MAKING FRIENDS MEET – NETWORK FORMATION WITH INTRODUCTIONS JAN-PETER SIEDLAREK A BSTRACT. Economic and social networks often combine high levels of clustering with short distances between pairs of agents. I propose a model of network formation with intermediaries and introductions, which is capable to explain both these features. Introductions, in which a common neighbour creates new connections between two players, are subject to a simple trade-off of gains from lower search cost and losses from lower intermediation rents for the central agents. Myopically stable networks are shown to have small world properties with a minimum level of clustering and limited network diameter. Analysing introductions from a forward looking perspective in a game of link formation, I identify conditions for a bilateral equilibrium in which efficient network structures are achieved. JEL Classification: D85 Keywords: network formation; intermediation; stability

Date: 23 October 2013. Department of Economics, University of Mannheim, Mannheim, Germany. I am grateful to Fernando Vega-Redondo for extensive discussions and advice. Thanks also to Peter Vida and seminar participants at the European University Institute, University of Mannheim, PET2013 Lisbon and the EEA2013 Goteborg for helpful comments and suggestions. All remaining errors are mine. 1

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1. I NTRODUCTION The structure of relationships between economic actors performs an important role in a variety of social and economic contexts, influencing the opportunities to cooperate, to trade and to interact. The size and structure of such networks has been shown to influence phenomena as diverse as finding a job, being promoted within a firm, R&D success and others.1 Given the importance of connections, the genesis of network structure – the question which relationships arise and how they come about – is of interest to economists and social scientist more generally. Two prominent features that are often observed in real-world networks are high levels of clustering and short distances, often created by agents bridging otherwise unconnected parts of the network. The first feature, clustering, describes a tendency for pairs of nodes to share common neighbours. In many real-world networks studied clustering is found to exceed levels expected from random link formation processes. For example, studies of co-authorship networks, which record collaborations between researchers, exhibit a very high probability that a given pair of cooperating scientists also both cooperate with a common third researcher (see Newman (2003) , Grossman (2002) and Goyal et al. (2006)). Similar features are observed for the cooperation of actors in movies (Watts, 2003) and web sites (Adamic, 1999). The second feature, bridging, is often discussed under the heading of “structural holes” referring to the seminal work of Burt (1992) who studies the phenomenon in the context of managerial networks in companies. It documents how agents connecting others that are otherwise unconnected can earn a return for such intermediation services and thus lead agents to create networks in which they fill structural holes and create bridges between otherwise unconnected agents.2 This paper proposes a specific network formation explanation for how structures with both bridging players with intermediation returns and high levels of clustering might arise. It starts from the observation that the conditions under which new links are formed depends on the existing network. New connections are mediated by the existing network. Specifically I study the impact of networks facilitating the formation of links between 1

See Jackson (2008) for an overview of network theory in economics and applications. For work on the resulting dynamics of jockeying for positions see Goyal and Vega-Redondo (2007) as well as Kleinberg et al. (2008). 2

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two nodes that have an existing connection to a common neighbour. This process is known as triadic closure in sociology and it has been observed to be of relevance in various applications: for example, in a Mayer and Puller (2008) a common neighbour turns out to be a strong predictor for new links in the social networks of a university. Here, I consider a specific form of triadic closure which I label “introduction”. Two parties which share a common neighbour can be invited to form a link by that neighbour. The cost for such a connection is lower than if the two parties were to seek each other out without assistance. A familiar manifestation of such a process is provided by the colloquial use of the term “networking” in friendship and business relationships. This often takes the form of social introductions in which a common acquaintance brings together two other parties. Similar processes in which the three parties involved are active can be seen in the formation of business relationships between firms and other organisations. For example, Uzzi (1996) in his study of the apparel industry in New York describes how new relationships between two parties often result from referrals by common business partners: “[...] ties primarily develop out of third-party referral networks and previous personal relations. In these cases, one actor with an embedded tie to two unconnected actors acts as their ‘go-between.’” (Uzzi, 1996, p. 48) The introduction concept thus combines a tendency for clustering with explicit consideration for the incentives arising from intermediation, generating a distinct trade-off: cost benefits from introductions create a tendency for link creation, whilst introducers may see intermediation rents threatened by the additional connections. In other words, introductions involve a classic trade-off between growing the pie and protecting one’s slice. The paper shows how based on this trade-off the introductions mechanic can explain both high clustering and bridges. In summary, this paper studies strategic network formation with introductions. The interplay of benefits from intermediation and cost advantages of introductions is shown to explain the coexistence of high clustering and structural holes bridging otherwise unconnected groups, thereby offering a simple joint explanation for two prominent features in real-world networks. The paper is structured as follows. Section 2 briefly discusses related literature. Section 3 presents the main model and payoff structures. Using this model, Sections 4 and 5

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characterise efficient networks and stable networks, respectively. Section 6 presents a dynamic perspective on introductions, studying a game of link formation. Finally, Section 7 concludes. 2. L ITERATURE C ONTEXT Before presenting the main model, this section briefly places the paper in the context of the existing literature. The paper provides a contribution to the literature on network formation and also in a wider sense to the literature on intermediation. The classic literature on network formation can usefully be grouped into two main categories: (i) random network formation and (ii) strategic network formation. Whilst the first approach analyses the outcome of an exogenous stochastic process of link creation aiming to explain observed features of real-world networks, the second explicitly studies the incentives of agents to form links amongst themselves. Network formation is then the outcome of individual payoff maximisation by agents. As such, the present paper falls into the second category, although I will in places refer to work from the random network formation tradition. I focus here on the work most closely related to the model studied here. The reader is referred to the comprehensive overview provided in Jackson (2008). The distinguishing features of the model relate to the introduction mechanic and the compensation of intermediate agents. Here, parts of the model overlap with some key papers in the existing literature. First, the nature of the introduction process implies that the ability to form new links is mediated by the structure of the existing network. Such network-based link formation has been little explored in the literature with Jackson and Rogers (2007) being a notable exception. There, the authors use a random network formation setup with a growing network in which new nodes first connect to a set of randomly chosen existing nodes and in a second process may connect to neighbours of those, i.e. connect to friends of friends. Such connections created by the second process are similar to those formed in an introduction as studied here. However, in contrast to this paper, Jackson and Rogers (2007) treat link creation as random ignoring the incentives for the participating players in creating the link. In this paper, I explicitly consider these incentives and in particular focus on the intermediate players that are key to introductions. Second, the introduction mechanic inherently requires a mechanism to facilitate compensation of the intermediating agent. As such, transfers and their ability to overcome

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externalities play a key role. The seminal contribution in this area is Currarini and Morelli (2000), who study a sequential setup with transfers and find that with relatively few restrictions on payoffs, efficient networks are formed in equilibrium. A generic analysis of various forms of transfers and their ability to implement efficiency in simultaneous network formation is found in Bloch and Jackson (2007). Third, the model analysed in this paper considers link creation from the perspective of players positioned in between others and in a position to provide introductions. Thus, the focus lies on the incentives for the intermediate players and their rents for intermediation. The potential benefits accruing to players in specific positions crucial to the network are discussed extensively by Burt (1992). There, the author investigates the rents available to individuals bridging so-called “structural holes” and the dynamics of jockeying for the positions required to access these rents. In the economics literature, a model which discusses issues most closely related to this paper is provided by Goyal and Vega-Redondo (2007). There, the authors consider network formation in the presence of intermediation benefits and analyse the interplay of three motivations: (i) access to the network, (ii) benefits from intermediation, and (iii) avoidance of sharing benefits with intermediaries. They find that in the absence of capacity constraints a star emerges, in which a single agent acts as intermediary for all transactions, receiving significant intermediation rents. Contrary to this paper, in their model, Goyal and Vega-Redondo (2007) focus on direct link creation and do not implement an introduction mechanic. A similar model is also explored in Kleinberg et al. (2008), again without allowing for introductions. 3. T HE M ODEL I study introductions in a variation of the canonical symmetric connections model proposed by Jackson and Wolinsky (1996), allowing for intermediaries that facilitate connections to capture a share of the surplus generated. Players are denoted by a finite set N = {1, 2, . . . , n} with n > 3. Players are connected

in a network g ∈ G characterised by the set of links L with typical element ij representing a link between players i and j. Players’ payoffs from connections in the network are a

combination of (i) benefits from direct connections, (ii) intermediation rents and (iii) link costs. I consider the case where players derive payoffs from each connection to another player, which may be direct or indirect, with payoffs from indirect connections decaying

