Structural holes in social networks

Transcription

1 Structural holes in social networks Sanjeev Goyal Fernando Vega-Redondo Preliminary version: July 2004 Abstract We consider a setting where every pair of players can interact (e.g. exchange goods) and this interaction yields a fixed surplus. An interaction can take place only if the players involved have a connection. This connection may be direct, in which case the two players split the surplus equally. But the interaction could also be indirect (i.e. involving more than two players in the path), and then every player essential to the connection is assumed to get an equal share of the surplus. In the model, the incentives to forming links derive from the possibilities they avail in three respects: () the creation of new interaction opportunities; (2) the rents that can be attained from essential intermediation; (3) the payment of rents that can be avoided by circumventing intermediaries. We focus on an equilibrium notion in which pairs of players may jointly coordinate on the (possibly simultaneous) creation and destruction of the links they control. Our main conclusion is that, so long as costs of forming links are not very small, all (non-empty and strict) equilibrium networks are stars. In such networks, a single agent is essential for all bilateral interaction and there is significant inequality in payoff distribution. We end the paper by showing that the main gist of the analysis is essentially maintained if commitment is ruled out and bilateral deviations are required to be internally consistent. Department of Economics, University of Essex, sgoyal@essex.ac.uk. Department of Economics, Universidad de Alicante, University of Essex, and Instituto Valenciano de Investigaciones Económicas, vega@merlin.fae.ua.es.

2 Introduction There is a growing consensus among social researchers that a proper understanding of many socio-economic phenomena cannot be dissociated from the underlying social network in which they are embedded. This pertains to a wide variety of problems ranging from labor markets (Granovetter, 973), to industrial R&D (Gulati et al., 2000), technological diffusion (Valente, 996), scientific collaboration (Newman, 200 and Goyal et al., 2004), modern inter-industrial trade (Kranton and Minehart, 200)), or even trade in historical times (Greif, 994). In all these cases, the aim of the research has been to understand how the underlying pattern of connections (i.e. the architecture of the social network) impinges on the outcome of the process employment, trade, or the scope for technological diffusion. Clearly, one of the features of network architecture that will have an important bearing on the network benefits accruing to the different individual concerns their relative importance in connecting others. Thus, for any particular agent, one would expect that the payoff she is able to muster would depend on how indispensable she is in bridging valuable connections among other individuals. These ideas are at the heart of the notion of structural holes introduced by Ronald Burt (992). Over the past decade, Burt and several others have elaborated on this notion in both theoretical and empirical terms. In particular, there is extensive empirical work done on this subject. This work may be summarized as follows. Even after controlling for individual characteristics, we are left with a substantial variance in individual performance in organizations (measured in terms of salaries and promotions). The differences in structural location of otherwise similar individuals in particular whether they bridge structural holes in the social network explains a significant part of this variance. Given the substantial rewards from network positioning, it seems natural to postulate that an individual will make investments to become a bridge (critical to the interconnectedness of the network) while other individuals have an incentive to prevent this from happening by forming alternative routes to interaction. In this paper, 2

3 we ask what will be the shape of networks when (ex-ante identical) individuals can rationally form links and therefore create positional advantages for themselves while others have an equal incentive to prevent them from acquiring this positional power. We develop a simple model of network formation to address this question. We consider a setting where it is costly to form links but every pair of connected players can interact (say exchange goods or communicate) and this yields a fixed surplus. The interaction can take place only if the players have a connection, direct or indirect. If the connection is direct (i.e. without intermediaries), the two players split the surplus equally. Otherwise, the interaction occurs through an indirect connection and other additional players are required to generate the surplus. This leads us to introducing the idea of essential players. A player k is said to be essential to an interaction between i and j if k lies on every path between i and j in the network. In general, of course, there may be more than one essential player on a path. Under the assumption that any essential player has the power to block the interaction, we invoke a standard bargaining approach to postulate that the surplus generated by some i and j is split equally between all the players (including i and j) essential for the connection. This formulation gives rise to three types of incentives. The first incentive is the standard access advantage: individuals would like to join the network so as to create exchange possibilities which in turn create surpluses. The second incentive is related to the rewards from being essential: players would like to place themselves between unconnected others in order to extract rents from intermediation. The third incentive arises out of the desire to avoid sharing surpluses with intermediaries; in other words, individuals will try to circumvent essential players to retain more of the surplus to themselves. In this setup we consider a model of network formation where links are formed by bilateral agreement and any pair of players can jointly deviate to form (and possibly destroy, simultaneously) the links that affect them. Our main result is that, as long as costs are not very low, all non-empty equilibrium 3

