability of efficient networks.

7

Concluding Remarks

The present paper provides a tractable non-cooperative model of agent-event network formation. We study the impact of prominent frictions on the tension between efficiency and stability and show that events can often reduce the potential conflict of interest between agents and thereby weaken the tension. Our analysis can be extended in several directions. For instance, a model where some agents endogenously choose to become event-entrepreneurs can be used to study the endogenous supply and equilibrium subscription pricing and differentiation of events.18 Furthermore, in line with Myerson (1980), we have assumed that all agents who subscribe to the same event are directly connected with each other. Future work can explore richer and more diversified patterns of connections among agents who subscribe to the same event. One can consider a model in which agents first determine whether to incur the costs to join a subset of events, with the expectation that in future periods, the general interactions facilitated by connections through events may provide useful information that guides agents’ decisions regarding whether to form ‘stronger’ links with a subset of other agents that can facilitate some richer interactions. A complete analysis of these considerations is beyond the scope of the present work and we leave it for future research. A large body of work on networks is devoted to understanding how network structure shapes 17

There exist inefficient networks that are stable (when c < 12 ), as in all the models considered in the paper.

18

Building up on the work reported in a previous version of our paper, Farinha (2014) explores one way to endogenize events.

28

the incentives of agents and affects behavior. Consequently, developing estimable models which can account for the endogenous formation of networks is an important and active area of research. The literature on agent-agent networks, in particular, has matured significantly and is beginning to offer concrete guidance for empirical work and generate policy insights. As highlighted by Jackson (2014), this maturity is reflected in two main developments. First, there exists a sufficiently nuanced understanding of how macro and micro measures of key network properties such as density, clustering, centrality and the lowest eigenvalue of the network affect a wide range of important economic behavior (see, for example, Banerjee et al. (2013), Ballester et al. (2006, 2010), and Bramoull´e et al. (2014)). Second, the last few years have witnessed a series of works that combine random and strategic considerations in judicious ways to come up with estimable agent-agent network formation models (Chandrasekhar and Jackson, 2014). Although there exists a sizeable literature on agent-event networks in other disciplines, the main difference in comparison with the literature on agent-agent networks is the absence of strategic considerations.19 A key reason for the interest in understanding agent-event networks is that the data about inter-connections between agents are often available in the agent-event form (Agneessens and Everett, 2013). If data needs to be collected, then as pointed out by Borgatti and Halgin (2011), the data about agent-event network structure often can be gathered “from a distance, without having to have special access to the actors”; in contrast, it is often difficult to gather data about agent-agent networks without actually surveying the agents. The growing interest in empirical applications of agent-event networks has been complemented by theoretical developments. Measures of micro and macro level properties have been proposed and utilized in empirical work (Latapy et al., 2008) and modeling their formation as a random process is a particularly active area of research (Wang et al., 2013). We believe tractable models that incorporate strategic considerations can add momentum to the development of the agent-event networks literature.

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A

Appendix

A.1

Proof of Proposition 6

Let |Cl | be the size of the component Cl . We say Cl is trivial if and only if Cl = 1. We omit the proof of part (a). We now prove part (b) and then part (c) of the proposition. Part (b). If the empty network is efficient or stable, then internal completeness is trivially satisfied. Let e be any non-empty stable and efficient network. Note that e must be minimal by part (a). Let J be an active event e. Suppose J is not internally complete. Then there exists an agent i ∈ NJ such that eJi = 0. The proof considers two mutually exclusive and exhaustive cases – either agent i and event J belong to the same component or to different components under e – and establishes a contradiction. Case 1, suppose agent i and event J belong to the same component under e. As i does not subscribe to J, there must exist another event J 0 6= J which lies on the unique path connecting agent i and event J, say, (J, . . . , J 0 , i). If i unsubscribes from event J 0 and subscribes to event J, the connectedness of the network cannot be affected but the subscription costs of agent i will reduce by (cH − cL ) > 0. This is because all the agents who are connected to agent i via event 32