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with geodesic distance between nodes.3 Formally, total benefit b(`(i, j; g)) generated by the connection between any pair of agents i and j in g is given by the function b:  = 0 if j ∈ / Ci ( g), (1) b(`(i, j; g)) > 0 if j ∈ C ( g). i

Ci ( g) denotes the set of players to which player i is connected in network g (the component of i in g) and `(i, j; g) denotes the length of the geodesic, that is the shortest path, connecting players i and j in g. As in the standard model, there is decay such that b(`) is decreasing in

`. I next turn to the allocation of surplus among the agents engaged. I follow Goyal and Vega-Redondo (2007) in assuming that intermediate agents benefit from their position by capturing a non-zero share of the surplus generated if and only if they are essential to the connection, that is if they are on every shortest path connecting the two trading nodes.4 The set of essential nodes is denoted E(i, j; g) and e(i, j; g) denotes its cardinality, in other words, the number of agents essential to connecting agents i and j. Total surplus is shared evenly between essential agents. Thus, the model allows for intermediaries to capture a share of the benefit if they are essential to the transaction. If there are alternative routes, I assume that intermediation rents are competed away in the spirit of Bertrand competition, which brings into focus the key strategic concern of intermediaries to avoid being circumvented.5 Finally, each link maintained by a given node i also incurs a cost c reflecting the cost of maintaining a relationship. In summary, total payoffs for player i from network g are given by (2)

πi ( g ) =

b(`( j, k; g)) I{i∈E( j,k)} b(`(i, j; g)) + ∑ − ηi ( g ) c 2 + e(i, j; g) j,k∈ N 2 + e( j, k; g) j∈C ( g)

∑ i

3See Bloch and Jackson (2007) for a detailed discussion of this family of models. 4The qualification of considering shortest paths only differs from Goyal and Vega-Redondo (2007). It is

imposed to reflect the notion that there is decay in this model and thus routes of different lengths arguably do not compete with each other. 5The assumption is consistent with a model of bargaining without replacement in the limit when bargaining frictions disappear (Siedlarek, 2012). See Kleinberg et al. (2008) for a different approach in which intermediation benefits decay with the number of alternative paths in a gradual way.

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where ηi ( g) denotes the degree of player i in network g. Profits are the sum of the following components: (a) benefits from direct and indirect connections to other agents; (b) intermediation benefits from indirect connections for which player i is essential; and (c) costs for connections maintained. I use this model to study introductions, a process which is inherently dynamic. Therefore in addition to static payoffs derived from network g itself, the model needs to allow for additional costs from the link formation process itself. Specifically, I distinguish direct link creation and link creation through introduction by an additional search cost d ≥ 0 that each in a pair of agents incurs when they form a direct link without assistance from a third party. Introductions have a cost advantage over search in that this cost does not apply in the case of a link created by introductions. The search costs d do not appear in the static payoff function πi ( g) in equation 2 above, but instead come into play when considering the implications of a transition between different networks as part of a stability analysis (Section 5) or a network formation game (Section 6). In summary, introductions create a simple trade-off between efficiency and distribution concerns in this setting. On the one hand, an introduction is an efficient way of creating connections as it circumvents search costs involved in non-intermediated link formation. On the other hand, links created by introductions can expose the introducing player to circumvention, threatening intermediation payoffs received from being essential. The analysis of this trade-off forms the core of the remainder of the paper, which begins by considering efficient network structures. 4. E FFICIENT N ETWORK F ORMATION WITH I NTRODUCTIONS Efficiency considerations in the model apply both to the process of network formation and the resulting structure of relationships. Generally speaking, introductions are more efficient than non-mediated link creation due to the search cost d. However, the requirement for agents to share a common neighbour before introductions are feasible implies some role for non-mediated link creation. Proposition 4.1 shows the interaction of these forces. The proof is closely related to results in the standard connections model and is provided together with all other proofs in the Appendix.

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Proposition 4.1. The efficient network structure in the model with introductions is: n o n −2 4 (1) The empty network if b(1) − 2c ≤ min 2d − 2 b(2), n d .

(2) The star network composed of non-mediated links and including all n players if d − n −2 2 b (2)

≤ b(1) − 2c ≤ b(2).

(3) The complete network if b(1) − 2c ≥ max

n

4 n d, b (2)

o

. The complete network is com-

posed of a star formed of non-mediated links and all remaining links being created through introduction.

The efficient structure is unique up to a permutation of agents. Here, as in subsequent analysis, the characterisation is based on the term b(1) − 2c,

which captures the net benefits of a direct connection, that is, total benefits less link maintenance costs. Note that Proposition 4.1 implies that a star can only ever be efficient

when maintenance costs c are sufficiently high. For 2c < b(1) − b(2), the efficient network is either empty or complete. Figure 1 illustrates the parameter regions characterised in

Proposition 4.1. Also, as search costs increase relative to maintenance costs c, the efficient network is either complete or empty - once link formation is productive at all, it pays to make maximum use of introductions and form the complete network. d

Empty Network

n 4 b(2)

Complete Network

Star

− n−2 2 b(2)

0

b(2)

b(1) − 2c

F IGURE 1. Efficient Network Configurations

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Note how at d = 0 the efficient structures correspond to those identified in Jackson and Wolinsky (1996).6 As search costs d increase, the area range in which the empty network is efficient grows, reflecting the additional costs to be incurred for a given benefit to be obtained. For low d first only the range for the star network is affected. Once d increases above n4 b(2), the star is never efficient and the range for which the complete network is efficient begins to shrink as well. The results given here provide a benchmark for the subsequent analysis which will approach the incentives for players to form networks from the perspective of stability (Section 5) as well as equilibrium play in an explicit game of link formation (Section 6). 5. S TABILITY A NALYSIS This section presents an analysis of networks stable under introductions. I consider a myopic stability notion employing a suitable extension of pairwise stability (Jackson and Wolinsky, 1996) before exploring dynamic approaches in a subsequent section (Section 6). As in Kleinberg et al. (2008), for this analysis I impose an additional assumption on the benefit function allowing positive payoffs to accrue only up to length two, that is, paths of length three or larger generate zero benefits. The restriction reflects empirical research on benefits from connections in organisations (Burt, 2007, 1992).

(3)

b (1) > b (2) > 0

(4)

b(`) = 0 for all ` ≥ 3

5.1. Myopic Stability with Introductions. This section analyses the strategic incentives for network formation in the connections model with introductions using a version of pairwise stability (Jackson and Wolinsky, 1996) suitably extended to allow for the dynamic introduction process that is the subject of this paper. Note that to study introductions effectively requires the inclusion of transfers to the introducing player in order to compensate that player for potentially lost intermediation benefits. To see this consider the payoff implications of the new link that is created. The new link shortens the distance between the two players being introduced to one step 6With the proviso that in their model benefits of b(`) accrue for each of the agents, whilst in this paper we

have one b(`) to divide.

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yielding the resulting benefits; however it does not generate additional benefits for the introducing player as he is already connected to both. Instead, the new link may result in the introducing player no longer being essential for the connection between the players introduced and thus he may lose out. Introductions by themselves are thus at best payoff neutral for the introducing player and transfers are necessary to make any introduction profitable for the introducer. As a result the stability concept I use here is one that considers deviations with transfers as proposed in Bloch and Jackson (2006) and adapted here to accommodate the introduction process. Definition 5.1 (Myopic stability under introductions with transfers). A network g is myopically stable under introductions if: a. (Destruction) ∀ ij ∈ g, πi ( g) + π j ( g) ≥ πi ( g − ij) + π j ( g − ij)

b. (Search) ∀ ij ∈ / g, πi ( g) + π j ( g) ≥ πi ( g + ij) + π j ( g + ij) − 2d c. (Introduction) ∀ j, k, l such that jk ∈ g, kl ∈ g and jl ∈ / g,

∑

i ∈{ j,k,l }

πi ( g ) ≥

∑

πi ( g + kl )

i ∈{ j,k,l }

The first two conditions correspond to those used for networks that are pairwise stable with transfers as analysed in Bloch and Jackson (2006), adapted for the introductions model. In particular, note that the conditions involve search cost d that is sunk in the sense that it is incurred when creating bilateral links and not recuperated by agents if a link is destroyed. In addition, the definition includes a third condition that requires that in a stable network there are no opportunities for profitable introductions conducted by a triplet of agents which form an open triangle. An introduction does not incur the search cost d. All three conditions allow for transfers between the agents involved by considering the sum of payoffs rather than individual payoffs. This also applies to link destruction in order to maintain symmetry between link creation and link destruction.7 Allowing for transfers in link destruction reduces the set of profitable deviations of this type: all link removals that are jointly profitable necessarily involve at least one agent for which it is unilaterally profitable; however, if one agent loses out from the removal of the link, Definition 5.1 requires that the damage done to the other side involved in the link not be too high. In this 7See footnote 5 in Bloch and Jackson (2006) for a discussion.

