MAKING FRIENDS MEET NETWORK FORMATION WITH INTRODUCTIONS

Transcription

1 MAKING FRIENDS MEET NETWORK FORMATION WITH INTRODUCTIONS JAN-PETER SIEDLAREK ABSTRACT. 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: 3 October 013. 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, PET013 Lisbon and the EEA013 Goteborg for helpful comments and suggestions. All remaining errors are mine. 1

2 MAKING FRIENDS MEET 1. INTRODUCTION 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 (003), Grossman (00) and Goyal et al. (006)). Similar features are observed for the cooperation of actors in movies (Watts, 003) 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 (199) 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. 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 (008) 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 (007) as well as Kleinberg et al. (008).

3 MAKING FRIENDS MEET 3 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 (008) 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 briefly discusses related literature. Section 3 presents the main model and payoff structures. Using this model, Sections 4 and 5

4 MAKING FRIENDS MEET 4 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.. LITERATURE CONTEXT 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 (008). 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 (007) 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 (007) 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

5 MAKING FRIENDS MEET 5 externalities play a key role. The seminal contribution in this area is Currarini and Morelli (000), 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 (007). 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 (199). 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 (007). 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 (007) focus on direct link creation and do not implement an introduction mechanic. A similar model is also explored in Kleinberg et al. (008), again without allowing for introductions. 3. THE MODEL 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,,..., 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

6 MAKING FRIENDS MEET 6 with geodesic distance between nodes. 3 Formally, total benefit b(l(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 / C i (g), (1) b(l(i, j; g)) > 0 if j C i (g). C i (g) denotes the set of players to which player i is connected in network g (the component of i in g) and l(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(l) is decreasing in l. I next turn to the allocation of surplus among the agents engaged. I follow Goyal and Vega-Redondo (007) 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 b(l(i, j; g)) () π i (g) = + e(i, j; g) + b(l(j, k; g))i {i E(j,k)} η + e(j, k; g) i (g)c j C i (g) j,k N 3 See Bloch and Jackson (007) for a detailed discussion of this family of models. 4 The qualification of considering shortest paths only differs from Goyal and Vega-Redondo (007). 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. 5 The assumption is consistent with a model of bargaining without replacement in the limit when bargaining frictions disappear (Siedlarek, 01). See Kleinberg et al. (008) for a different approach in which intermediation benefits decay with the number of alternative paths in a gradual way.

7 MAKING FRIENDS MEET 7 where η i (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 π i (g) in equation 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. EFFICIENT NETWORK FORMATION WITH INTRODUCTIONS 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.

8 MAKING FRIENDS MEET 8 Proposition 4.1. The efficient network structure in the model with introductions is: { (1) The empty network if b(1) c min d n b(), n }. 4 d () The star network composed of non-mediated links and including all n players if d n b() b(1) c b(). { } (3) The complete network if b(1) c max 4n d, b(). The complete network is composed 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) c, 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 c < b(1) b(), 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() Complete Network Star n b(1) c b() 0 b() FIGURE 1. Efficient Network Configurations

9 MAKING FRIENDS MEET 9 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 n 4 b(), 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. STABILITY ANALYSIS 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. (008), 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, 007, 199). (3) (4) b(1) > b() > 0 b(l) = 0 for all l 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 6 With the proviso that in their model benefits of b(l) accrue for each of the agents, whilst in this paper we have one b(l) to divide.

10 MAKING FRIENDS MEET 10 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 (006) 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, π i (g) + π j (g) π i (g ij) + π j (g ij) b. (Search) ij / g, π i (g) + π j (g) π i (g + ij) + π j (g + ij) d c. (Introduction) j, k, l such that jk g, kl g and jl / g, π i (g) π i (g + kl) i {j,k,l} 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 (006), 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 7 See footnote 5 in Bloch and Jackson (006) for a discussion.

11 MAKING FRIENDS MEET 11 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.) 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.. 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. 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) c is in the interval [ 1 b(), b()] and in addition search costs are sufficiently high (d > 1 [b(1) c]), 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. Network Efficient Myopically Stable { } Empty b(1) c min d n b(), n 4 d b(1) c d Star b(1) c d 1(n )b() b(1) c 3 (n )b() b(1) c b() b(1) c b() { } b(1) c d + 3 b() Complete b(1) c max 4n d, b() b(1) c 1 3 b() TABLE 1. Parameter Ranges for Efficient and Myopically Stable Networks 8 Note 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.