J 0 would remain connected to agent i in the resulting network via the path (i, J, . . . , J 0 ), or a part thereof; and, the connectivity between agents who are connected to agent i via events other than J 0 would remain unaffected. Thus, we have Πi (e + iJ − iJ 0 ) > Πi (e), which contradicts the stability of e. Furthermore, since no agent loses any connection in the transition from network e to network e + iJ − iJ 0 , the payoffs of all agents other than i will remain unaffected. Therefore, Π(e + iJ − iJ 0 ) > Π(e), which contradicts the efficiency of e. Case 2, suppose agent i and event J do not belong to the same component under network e. Let C1 be the component which contains J and C2 be the component which contains i (which may or may not be trivial). The sum of benefits generated by the components C1 and C2 will be |C1 |(|C1 | − 1) + |C2 |(|C2 | − 1) where |Cl | is the size of component l ∈ {1, 2}. Since e is efficient, the total benefit generated by component C1 must exceed the lowest total cost required to construct this component which is |C1 |cL . Thus, we must have |C1 |−1 ≥ cL . If agent i subscribes to event J, then the two components will be connected in the resulting network e+iJ. The total benefit generated by the resulting component will be (|C1 |+|C2 |)(|C1 |+|C2 |−1). It will generate a higher value than C1 and C2 if and only if (|C1 |+|C2 |)(|C1 |+|C2 |−1)−|C1 |(|C1 |−1)−|C2 |(|C2 |−1) ≥ cL , or if 2|C1 ||C2 | ≥ cL . Clearly, 2|C1 ||C2 | ≥ 2|C1 | > |C1 |−1 ≥ cL . Thus, connecting C1 and C2 leads to a strict increase in total value since all the other components in e cannot be affected by the additional subscription by i to J. Hence, Π(e + iJ) > Π(e) which contradicts the efficiency of e. On there the hand, observe the maximum feasible value an agent in component C1 can derive is |C1 | − 1 − cL . Stability of e implies it must be positive. If agent i subscribes to event J, then he can gain connections to all the agents in C1 at a cost of cL . Since Πi (e+iJ)−Πi (e) = |C1 |−cL > |C1 |−1−cL ≥ 0, e cannot be stable. Part (c). Let e be any non-empty efficient network which is disconnected. We first show that every active agent in e subscribes to her internal event. This, in combine with parts (a) and (b), allows us to claim that every component in e is itself an m-ICMC network with m now refers to the number of events in the component. Then we will show every active agent in e subscribes only to her internal event, which is equivalent to saying that there is no external subscription in e. First, suppose there exists an active agent i who does not subscribe to her internal event (or subscribes only to external events) and let C1 be the non-trivial component in e which contains i. Consider the network which removes all the subscriptions due to i. If di (e) = 1, then the change of total value due to the removal of i is given by cH − 2(|C1 | − 1). If di (e) > 1, then by minimality the removal of i will break C1 into di (e) components in the resulting network where di (e) is the number of external subscriptions by i. However, those di (e) components can potentially be reconnected by di (e) − 1 external subscriptions initiated by any of the other agents in C1 . Thus the change of the total value from e to e − i under such a situation is also cH − 2(|C1 | − 1). Efficiency will require this expression to be negative which implies cH < 2(|C1 | − 1). Since e is disconnected, there must exist another component C2 in e. Allowing an agent from C2 to subscribe to an event in C1 will incur a cost which is at most cH and generate an additional total benefit of 2|C1 ||C2 |. 33