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sense the solution concept with transfers is weaker than without as far as link destruction is concerned.8 In the following I consider (i) the stability properties of the efficient configurations (Section 5.2) and (ii) characterise properties of stable networks in general, namely regarding the stability of unused introduction opportunities (Section 5.3) and overall connectedness (Section 5.4). 5.2. Stability of Efficient Networks. Here I briefly consider the stability properties of the configurations which are possible efficient arrangements for some parameter ranges. The analysis starts with one of the possible efficient configurations – the empty network, the star and the complete network – and characterises the parameter restrictions necessary for each configuration to be stable. The derivation has been relegated to Appendix B. The thresholds are illustrated in Figure 2. Table 1 lists the conditions, next to the corresponding thresholds for efficiency derived in Section 4. The conditions imply that for certain parameter ranges multiple configurations can be myopically stable. For example, if surplus decays slowly such that b(1) − 2c is in the interval [ 12 b(2), b(2)] and in addition 1 2

[b(1) − 2c]), then all three configurations are stable. As seen in Section 4, only one of these would be efficient at the same time. Multiplicity of stable networks is a generic feature of the model and a function of both the myopic stability concept and the search cost d. search costs are sufficiently high (d >

Network Empty Star

Efficient

Myopically Stable n

b(1) − 2c ≤ min 2d −

b(1) − 2c ≥ d − b(1) − 2c ≤ b(2)

n −2 4 2 b (2), n d

1 2 ( n − 2) b (2)

Complete b(1) − 2c ≥ max

n

4 n d, b (2)

o

o

b(1) − 2c ≤ 2d

b(1) − 2c ≥ − 32 (n − 2)b(2) b(1) − 2c ≤ b(2) b(1) − 2c ≤ 2d + 32 b(2)

b(1) − 2c ≥ 13 b(2)

TABLE 1. Parameter Ranges for Efficient and Myopically Stable Networks

8Note also that Definition 5.1 includes transfers implicitly by comparing sums of payoffs rather than

specifying explicitly the amounts exchanged between players. The implicit approach is more concise and sufficient for the myopic case. However, the farsighted approach discussed in Appendix C requires us to explicitly keep track of amounts transferred.

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d Empty, Star & Complete

Empty

Empty & Star

Empty & Complete

Complete

− 23 (n − 2)b(2)

0 Star

1 2 b(2)

b(2)

b(1) − 2c Star & Complete

F IGURE 2. Illustration of Parameter Ranges for Stability of Empty, Star and Complete Network 5.3. Clustering. In this section I consider the impact of the possibility for introductions on the structure of myopically stable networks. In particular, I ask which possible introductions will remain unutilised in stable networks. Consider the case with b(1) − 2c > b(2) in which a direct connection generates more

surplus than an indirect one. In this case, any introduction that is conducted is increasing

total surplus from an efficiency perspective and thus should be conducted. I ask to what extent introductions are conducted in myopically stable networks. First, Proposition 5.1 establishes that in a stable network an open introduction necessarily involves an introducing node being essential to the connection between the two nodes that could be connected. Proposition 5.1. Consider a network g that is myopically stable with introductions and includes an unused opportunity for k to introduce i and j. Then k is essential to the connection between i and j. Next, consider the structural characteristics of introduction opportunities with essential introducers may remain unused in the set of stable networks. It can be shown that there is a lower bound on the number of nodes that are linked to both the introducer and exactly one of the nodes being introduced. Figure 3 shows some example configurations to illustrates

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this insight. In all cases, a connection between i and j is possible through introduction by k with k being essential to the connection prior to introduction. l

l

j

j

i

i

i

k

( A ) No neighbours

j

k

( B ) Unshared neighbour

k

( C ) Shared neighbour

F IGURE 3. Configurations with introduction opportunities Consider first the isolated introduction opportunity in Figure 3a, in which the three nodes have no neighbours. Prior to the introduction, the end-nodes i and j receive benefits b (1) 2

from the connection with k and

b (2) 3

from the connection with each other, which is b (1) 2

+ b(32) , from direct links to i and j as well as intermediating the indirect connection between these two. The new link replaces the indirect connection between i and j with a direct one. Total surplus available thus increases. The new link also makes k non-essential for the connection between i and j, implying a loss of the intermediation benefits to k. However, as Definition 5.1 allows for transfers, this loss can be compensated for by the gains of i and j. In total, the net affect of the introduction on total payoffs of i, j and k is b(1) − b(2) > 0, and thus the configuration in Figure 3a is not pairwise stable. The second configuration in Figure 3b shows an additional node l connected to k. What is the impact of this additional node on the payoff from the introduction? The effect remains as before in the isolated case as the introduction affects neither the length of any connections involving l nor the essentiality of agent k as regards connections to l. The third configuration in Figure 3c shows i and k both being connected to l. In this setting in which a neighbour is shared between the introducer and one introduced agent, the introduction of i and j affects the distribution of payoffs emanating from l as k is no b (2) longer essential to the connection of i and l, leaving k with an additional loss of 3 . The saved intermediation rent is shared between i and l, with both receiving an incremental b (2) 6 . Now, in the context of Definition 5.1, transfers are only feasible between the three intermediated by k. k receives a payoff 2

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agents involved in the introduction, leaving the group with a loss of

b (2) 6

relative to the

other configurations without a shared neighbour. Thus, the network in Figure 3c can b (2) 6

≤ 0, which is equivalent to b(1) ≤ 67 b(2). In summary, if benefits decay sufficiently slowly, the network in Figure 3c is stable to introductions. This insight can be generalised to the following proposition. be stable if b(1) − b(2) −

Proposition 5.2. Consider a network g which is pairwise stable with introductions and includes introduction opportunities that are unused. Then for each such opportunity the introducer is essential and the total number of nodes  connected  to both the introducer and exactly one of the parties to be introduced is at least 6

b(1)−2c b (2)

−1 .

Note that the result implies that there is a lower bound on clustering: where open triangles are observed in the model, they are combined with a minimum number of adjacent closed triangles. For example, with b(1) = 1 and b(2) = 2/3, the minimum individual clustering of a node in a position to facilitate an introduction is at least 3/4. Furthermore, Proposition 5.2 implies that in a stable network, every node is connected to at most a single end node, excluding inter alia stars as possible outcomes. 5.4. Connectedness. This section considers the connectedness of networks that are pairwise stable with introductions. I identify an upper bound on the number of connections of the two highest degree nodes that are not in the same component. Proposition 5.3. Let g be a network which is pairwise stable with introductions and has more than one component. Let i be the node with highest degree ηi ( g) in g. Let j be a the highest degree node of the set of nodes not in Ci ( g). Label j’s degree η j ( g). Then: (5)

ηi ( g ) + η j ( g ) ≤ −

3 [b(1) − 2c − 2d] 2b(2)

Proposition 5.3 implies that for search costs d < 23 b(2) + one component.

b (1) 2

− c there will be at most

5.5. Discussion. In summary, the study of the model with introductions generates some useful insights. It reveals how the possibility of introductions creates a lower bound on the level of clustering, thereby matching the observed regularity of significant clustering

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in real-world networks. In addition, it finds that distinct parts of a network tend to be connected, which is consistent with the short distances often found in networks. The analysis does, however, expose certain limitations inherent to myopic stability analysis, in particular concerning the lack of foresight when considering changes to the network structure. Introductions are inherently dynamic, in particular through the inclusion of search costs for initial links which can then form the foundation for a searchcost free introduction. A myopic approach does not account for the possible investment in search costs leveraged at a later stage as the network develops. From a myopic perspective, the empty network can be stable even if more connected structures will eventually be profitable for all agents involved. In Section 6 I will address this limitation by adopting an explicitly dynamic perspective and allowing agents to be forward-looking in a dynamic game of link formation (Section 6). A brief discussion of a stability analysis using farsighted stability notions is attached in Appendix Section C. 6. L INK F ORMATION G AME Whilst providing some useful insights the preceding analysis ignores the question how links arise in the first place. In this section I will explore as a complementary exercise a simple dynamic model of link formation incorporating both direct links and introductions to analyse the question to what extent players’ expectation of introduction shape their strategic link formation. 6.1. Game Description. I consider a link formation game with two periods of link formation. In the first period, players can form links with any other player by engaging in costly search. The outcome of this period is an intermediary network which then creates the possibility of introductions in the following period. In the second period players who share a common friend can connect to each other through introduction. Players conducting an introduction receive fixed transfers from those being introduced. The basic two stage structure is illustrated in Figure 4. 