12 MAKING FRIENDS MEET 1 d Empty, Star & Complete Empty Empty & Star Empty & Complete Complete b(1) c 3 (n )b() 0 1 b() b() Star Star & Complete FIGURE. 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) c > b() 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

13 MAKING FRIENDS MEET 13 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 j i i i k (A) No neighbours (B) Unshared neighbour k (C) Shared neighbour k FIGURE 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) from the connection with k and b() 3 from the connection with each other, which is intermediated by k. k receives a payoff b(1), from direct links to i and j as well + b() 3 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() > 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 longer essential to the connection of i and l, leaving k with an additional loss of b() 3. The saved intermediation rent is shared between i and l, with both receiving an incremental b() 6. Now, in the context of Definition 5.1, transfers are only feasible between the three

14 MAKING FRIENDS MEET 14 agents involved in the introduction, leaving the group with a loss of b() 6 relative to the other configurations without a shared neighbour. Thus, the network in Figure 3c can be stable if b(1) b() b() 6 0, which is equivalent to b(1) 6 7 b(). 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. Proposition 5.. 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 b(1) c parties to be introduced is at least 6 1. b() 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() = /3, the minimum individual clustering of a node in a position to facilitate an introduction is at least 3/4. Furthermore, Proposition 5. implies that in a stable network, every node is connected to at most a single end node, excluding inter alia stars as possible outcomes 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 η i (g) in g. Let j be a the highest degree node of the set of nodes not in C i (g). Label j s degree η j (g). Then: (5) 3 [b(1) c d] η i (g) + η j (g) b() Proposition 5.3 implies that for search costs d < b(1) 3b() + c there will be at most one component 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

15 MAKING FRIENDS MEET 15 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. LINK FORMATION GAME 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 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 a i = [ aij ] j N\{i}, where a ij {0, 1} and a ij = 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. a ij = a ji = 1 (see Myerson (1977)). The links resulting from this stage form an intermediate

16 MAKING FRIENDS MEET 16 j Stage 1: Link Anouncement j Stage : Introductions j k k k i i g i g FIGURE 4. Two Stage Game of Link Formation network g, where I denote g ij = g ij = 1 to indicate a connection between i and j in this network. Following the link announcement period, players can conduct introductions in period two in a two-stage subgame based on the intermediate network g. Introductions can take place where an opportunity in form of an open triangle exists under g. In such cases, the 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 allow for proposals conditional on g, the outcome of Stage 1. 9 Player k can propose 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 p k ( 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 r i,{(i,j),k} {Accept, Reject}. A proposed introduction {(i, j), k} is conducted if and only if (6) r i,{(i,j),k} = r j,{(i,j),k} = {Accept} 9 NB: 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.

17 MAKING FRIENDS MEET 17 I label the set of conducted introductions I with typical element {(i, j), k} and the resulting final network g. In summary, a strategy s i for player i is defined by: (7) s i = {a i, p i ( g), r i ( g, P)} 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 t in compensation. 6.. 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 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: If k is not essential to the connection between i and j in g, then for the connection ij, payoffs to each player change by b(1) c b() 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. 10 See also the discussion in Jackson (008, Chapter 6.1).

18 MAKING FRIENDS MEET 18 If k is essential to the connection between i and j in g, then in addition to a shorter 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

19 MAKING FRIENDS MEET 19 Vega-Redondo, 007) 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 s i ins i, Π i (s ) Π i (s i, s i ); and b. for any pair of players i, j N and every strategy pair (s i, s j ) S i S j, Π i (s i, s j, s i j ) > Π i(s i, s j, s i j ) Π j(s i, s j, s i j ) < Π j(s i, s j, s i j ) Under this equilibrium concept, for t < b(1) b() c two players rejecting a proposed introduction does not constitute an equilibrium - the bilateral deviation to accepting would be profitable for both players. Lemma 6.1. For each subgame starting at the response stage, if t < b(1) b() 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. 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 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: 11 This concept is closely related to that of pairwise Nash equilibrium described in Calvó-Armengol and İlkılıç (009), 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.

20 MAKING FRIENDS MEET 0 Proposition 6.1. In any bilateral equilibrium of the proposal subgame starting with an intermediate network g, if 0 < t < b(1) b() 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 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() 3. Proposition 6.. In any bilateral equilibrium of the proposal subgame starting with an intermediate network g, if b() 6 < t < b(1) b() 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. 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.. 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. 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. 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 network g which involves stars as subnetworks, if b() 6 < t < b(1) b() c, in each star all possible introductions will be proposed resulting in a completely connected subnetwork.

21 MAKING FRIENDS MEET 1 Finally, I can establish the following property regarding the change in payoffs experienced by players in the introduction stage. Lemma 6.. 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 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 η i (g) in g. Let j with degree η j (g) be a the highest degree node in any other component. Then: (8) 3 [b(1) c d] η i (g) + η j (g) b() 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 { complete } network arrived at through the star is efficient if and only if b(1) c max 4n d, b(). 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

22 MAKING FRIENDS MEET 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., this link will be closed for transfers in the relevant range, leading to a complete network in the final stage. This deviation is thus profitable whenever d > t and in any case profitable if d > t max = b(1) c b(). 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. ( ) Proposition 6.4. Assume n 4 and sufficient decay such that b() < 3 b(1) 4 c. [ ] Then if d b() b(), b(1) c 6(n ), 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. 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) c. 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. CONCLUSION 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.

23 MAKING FRIENDS MEET 3 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.