Since 2|C1 ||C2 | > 2(|C1 | − 1) ≥ cH , such a move will always result a network with a higher value and hence violates the efficiency of e. Thus every active agent in e subscribes to her internal event. Second, let C1 be a non-trivial component which contains some external subscriptions. The conclusion we have made earlier, in combine with parts (a) and (b), dictate that C1 is minimally internally complete and contains exactly a − 1 external subscriptions where a is the number of events in C1 . We now consider two exhaustive cases. Case 1. Suppose, except C1 , there exists another component C2 in e which is also non-trivial. Since the number of events is less than the number of external subscriptions, there exists at least one event, say J, such that it contains at most one agent who has subscribed to more that one event. Let i be the only agent who has subscribed to J and does not belong to NJ . (It is also possible that i belongs to NJ and subscribes to another external event, say J 0 , which connects J to other events in C1 . If this is the case, then replace iJ with iJ 0 will lead to the same conclusion.) Denote e − iJ by e0 . Let C11 and C12 (where C12 contains only the internally complete event J) be the two components obtained by C1 due to the removal of iJ. Then the values of C1 , C11 and C12 can be written as a¯ n[a¯ n − 1] − a¯ ncL − (a − 1)cH , (a − 1)¯ n[(a − 1)¯ n − 1] − (a − 1)¯ ncL − (a − 2)cH and n ¯ (¯ n − 1) − n ¯ cL respectively. Since e is efficient, the value of C1 cannot be lower than the sum of the values of C11 and C12 . Simplifying Π(e) ≥ Π(e0 ) will give us 2a¯ n2 − 2¯ n2 ≥ cH . Similarly, construct another network e00 which is same as e except that e00 contains an extra external subscription which connects C1 and C2 . Then the value of the component (which contains all the agents in C1 and C2 ) will be given by (a+b)¯ n[(a+b)¯ n −1]−(a+b)¯ ncL −(a+b−1)cH where b is the number of events in C2 . Simplifying Π(e00 ) > Π(e) will give us 2ab¯ n2 > cH . Since 2ab¯ n2 ≥ 2a¯ n2 > 2a¯ n2 − 2¯ n2 ≥ cH , we have Π(e00 ) > Π(e) which contradicts the efficiency of e. Case 2. Suppose, except C1 , there does not exist another non-trivial component in e. Consider a network e0 which is same as e except that e0 contains only a − 1 active events and a − 2 external subscriptions. That is e0 has n ¯ more inactive agents than e. Since all the other components in e and e0 are trivial and has no value, the total value derived from e and e0 will be given by a¯ n(a¯ n − 1) − a¯ ncL − (a − 1)cH and (a − 1)¯ n[(a − 1)¯ n − 1] − (a − 1)¯ ncL − (a − 2)cH respectively. Since e is efficient, we must have Π(e) ≥ Π(e0 ). Simplifying will give us 2a¯ n2 − n ¯ (¯ n + 1) ≥ n ¯ cL + cH . Construct another network e00 which is same as e except that its non-trivial component contains a + 1 active events and a external subscriptions. (Such a network is always feasible because we have proven earlier that C1 does not contain any agent whose internal event is not in C1 . Thus there must exist a group of agents in which all of them are inactive.) Then the value of e00 will be given by (a + 1)¯ n[(a + 1)¯ n − 1] − (a + 1)¯ ncL − acH . Simplifying Π(e00 ) > Π(e) will give us 2a¯ n2 + n ¯ (¯ n − 1) > n ¯ cL + cH . Since 2a¯ n2 + n ¯ (¯ n − 1) > 2a¯ n2 − n ¯ (¯ n + 1) ≥ n ¯ cL + cH , we have Π(e00 ) > Π(e) which contradicts the efficiency of e. Thus non-trivial components cannot contain any external subscription. We now turn to stability. Let e be any non-empty stable network. Suppose e is disconnected and contain some external subscriptions in it. Let i be an agent in e who subscribes to an event J externally. Suppose C1 is the component which contains i and J. Then the value that i can 34

derive from C1 cannot exceed |C1 | − 1 − cH . By the stability of e, we must have |C1 | − 1 ≥ cH otherwise i would not have subscribed to J. Let k be any agent who is not contained in C1 . If k subscribes to J, then his total benefit will be increased by |C1 | with a cost of at most cH . Since |C1 | > |C1 | − 1 ≥ cH , we have Πk (e + kJ) > Πk (e), which contradicts the stability of e.