Formally, in the first stage, each player i ∈ N announces a set of intended links ai =  aij ∀ j∈ N \{i} , where aij ∈ {0, 1} and aij = 1 implies an announced linking of i with j. A

connection between i and j is formed if and only if both players intend to create the link, i.e. aij = a ji = 1 (see Myerson (1977)). The links resulting from this stage form an intermediate

MAKING FRIENDS MEET Stage 1: Link Anouncement j

Stage 2: Introductions

j

j

k i

16

k i

k i

g˜

g

F IGURE 4. Two Stage Game of Link Formation ˜ where I denote g˜ij = g˜ij = 1 to indicate a connection between i and j in this network g, network. Following the link announcement period, players can conduct introductions in period ˜ Introductions can take two in a two-stage subgame based on the intermediate network g. ˜ In such cases, the place where an opportunity in form of an open triangle exists under g. central player can propose an introduction between the other two agents. All possible introduction proposals in g˜ take place simultaneously. Following the proposal round, all players involved in a proposed introduction then simultaneously accept or reject the introduction. Formally, as the proposal stage follows the announcement stage, a strategy has to ˜ the outcome of Stage 1.9 Player k can propose allow for proposals conditional on g, to introduce players i and j if and only if g˜ik = g˜ jk = 1 and g˜ij = 0. I label the set of proposed introductions for player k conditional on g˜ by pk ( g˜ ) with typical element

{(i, j), k} representing a proposal by player k to introduce players i and j. The set of proposals made by all players is labelled by P. Following the proposal stage, each player then accepts or rejects the proposals in which she is involved. Formally, player i has to provide a response for all proposals {(l, m), k } where either l = i or m = i. A response is labelled by ri,{(i,j),k} ∈ { Accept, Reject}. A proposed introduction {(i, j), k } is conducted if and only if (6)

ri,{(i,j),k} = r j,{(i,j),k} = { Accept}

9NB: This requires strategies to be consistent for announcement stages that yield the same g. ˜ In principle,

a given intermediate g˜ can be achieved with multiple stage 1 announcements and players may want to condition their further play on these announcements. However, I abstract from this source of multiplicity and focus on strategies that condition on g˜ only.

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I label the set of conducted introductions I with typical element {(i, j), k } and the resulting final network g.

In summary, a strategy si for player i is defined by: (7)

˜ P)} si = { ai , pi ( g˜ ), ri ( g,

including a complete description of link announcements, a proposal function for each ˜ feasible g˜ ∈ G, and a response function for each possible combination of P and g.

Payoffs are collected after the introductions stage is complete and consist of payoffs

from the final network g as described in Section 3 above plus any transfers from accepted introductions. For each successful introduction, the agent proposing the introduction receives t from each of the introduced players for a total of 2t in compensation. 6.2. Equilibrium Analysis. The aim of this section is to characterise equilibrium play in the given model of network formation. I focus attention on subgame perfect Nash equilibria in pure strategies. Furthermore, I consider an equilibrium concept robust to pairwise deviations to address the fact that both the link announcement stage and the response stage require pairwise coordination successfully to create links.10 I employ backward induction and commence by considering the response stage subgame starting with an intermediate network g˜ and a set of proposals P. 6.2.1. Response Stage. Each configuration at the response stage characterised by the intermediate network g˜ and the set of proposals P describes a proper subgame of the overall game. I consider equilibrium behaviour of this subgame by analysing the best response of a player i faced with a set of proposed introductions P such that i is involved in at least one. I consider first a proposed introduction {(i, j), k } and response r j,{(i,j),k} . If both i and j

accept, a link is created between j and i for a direct connection where previously they were indirectly connected. Payoff implications for i and j are as follows: ˜ then for the connection ij, • If k is not essential to the connection between i and j in g, b (1) b (2) payoffs to each player change by 2 − c − 2 − t. In addition, the link may reduce the path length for other connections of i and j, leading to additional benefits. Other than the cost for the new link c, an introduction has no negative effects. 10See also the discussion in Jackson (2008, Chapter 6.1).

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˜ then in addition to a shorter • If k is essential to the connection between i and j in g, connection between i and j, the introduction will also reduce intermediation rents paid to k and provide benefits above and beyond those for the case of the nonessential k.

Note that the payoff effects described above are only applicable to the first introduction between i and j that is accepted. Each subsequent introduction will incur transfer costs t but no incremental benefits. Thus, in the response stage, if more than one introduction is proposed for the same pair of agents, then in any equilibrium, at most one of these redundant introductions will be accepted. More generally, the specific payoff implications will depend on the pattern of proposed introductions and network structure. The optimal acceptance decision will select from the set of proposed introductions that subset which leads to the highest payoff given other players strategies. However, identifying this set in the general case quickly becomes untractable, in particular because there are externalities between different introductions affecting a player. It is however the case that if transfers do not exceed the benefits from the direct impact on the connection between i and j, then accepting all non-redundant proposed introductions will be optimal. A deviation from acceptance to reject would result in a loss of payoffs and acceptance of all non-redundant introductions can thus be part of an equilibrium. Note that this argument does not establish that acceptance is the unique Nash equilibrium for the response subgame in the relevant parameter range. Indeed, as introduction proposals require two parties to accept for implementation, there are additional Nash equilibria in which neither player accepts any subset of the proposed introductions. This play can be part of an equilibrium because a unilateral deviation by one player will not change the outcome of the link not being formed. This issue is commonly encountered in network formation models and also played a role in the stability analysis of Section 5. In order to address this issue, I will consider a concept of equilibrium which allows for pairwise deviations. I adopt the notion of “strict bilateral equilibrium” (Goyal and

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Vega-Redondo, 2007) which explicitly requires an equilibrium strategy profile to be robust to deviations by pairs of players.

11

Definition 6.1. A strategy profile s∗ is a bilateral equilibrium (BE) if the following two conditions hold: a. for any i ∈ N and every si inSi , Πi (s∗ ) ≥ Πi (si , s∗−i ); and

b. for any pair of players i, j ∈ N and every strategy pair (si , s j ) ∈ Si × S j , Πi (si , s j , s∗−i− j ) > Πi (si∗ , s∗j , s∗−i− j ) → Π j (si , s j , s∗−i− j ) < Π j (si∗ , s∗j , s∗−i− j ) b(1)−b(2) 2

− c two players rejecting a proposed introduction does not constitute an equilibrium - the bilateral deviation to accepting would be profitable for both players. Under this equilibrium concept, for t <

b(1)−b(2) 2

− c, a strategy profile s is a bilateral equilibrium if and only if all proposed non-redundant introductions are accepted and exactly one proposal is accepted for pairs ij which are faced with multiple redundant proposals. Lemma 6.1. For each subgame starting at the response stage, if t <

In summary, accepting all non-redundant proposals forms part of every BE in the subgame starting from a set of proposals. Where there are redundant proposals a BE involves accepting exactly one proposal from each set of redundant proposals. The BE is non-unique to the extent that redundant introductions are proposed - in this case accepting any one of the redundant introductions can be part of an equilibrium. ˜ 6.2.2. Proposal Stage. I now consider the proposal stage for a given intermediate network g. The key question I will consider in this section is to what extent proposing introductions can be part of equilibrium play. First, recall the implications for introductions of whether or not an introducing node is essential or not to the connection between agents being introduced discussed in Section 5.3 of the stability analysis. Proposition 5.1 establishes that in any stable network open introduction opportunities correspond to introducing agents that are essential. This insight is replicated in the link formation game in the following proposition: 11This concept is closely related to that of “pairwise Nash equilibrium” described in Calvó-Armengol and

˙ Ilkılıç (2009), with the main difference being that the latter explicitly considers the addition of links whilst the notion of bilateral equilibrium is defined on the basis of deviations in strategies.

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Proposition 6.1. In any bilateral equilibrium of the proposal subgame starting with an intermedib(1)−b(2) 2

− c, then all pairs of agents i and j which can be introduced, will be offered at least one introduction unless the only feasible introducer k is essential to the connection ij. ˜ if 0 < t < ate network g,

If an introduction proposal is made by an agent essential to the connection, there is a loss of intermediation benefits for the introducing agent. In this case, for introductions to happen, transfers between the parties to the introduction need to be sufficiently high to compensate the introducing agent. Specifically, for an isolated introduction to occur, the transfers received need to cover the loss of benefits of

b (2) 3 .

Proposition 6.2. In any bilateral equilibrium of the proposal subgame starting with an intermedi-

< t < b(1)−2 b(2) − c, all possible introductions will be proposed unless the opportunity (i, j, k) involves a proposing agent k that is (a) essential to the connection i, j, and (b) subject to at least one incoming introduction. ˜ if ate network g,

b (2) 6

This result has two elements which deserve discussion in light of the results derived in the stability analysis. First, whether or not introductions occur does not depend on the number of shared agents for which intermediation benefits might be lost, contrary to Proposition 5.2. This is the result of the property that in the game theoretic analysis the introducing agent can make introduction proposals simultaneously. Where an introduction threatens the intermediation benefits derived from connecting one of the parties to be connected and a third, non-involved party, the introducing agent can preemptively propose introductions to all those agents and recover a transfer from all. Second, in Proposition 6.2 it is incoming introductions that determine whether or not an introduction is being made. The intuition for this is that those incoming introductions open new opportunities for intermediation benefits but those benefits cannot be recovered through subsequent introductions as there is only one round of introductions. As a direct corollary to Proposition 6.2 it can be shown that for all intermediary networks which involve stars the transition to the complete network for each of these is the unique bilateral equilibrium outcome. Corollary 6.1. In any bilateral equilibrium of the proposal subgame starting with an intermediate

< t < b(1)−2 b(2) − c, in each star all possible introductions will be proposed resulting in a completely connected subnetwork. network g˜ which involves stars as subnetworks, if

b (2) 6

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Finally, I can establish the following property regarding the change in payoffs experienced by players in the introduction stage. Lemma 6.2. In any bilateral equilibrium of the link formation game, the payoffs for all players received at the end of the introduction stage (i.e. payoffs from the final network g plus transfers) are ˜ at least as large as those derived from the intermediary network g. 6.2.3. Announcement Stage. I now turn to the announcement stage to consider which links are formed through search in equilibrium. Players explicitly take account of the subsequent stages and the fact that introductions can be conducted. In this sense the analysis is similar to the farsighted stability analysis in Section C. However, whilst I considered network formation paths with many steps in that section, here there is exactly one link formation stage followed by exactly one introduction stage. First I consider the feasibility of equilibrium networks to consist of more than one component before considering under what conditions the complete network arises as the final network of equilibrium play in the link formation game. Proposition 6.3. Let g be a network with more than one component formed in a bilateral equilibrium in the game of link formation with introductions. Let i be the node with highest degree ηi ( g) in g. Let j with degree η j ( g) be a the highest degree node in any other component. Then: ηi ( g ) + η j ( g ) ≤ −

(8)

3 [b(1) − 2c − 2d] 2b(2)

The result as well as the proof (see Appendix) mirrors Proposition 5.3. The reason behind the close similarity is the symmetry in the payoffs for the players involved in link creation through search, which imply that the condition for individual profitability mirrors that for joint profitability. Next, I will focus on the complete network as the outcome of equilibrium play. Recall that the n complete o network arrived at through the star is efficient if and only if b(1) − 2c ≥ max

4 n d, b (2)

. I consider two distinct equilibria strategy profiles which result in a

complete network: (i) the complete network formed in the first stage; and (ii) the complete network arrived at in the second stage through the star as first stage outcome. Now, the complete network as outcome in the first period will involve all agents announcing to form all links in the first period and then no introductions in the second

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period. A deviation to consider is for an agent not to announce all links in the first period, resulting in a missing link. Now, given Proposition 6.2, this link will be closed for transfers in the relevant range, leading to a complete network in the final stage. This deviation is

− c − b(22) . Thus, the complete network arrive at in the first stage is an equilibrium outcome only if d is sufficiently small and less than the transfer paid for an introduction. Next, I consider whether the complete network arrived at through a star network as an intermediate configuration can arise as an equilibrium outcome of the game.   b (1) Proposition 6.4. Assume n ≥ 4 and sufficient decay such that b(2) < 34 2 − c . h i b (2) b (2) Then if d ∈ 6(n−2) , b(1) − 2c − 2 , there exists a transfer parameter t for which the strategy profile which creates an intermediary star network followed by transition to the complete network with all feasible introductions is a bilateral equilibrium. thus profitable whenever d > t and in any case profitable if d > tmax =

b (1) 2

The condition can be verified by considering possible deviations and verifying that a transfer payment t can be found which ensures they are not profitable. For details, see the appendix. The result implies that for the equilibrium configuration to consist of an intermediary star followed by the complete network, the search cost d cannot be too high relative to the benefit of a direct link b(1) − 2c. The upper bound on d is strictly less

than that for the configuration to be efficient, implying that for relatively high d, the star followed by the complete network is efficient but not a bilateral equilibrium. The relevant equilibrium condition that is violated relates to the need for the hub to invest ex ante in a large number of connections incurring search costs, for which subsequent compensation on a connection by connection basis is insufficient. 7. C ONCLUSION This paper presents an analysis of network formation where the creation of new links is mediated by existing connections. Specifically, I consider a setting in which players that are unconnected but share a direct neighbour can be “introduced” by the neighbour. The paper presents an analysis of myopic and farsighted stability as well as a game of link formation to study the incentives involved in introductions and the impact on network outcomes.

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I find that myopically stable networks include efficient configurations for some parameter ranges but that the concept is also affected by significant multiplicity. Nonetheless, predictions can be made regarding the level of connectedness and in particular regarding the extent to which introductions in the model are used by agents to create networks with significant levels of clustering that mirror those observed in real world networks. Adopting a forward looking perspective the paper then presents a simple game of link formation which explores how networks might take shape starting from a set of unconnected nodes. In a two-stage setup, I find equilibria in which agents invest in costly links and leverage those connections in a subsequent round of introductions.

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A PPENDIX A. Proofs. A.1. Proof of Proposition 4.1 - Efficient Network Configurations. The proof adapts the efficiency results for the standard connections model described in Jackson (2008, Chapter 6.3). Proof. First, consider the case with b(1) − 2c < b(2). I will argue that the star is the efficient

configuration to connect k nodes. A star network of k agents incurs search costs of exactly

(k − 1)2d and in the model generates a benefit of: (k − 1)(k − 2) b(2) − (k − 1)2d 2 Now any other configuration connecting k nodes with m ≥ k − 1 links will generate at most   k ( k − 1) m (b(1) − 2c) + (10) − m b(2) − (k − 1)2d 2 (9)

(k − 1) (b(1) − 2c) +

where the first component represents the benefits from direct links and the second component reflects the upper bound of benefit b(2) that can be derived from any indirectly connected agents. The third component reflects the minimum amount of search costs incurred. Note that to connect k agents requires at least k − 1 connections to be created by search and thus incurs a total cost of at least (k − 1)2d. Subtracting the second equation

from the first and rearranging yields the payoff advantage for the star relative to any other configuration of k agents: (11)

(k − 1 − m) (b(1) − 2c − b(2))

As b(1) − 2c − b(2) < 0, this implies that the payoff advantage is minimised at m = k − 1. Any other configuration of k − 1 links connecting k nodes that is not a star will incur

benefits that are strictly less than those from the star as only the star has all nodes that are not directly connected at distance two. Any other configuration has at least one pair that is not directly connected at distance 3 or above, yielding strictly lower payoffs. It is thus established that if b(1) − 2c < b(2), efficient networks consist of stars and

isolated nodes. Next, restrict the set of candidate efficient networks for b(1) − 2c < b(2)

further by establishing that there is either a single star of all nodes or an empty network.