A.2

Proof of Proposition 7

From Proposition 6 we have already known that disconnected efficient networks are a-ICA and connected efficient networks are a-ICMC. We will now further prove that (i) if an efficient network is a-ICA, then a = m except for cL = n ¯ ; and (ii) if an efficient network is a-ICMC, then a = 1 or a = m except for

cH cL

=

n ¯ n ¯ −1 .

First, let e be an a-ICA network which is also efficient. Suppose not all the events in e are active. Let J be an inactive event in e. The efficiency of e requires n ¯ (¯ n − 1) ≥ n ¯ cL . Allow all the agents belong to NJ to subscribe to J will increase the value of the network by n ¯ (¯ n − 1) − n ¯ cL , which violates the efficiency of e unless cL = n ¯ − 1. Therefore, every event in e must be active which implies e is m-ICA except for cL = n ¯ − 1. Second, let e be an a-ICMC network which is also efficient. Observe that the total benefit derived from a connected network always equals to n(n − 1), thus we can restrict our attention only to subscription costs. Consider the total cost function for a-ICMC networks a¯ ncL + (n − a¯ n)cH + (a − 1)cH It is straightforward to show that the above function is strictly increasing in a if decreasing in a if

cH cL

>

n ¯ n ¯ −1 ,

and invariant to a if

dominates all other a-ICMC architectures if other a-ICMC architectures if value if

cH cL

=

n ¯ n ¯ −1 .

cH cL

>

n ¯ n ¯ −1 ,

cH cL

cH cL

<

=

n ¯ n ¯ −1 ,

n ¯ n ¯ −1 .

cH cL

<

n ¯ n ¯ −1 ,

strictly

Therefore, the 1-ICMC architecture

the m-ICMC architecture dominates all

and all the a-ICMC architectures generate the same total

Hence e has to be either 1-ICMC or m-ICMC except for

cH cL

=

n ¯ n ¯ −1 .

We have shown that an efficient network is either m-ICA, 1-ICMC or m-ICMC except for the two special cases (i.e. cL = n ¯ and

cH cL

=

n ¯ n ¯ −1 ).

Observe that the total value of the four networks

are given by 0, n(¯ n − 1) − ncL , n(n − 1) − [(n − n ¯ )cH + n ¯ cL ] and n(n − 1) − [(m − 1)cH + ncL ]. Solving these four values will give us the result.

A.3

Proof of Proposition 8

From Proposition 6 we have already known that disconnected stable networks are a-ICA and connected stable networks are a-ICMC. Let e be any stable network, we now prove parts (a), (b) and (c) one by one. (a) When cH is higher than n − 1, then no external subscription will be profitable. Thus e cannot be a-ICMC. When cL is higher than n ¯ − 1, no internal subscription will be profitable unless the event has a size which is larger than n ¯ (i.e. there exists some external subscriptions from agents who belong to other groups). Since we have shown that no external subscription is present given cH > n − 1, e cannot be a-ICA either. (b) When cH is lower than 35

n − 1, the 1-ICMC networks which is a subset of a-ICMC networks are stable. On the other hand, when cH > n − 1, no external subscription can be profitable and hence e cannot be a-ICMC. Thus a-ICMC networks are stable if and only if when cH ≤ n − 1. (c) Let J be an active event in e. When cH < n ¯ , for any agent i who is not connected to J, subscribing to J will yield a benefit of at least n ¯ with a cost of cH < n ¯ . Thus e cannot be a-ICA if cH < n ¯ . Now suppose e is a-ICA, if cH ≥ n ¯ , then no additional subscription will be profitable. If cL ≤ n ¯ − 1, then no agent will have a strict incentive to unsubscribe from her internal event. Thus if e is a-ICA then cH ≥ n ¯ and cL ≤ n ¯ − 1.

36