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Assume a candidate network consists of two stars with k1 ≥ 1 and k2 ≥ 2 nodes yielding

positive utility each. Then total payoff is: (12)

   b (2) b (2) (k1 − 1) b(1) − 2c − 2d + (k1 − 2) + (k2 − 1) b(1) − 2c − 2d + (k2 − 2) 2 2 

Reconfiguring the nodes into a single star yields:   b (2) (13) (k1 + k2 − 1) b(1) − 2c − 2d + (k1 + k2 − 2) 2 Now, subtracting the first equation from the second and simplifying yields:

[b(1) − 2c − 2d] + (2k1 k2 − 2)

(14)

b (2) 2

which is strictly positive if each separate star yields positive utility as 2k1 k2 > k1 and 2k1 k2 > k2 . Thus, a network with more than one separate stars (including the case where all but one star are single disconnected nodes) yields lower utility than one in which the nodes involves are combined into a single star. The case where b(1) − 2c < b(2) can be closed by comparing payoffs of a single star

involving n nodes and the empty network. As the latter derives zero utility, the star is efficient if: (15)

(n − 1) (b(1) − 2c) +

(n − 1)(n − 2) b(2) − (n − 1)2d ≥ 0 2

which reduces to:

( n − 2) b (2) 2 Next consider the case with b(1) − 2c ≥ b(2). In this case, adding a link increases utility before search costs. Thus, any component of connected nodes has to be completely connected, a clique. For a clique of k nodes, the minimum number of links created through search is k − 1. All other links can be created through introductions without search costs. Now, for the case where there are more than one such cliques an argument analogous to that for two stars shows that utility increases in a single connected clique. Thus, if b(1) − 2c ≥ b(2) the efficient network is either complete or empty. Utility from the (16)

b(1) − 2c ≥ 2d −

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complete network with n − 1 links from search is higher if: (17)

( n − 1) n (b(1) − 2c) − (n − 1)2d ≥ 0 2

which reduces to (18)

b(1) − 2c ≥

4d n

 A.2. Proof of Proposition 5.1. Proof. The proof is by contradiction. Assume a network g which is stable and with open triangle ikj in which k is not essential to the connection between i and j. Consider an introduction between i and j facilitated by k and its impact on the payoffs for all three parties involved. Note first that the new link does not affect the set of connections for which any of the three nodes is essential as it generates a new connection between i and j between which there are at least two indirect paths in g by k being non-essential. Now, for i and j, the surplus form their connection with each other changes by b(1) − 2c − b(2)

which is positive by assumption. For connections of i or j to nodes uninvolved in the

introduction, the introduction may decrease the path length in a similar fashion, potentially increasing available surplus. Thus, the payoff impact of the introduction on i and j is strictly positive. Finally, the impact on k is zero as a link between i and j does not affect the length of the shortest path to any connections of k. In sum, an introduction of i and j by k thus increases payoffs to the three agents involved and thus g is not pairwise stable. This delivers the contradiction.

 A.3. Proof of Proposition 5.2. Proof. The proof is by contradiction. Assume network g is a myopically stable network and has unused introduction opportunities involving an introducer essential to the connection such that the number of nodes connected toboth the introducer and exactly one of the  b(1)−2c b (2)

−1 . Consider the payoff implications of conducting an introduction on the three agents involved. The direct gain from replacing an indirect connection with a direct one is at least

parties to be introduced is µ < 6

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b(1) − 2c − b(2). Now consider possible losses from the introduction to the introducing

agent. Losses arise from being circumvented on connections of length two for which the

agent is essential pre-introduction but not essential after the introduction. This necessarily requires that pre-introduction the node is (a) connected directly to the introducer and (b) connected either directly or indirectly to one of the nodes to be introduced. Where that connection is indirect, this would leave an introduction opportunity with the introducer being inessential, thus violating pairwise stability. Thus, these nodes are connected directly to one of the nodes being introduced. For each such node, the loss from missing out on intermediation benefits is

b (2) 6 .

Total payoff is thus at least:

b(1) − 2c − b(2) − µ

(19)

b (2) >0 6

where the inequality arises from the assumption on µ. Total change in payoffs from the



introduction is thus positive, violating stability. A.4. Proof of Proposition 5.3.

Proof. The proof is by contradiction. Assume network g with more than one component and myopically pairwise stable with introductions. Further, assume that there are i, j violating 5 such that Ci ( g) 6= Ci ( g) and ηi ( g) + η j ( g) >

3[b(1)−2c−2d] 2b(2)

Consider a new connection being formed between i and j. The new link will create

a direct connection with direct net benefit b(1) − 2c at search cost 2d. In addition there

are ηi ( g) + η j ( g) new connections of length two and total benefit b(2). For each such connection, i and j form an endnode and an essential intermediary, respectively, and thus capture a benefit of 23 b(2). Total payoff to i and j is thus (20) (21) (22)

  2 b(2) ηi ( g) + η j ( g) + [b(1) − 2c − 2d] 3 > − [b(1) − 2c − 2d] + [b(1) − 2c − 2d]

=0

and thus the additional link is jointly profitable and the network g not pairwise stable, delivering the contradiction.



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A.5. Proof of Lemma 6.1. Proof. A subgame at the response stage is defined by the intermediate network g˜ and a set of proposals P. Players active in the subgame are those i ∈ N which are involved in at least one proposal in P.

First, for the “if” part, it can be established that it is a bilateral equilibrium strategy profile to accept all non-redundant introductions and exactly one introduction where multiple redundant introductions are proposed. Now, any unilateral or bilateral deviation to profile s0 will create at least one of the following: a. A pair i, j with a rejected non-redundant introduction proposal. Then if 2t < b(1) −

b(2) − 2c, both i and j are strictly worse off under the deviation and s0 is non-profitable.

b. A pair i, j of agents who accept multiple redundant introductions. Then the resulting final network will remain unchanged but i, j will pay multiple sets of transfers t, leaving i, j strictly worse off and s0 is non-profitable.

c. A pair i, j of agents who accept a different proposal from a set of redundant proposals. Then the resulting network will remaind unchanged as will transfers paid by i, j, leaving payoffs unchanged. Thus s0 is not strictly profitable. Second, for the “only if” part, it needs to be established that any strategy profile with at least one open introduction or at least one occasion where multiple redundant proposals are accepted cannot be a bilateral equilibrium. The proof is by contradiction. Consider a putative bilateral equilibrium strategy profile s0 with at least one rejected non-redundant introduction proposal or at least one occasion where multiple redundant proposals are accepted. Then following the previous argument, agents can be better off by deviating to a strategy profile in which all non-redundant proposals are accepted and exactly one proposal is accepted where multiple are proposed.

 A.6. Proof of Proposition 6.1. Proof. The proof is by contradiction. Assume a bilateral equilibrium of a proposal subgame starting with any intermediate network g˜ and a possible introduction opportunity between i and j that is not proposed and where at least one possible introducer k is not essential to the connection i, j.

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Consider a deviation by k involving to propose the introduction of i and j. Then given b(1)−b(2) 2

− c and Lemma 6.1, this proposal would be accepted, yielding a positive transfer payment 2t > 0 for k. As the new link cannot reduce payoffs to k unless k is essential, the deviation would be profitable, and thus the strategy profile of the subgame would not constitute a bilateral equilibrium.  t <

A.7. Proof of Lemma 6.2. Proof. Introductions create additional connections. Additional connections can only have a negative impact on agents if the new connection changes the status of those agents from essential to non-essential for at least one connection. This is not possible for agents that are not involved in the introduction concerned as the new connection creates a new direct link between a pair that were already indirectly linked. Thus, but for the agent facilitating the introduction, the pattern of pattern of essentiality remains unchanged. Now for agents involved in an introduction, assume the payoff implication was negative. Then agents would be better off by changing their proposals or response to a proposal and thus the initial strategy cannot have been part of an equilibrium.



A.8. Proof of Proposition 6.3 - Connectedness in the Link Formation Game. Proof. The proof is by contradiction. Assume an equilibrium strategy profile s˜ such that the resulting network has multiple components with highest degree nodes i and j such that Condition 8 is violated. Now consider a deviation by i and j involving the creation of a link ij at the link announcement stage. For each agent, this incurs search costs d whilst giving direct
b (1) benefits of 2 − c as well as intermediation rents ηi ( g) + η j ( g) . If Condition 8 holds, the

marginal effect of this deviation on payoffs is positive and thus the deviation is profitable. s˜ is thus no equilibrium strategy profile.

 A.9. Proof of Proposition 6.4 - Empty-Star-Complete. Proof. In equilibrium two types of nodes can be distinguished, based on their location in the intermediate network. A single “hub” node that forms the centre of the star, and n − 1

“spoke” nodes that form the periphery. I begin by considering the equilibrium payoffs for

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each type, as illustrated in Table 2. Note that both types receive equivalent payoffs from the network reflecting the symmetry in the final configuration. The two types differ in the share of search costs paid and in terms of whether they receive or pay transfers.

Player Payoff Hub Spoke



b (1)  2 (n − 1) b(21)

( n − 1)

 − c − (n − 1) d + (n−1)(2 n−2) · 2t  − c − d − ( n − 2) · t

TABLE 2. Payoffs in the Empty-Star-Complete Equilibrium

Now, to verify whether the strategy profile is indeed a bilateral equilibrium consider possible deviations. These are feasible in the link announcement stage or in the introduction stage. Regarding the proposal stage, the transition from an intermediary star to a final complete network is provided by Proposition 6.2. Regarding the announcement stage, I distinguish available deviation by type of agent. The hub can deviate by announcing fewer connections, in consequence disconnecting individual spokes from the network. Consider the disconnect of 0 < k ≤ n − 1 nodes. The

payoff implications of this deviation are given by: k

(23)

k · d − ∑ (n − i ) · 2t i =1

Note that the marginal effect of removing an additional link is increasing, as at higher k an additional link leads to a smaller number of transfer payments lost whilst saving the same amount of search costs d. Thus, the optimal deviation is to remove all links at k = n − 1,

which leads to a zero payoff. This deviation is profitable if:   b (1) (n − 1)(n − 2) (24) − c − ( n − 1) d + · 2t < 0 ( n − 1) 2 2   b (1) (25) − c − d + ( n − 2) · t < 0 2

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Condition on (n − 2)t   b (1) t ≥ d− 2 −c t t t t

≥ b(62) ≤ b(21) − c − b(21) ≤ (n − 2)d  ≤ (n − 1) b(21) − c − d

TABLE 3. Restrictions on Transfers Next consider deviation opportunities for spokes. First, consider the deviation to destroy a link at the link announcement stage. This is profitable if:   b (1) (26) − c − d − ( n − 2) · t < 0 ( n − 1) 2 Second, consider the deviation of a pair of spokes to create a link at the link announcement stage. This deviation for each spoke replaces connecting to all other notes through a hub with connecting to all by search. The deviation is profitable if: d−t > 0

(27)

These equations provide conditions on the transfer payment t in terms of the other parameters of the model. It remains to be shown that these conditions, together with the conditions in Proposition 6.2, can be reduced to those in Proposition 6.4. Now, rearrange for (n − 2)t and separate conditions on upper  and lower bounds for

transfers as in Table 3. Then, given n > 4 and b(2) < positive measure as long as

b (2) 6( n −2)

≤ d ≤ b(1) − 2c −

3 b (1) 4 2 − c , there b (2) 2 as required.

is an interval of



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B. Stability of Efficient Network Configurations. This section discusses the characterisation of parameter restrictions for the stability of the three network configurations which can be efficient. Empty Network: In the empty network, no links exist implying πi ( g) = 0 ∀ i and

thus the only condition to check is bilateral link creation, yielding the stability condition: b(1) − 2c ≤ 2d

(28)

Thus the empty network is stable if net benefits from a single new link do not outweigh the search costs. Star Network: In the star network, the set of possible deviations to consider applies to two types of nodes (hub and spoke) and there are opportunities to destroy links as well as create links through search and introductions: (a) The network is stable against link destruction by the hub and one peripheral node if: 2 b(1) − 2c ≥ − (n − 2)b(2) 3

(29)

(b) The network is stable against link creation by search of two peripheral nodes if: 2 b(1) − 2c ≤ 2d + b(2) 3

(30)

(c) The network is stable against introduction of a pair of peripheral nodes through the hub if: b(1) − 2c ≤ b(2)

(31)

Complete Network: Total net payoffs from the complete network before search costs and transfers are given by: b (1) π ( g ) = ( n − 1) −c 2 

(32)



As all links are in place, the only deviation to consider is whether a pair of agents would find it profitable to destroy a link. In this case, a single direct link is replaced with a set of n − 2 indirect connections, one through each of the other agents that remain connected to i. For n > 3, this implies that there are at least two

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intermediaries and the two disconnecting agents capture 12 b(2). The complete network is stable against link destruction if: (33)

b(1) − 2c ≥

1 b (2) 3

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C. Farsightedly Stable Networks with Introductions - An Example. This section presents an analysis of the model of network formation with introductions using a forward looking notion of stability, which allows players to recognise the potential future benefits from deviations they initiate. The concept I employ is based on Herings et al. (2009) with adaptations to allow for introductions and, in particular, the transfers involved. C.1. Definitions. Farsighted stability in Herings et al. (2009) is based on the notion of farsightedly improving paths, that is, transitions between networks which agents engage in willingly because the transition leads to a final outcome which improves their payoffs. Now, as noted above, the analysis of stability with introductions necessarily requires the inclusion of transfers and this was implicit in Definition 5.1. There are different plausible approaches to include transfers in a farsighted setting. For example, one may consider transfers to be paid at the time of a move or once only, e.g. at the end. One may also consider transfers that are feasible across all agents or only across those that are moving at a particular point. Finally, transfers may be made conditional on specific links being formed – or indeed not formed. The approach adopted in Definition .2 is one which requires transfers to be paid at the time of a move and restricts them to the parties involved in a move. This approach may be most plausible for applications in which complex contracts are not feasible or enforceable and thus agents’ ability to fully coordinate their actions is restricted.12 The definition makes use of the concept of farsightedly improving paths. Farsightedly improving paths are sequences of networks g1 , g2 , . . . , gK along which agents find it beneficial to conduct the actions require to step from one network to the next at each point on the basis that they will be better off at the end of the sequence. Let the transfers along the way be defined as follows: Tik is the amount agent i receives in the transition from gk to gk+1 , a negative transfer implying a loss. T k is the vector of transfers at step k. Ti the sum of transfers to player i along the full path. A farsightedly improving path can then be defined as follows: 12I conjecture that in the extreme case where transfers are feasible between all agents and contracts can

be written to make payments be fully conditional on the resulting network, the efficient network can be achieved in a way similar to the results in Bloch and Jackson (2007). That paper offers a comprehensive discussion of the interaction of transfers and efficiency in network formation.

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Definition .1 (Farsightedly improving path). A farsightedly improving path in the model with introductions from a network g to network g0 6= g is a sequence of graphs g1 , g2 , . . . , gK with associated sequence of transfers T 1 , T 2 , . . . , T K such that for any k ∈ {1, 2, . . . , K − 1} one of the following holds:

i. (Destruction) gk+1 = gk − ij for some ij such that a. πi ( gK ) + ∑rK=−k1 Tir ≥ πi ( gk ),

b. π j ( gK ) + ∑rK=−k1 Tjr ≥ π j ( gk ), c. Tik + Tjk = 0, and

d. Tlk = 0 ∀ l ∈ / {i, j}

ii. (Search) gk+1 = gk + ij for some ij such that a. πi ( gK ) + ∑rK=−k1 Tir − d ≥ πi ( gk ),

b. π j ( gK ) + ∑rK=−k1 Tjr − d ≥ π j ( gk ), c. Tik + Tjk = 0, and

d. Tlk = 0 ∀ l ∈ / {i, j}

iii. (Introduction) gk+1 = gk + ij for some ij such that ∃m ∈ ηi ( gk ) ∩ η j ( gk ) such that a. πi ( gK ) + ∑rK=−k1 Tir ≥ πi ( gk ),

b. π j ( gK ) + ∑rK=−k1 Tjr ≥ πi ( gk ),

c. πm ( gK ) + ∑rK=−k1 Tmr ≥ πm ( gk ),

d. Tik + Tjk + Tmk = 0, and e. Tlk = 0 ∀ l ∈ / {i, j, m}

In this definition of a farsightedly improving path agents assess the benefits of a move at the time of the move taking into account the final network as well as all transfers accumulated on the path. Transfers are possible only between agents active during a move from one network to the next and have to balance within each period. Borrowing notation from and building on Herings et al. (2009) let F ( g) be the set of networks for which a farsightedly improving path exists from network g. Farsighted stability with introductions can then be defined as follows: Definition .2 (Farsightedly stable set with introductions). A set of networks G ⊆ G is a farsightedly stable set with introductions if (1) For all g ∈ G,

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(a) (Destruction) ∀ ij ∈ g such that g − ij ∈ / G, ∃ g0 ∈ F ( g − ij) ∩ G with associated transfers T along the path such that πi ( g0 ) + π j ( g0 ) + Ti + Tj < πi ( g) + π j ( g)

/ g such that g + ij ∈ / G, ∃ g0 ∈ F ( g + ij) ∩ G such that πi ( g0 ) + (b) (Search) ∀ ij ∈ π j ( g0 ) + Ti + Tj − 2d ≤ πi ( g) + π j ( g)

(c) (Introduction) ∀ ij ∈ / g such that g + ij ∈ / G and ∃m ∈ ηi ( gk ) ∩ η j ( gk ), ∃ g0 ∈ F ( g +

ij) ∩ G such that πi ( g0 ) + π j ( g0 ) + πm ( g0 ) + Ti + Tj + Tm ≤ πi ( g) + π j ( g) + πm ( g)

(2) For all g0 ∈ G \ G, F ( g0 ) ∩ G 6= ∅

(3) @ G 0 * G such that G 0 meets the previous two conditions.

Intuitively, the first condition checks that for each network in the stable set each possible deviation is deterred by a farsightedly improving path ending up in a network that makes the deviating players jointly worse off. The joint assessment of profitability means that transfers amongst players involved in a deviation are allowed for as in the myopic stability of Definition 5.1. The second condition verifies that from all networks outside the stable set there is a farsightedly improving path ending up in it. The third condition then simply aims to identify the smallest set that is stable with respect to the first two conditions. C.2. Characterising Farsightedly Stable Networks - The Three Player Case. This section analyses the implication of Definition .2 using the three player case. Figure 5 lists the available configurations. Payoffs for all three players are tabulated in Table 4. Note that the listed payoffs for each configuration exclude any sunk search costs incurred to arrive at the network as well as any transfers. I will focus on the question to what extent agents are able to overcome sunk search costs through the farsighted stability approach outlined in Definitions .1 and .2 in those cases where a movement to the complete network g7 would be efficient. This implies that the net benefit of a direct connection (b(1) − 2c) is positive and sufficiently large to make it more efficient than an indirect connection and to make it worthwhile to invest in search cost in the first place: (34)

b(1) − 2c > b(2) > 0

(35)

b(1) − 2c >

4 d 3

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g0

g1

1

3

2

g4

3

2

g5

1

3

g2

1

2

3

2

g3

1

3 g6

1

37

2

3

3 g7

1

2

1

2 1

3

2

F IGURE 5. Network configurations for the three player case π1 ( g )

Network g g0 g1 g2 g3 g4 g5 g6 g7

0 −c −c 0o −c +

b (1) 2 b (1) 2

2

n

π2 ( g )

π3 ( g )

0 −c 0 b (1) 2 −c b (1) − c + b(32) n2 o b (1) b (2) 2 2 −c + 3

0 0 b (1) 2 −c b (1) 2 −c b (2) b (1) 2 −c+ 3

b (1) 2

b (1) b (2) 2 3 b (1) b (2) 2 −c+ 3 b (1) b (2) 2n− c + o 3 b (1) 2 2 −c

b (1) b (2) 2n− c + o 3 b (1) 2 2 −c

b (1) − c + b(32) 2 n o b (1) b (2) 2 2 −c + 3 n o b (1) 2 2 −c

TABLE 4. Network payoffs for the three player case

In this context g4 , g5 and g6 can never be stable in the sense of Definition 5.1 as the deviation to g7 is profitable. Now, g0 may be myopically stable if direct benefits are not sufficiently large, namely: (36)

b(1) − 2c < 2d

Furthermore, g1 , g2 and g3 are stable with respect to a deviation by destruction and may also be stable with respect to a deviation to three-player stars (g4 , g5 , g6 ) if: (37)

2 b(1) − 2c < 2d − b(2) 3

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which implies the previous condition. I will now consider the farsighted stability of the possible configurations by analysing the existence of farsightedly improving paths from different starting position. It is immediate that F ( g7 ) = ∅ as all other configurations have fewer links, leading to lower payoffs. For g4 , g5 and g6 there is a farsightedly improving path to g7 through introduction by condition 34. There are no farsighted improving paths to less connected networks. Thus F ( g4 ) = F ( g5 ) = F ( g6 ) = { g7 } .

Next, consider paths starting from g1 , g2 or g3 . Moves to g0 are not profitable by

condition 34. Moves to g4 (or g5 or g6 , respectively) involve a single step of link creation by search. Thus, there are farsightedly improving paths to the two link configuration if condition 37 is violated. Now consider moves to g7 , the complete network. This involves two steps: first, the creation of an additional link through search, followed by the closing of the triangle to create a third link, which can be done by introduction. Under what conditions is g7 an element of F ( g1 )? Definition .1 requires the existence of at least one combination of steps and sequence of transfers to make the transition to the final outcome profitable for the active players at each intermediate step. As the assessment at each step includes future transfers, I proceed by backward induction and consider first the introduction to g7 . This is jointly profitable for all three players by condition 34 and thus transfers can be constructed to make the move acceptable to each player. The change in payoffs for each player from staying at the starting point of the step and progressing to g7 can be compute as follows. Row 2 in Table 5 shows the differences for the final step through introduction. Row 3 shows the corresponding information for the step starting at g1 , showing the amounts accrued to each player from g1 and progressing on the route towards g7 . No information is given for Player 2 to illustrate that this player is not active at the beginning of the route. Row 4 then applies the same logic for the transition from g0 , excluding Player 3. Now to assess whether a farsighted improving path from g1 to g7 can be constructed, for each step it needs to be verified that players can receive sufficient transfers to provide incentives whilst keeping transfers balanced within each period and between the players active at the time. Consider first g4 to g7 . This move involves all three players. Balance requires that total transfers between 1, 2 and 3 add to zero. Adding up the total payoff changes across the

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Period Transition Player 1 3

g4 to g7

2

g1 to g7

1

g0 to g7

b (2) 3  b (1) − c −d 2  b (1) 2 2 − c − 2d

39

Player 2   b (1) − c − 2

b (2) 3

0 2



b (1) 2



Player 3   b (1) − c − b(32) 2   b (1) − c −d 2

− c − 2d 0

TABLE 5. Minimum transfer requirements for farsighted improving paths to g7 active players, it can be assessed whether joint benefits are positive and thus each player can be made indifferent whilst maintaining balance:   b (1) (38) 2 − c − b (2) ≥ 0 2 This is condition 34. Looking at the move from g1 to g4 , note first that player 2 does not act in this period and thus I need to check transfers between 1 and 3. Total change in payoffs for active players is 3 2

(b(1) − 2c) − 2d. This sum can be composed of transfers in this period and transfers to 1 and 3 in the next period. Thus the condition is:   b (1) (39) − c − 2d + T12 + T13 + T32 + T33 ≥ 0 3 2 Now, the maximum available to these players in the final period is achieved by paying player 2 the minimum transfer at this point, leaving T13 + T33 = Substituting in and using T12 + T32 = 0 yields:   b (1) b (2) (40) 4 − c − 2d − ≥0 2 3

b (1) 2

−c−

b (2) 3

for 1 and 3.

The search cost incurred 2d has to be recovered through the future benefits described by two new direct links - one through search and one through introduction, less

b (2) 3 .

Note

that this condition is similar to checking whether a move from g1 to g7 is efficient, but for the inclusion of the term

b (2) 3 .

This reflects the positive externality to Player 2 of the move

from g1 to g4 . This gain is sunk at the time 2 enters the table and thus is not available for transfer. Using a similar approach, I can now construct a farsighted path from g0 to g7 via g1 and g4 . Note that the steps from g1 onwards are as before and the conditions identified

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above apply. What remains to be shown is the criteria for the first step g0 to g1 . The active players in this move are 1 and 2. The total change in payoffs for these two players is  b (1) 4 2 − c − 3d. Thus, a path is feasible if:   b (1) 4 − c − 3d + T11 + T12 + T13 + T21 + T22 + T23 ≥ 0 (41) 2 Note that T22 = period and set n 0. Furthermore,  oset Player 3 to be indifferent inthe previous  b (1) b (1) T32 + T33 = − 2 2 − c − d . Balance then yields T12 = 2 2 − c − d + T33 . Finally, substitute T13 + T33 from above. This then yields:   b (1) 6 (42) − c − 4d ≥ 0 2

Note that this condition, contrary to the path starting at g1 is equivalent to the efficiency condition. The reasoning here is that at the moment the move is undertaken, there are no externalities to players who are not involved. Specifically, payoffs for Player 3 who is excluded at the time, are unaffected. Any later impacts are captured by the participation of that and the subsequent inclusion in the transfers. C.3. Discussion. In this section I have applied the notion of farsighted stability to the model of introductions. The analysis shows the potential for farsighted solution concepts to allow agents to capture the benefits arising from introduction opportunities more fully than under myopic approaches. This is shown in particular in the fact that a farsightedly improving path from the empty network to the complete network exists if and only if the complete network is efficient.

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