4 networks define a star, with a single individual acting as an essential intermediary to all bilateral connections. There are two aspects of the result that we would like to stress: the uniqueness of the equilibrium network and the significant inequality it entails among the payoffs of players who are ex-ante identical. Thus an extreme version of strategic positioning with attendant payoff differences is the only possible outcome in a setting with ex-ante identical individuals. We now briefly sketch the main logic underlying this finding. The basic arguments derive from the incentives mentioned above. A first simple insight is that, due to the role played by access benefits, an equilibrium network must be either empty or connected (i.e. display a single component). Thus, among non-empty networks, we only need to check which connected networks are stable to both unilateral and bilateral deviations. If the network is minimally connected (i.e. has no cycles), long paths cannot be stable. This happens because extremal players benefit from connecting to a central player in order to save on intermediation costs (cutting path lengths) and, on the other hand, a central player is also ready to incur the cost of an additional link because this enhances her intermediation payoffs. Thus there is a natural tendency for minimally connected networks to become agglomerated around central players. But what about the possible existence of cycles? In principle, a cycle containing all players appears to be very attractive. No player is essential in such a network and each of them can access every other. Such a cycle network, however, is vulnerable to the pressures towards becoming essential and reaping the entailed intermediation benefits. To see this, consider two players that are far apart in the cycle and establish a direct link. By simultaneously breaking one link each, they can produce a line and become central in it. Certainly, this means that they must then pay some intermediation costs (which were previously absent, since no player is essential in a full cycle), but their prominent centrality more than offsets those costs through even larger intermediation benefits. A similar argument can be used to break any cycle, thus leaving the star network (i.e. the minimally connected network with shortest distances) as the unique non-empty equilibrium network. 4

5 Our paper is related to research conducted in both economics and sociology during the last decades. In particular, much attention has been recently devoted by economists and game theorists to the theory of network formation. In this respect, the principal contribution of the present paper is to provide a new model of network formation that embodies two novel ideas, introduced for concreteness within the general framework of the connections model (Jackson and Wolinsky, 996; Bala and Goyal, 2000). The first idea is that the total value generated in the network must be distributed in a rival fashion among different individuals, i.e. a larger share enjoyed by one individual must be obtained at the expense of what others get. The second idea pertains to the notion of essentialness. This notion determines which players are critical for a certain fruitful interaction and allocates to them the value thus generated. We now briefly relate our paper to the work in sociology on structural holes. The large body of empirical work on economic advantages of networks positions which we cited earlier generated a natural question: how is it that such advantages arise in organizations and that they remain stable over time in a setting with rational actors who can figure out the rewards from structural positions? Our paper shows that this state of affairs is consistent with a model of rational players who strategically seek to create positional advantages for themselves and also have an incentive in preventing others from becoming central. To the best of our knowledge our model is the first formal model to illustrate the strategic stability of structural holes and the large inequality in payoffs that go with it. [[The paper is organized in four sections. The next section lays out the basic model. Section 3 presents the main results on equilibrium networks where both single players as well as any pair of players can implement any deviation. Section 3 briefly explores robustness of the main results to an alternative notion of strategic stability where bilateral commitment is not presumed. Section 4 concludes.]] Early work includes Aumann and Myerson (988); for more recent work see Dutta, van Den Nouweland and Tijs (995), Jackson and Wolinsky (996), and Bala and Goyal (999). 5

6 2 The Model 2. Link formation game Let N = {, 2,..., n} denote a finite set of ex-ante identical players. We shall assume that n 3. Every player makes an announcement of intended links. An intended link s i,j {0, }, where s ij = means that player i intends to form a link with player j, while s ij = 0 means that player i does not intend to form such a link. Thus a strategy of player i is given by s i = [s ij ] j N\{i}. Let S i denote the strategy set of player i. A link between two players i and j is formed if and only if s ij = s ji =. We denote the formed (undirected) link by g i,j g j,i = and the absence of a link by g i,j g j,i = 0. A strategy profile s = (s, s 2,..., s n ) therefore induces a network g(s). For expositional simplicity we shall often omit the dependence of the network on the underlying strategy profile. A network g = {(g i,j )} i,j N is a formal description of the pair-wise links that exist between the players. We let G denote the set of all possible networks (i.e. all undirected graphs with n vertices). We also let N i (g) = {j N : j i, g ij = } be the set of players with whom player i has a link in the network g, and have η i (g) = N i (g) denote the cardinality of this set. Given a network g, g + g i,j denotes the network obtained by replacing g i,j in network g by g i,j =, while g g i,j denotes the network obtained by replacing g i,j in network g by g i,j = 0. There exists a path between i and j in a network g if either g i,j = or if there is a distinct set of players {i,..., i n } such that g i,i = g i,i 2 = g i2,i 3 = = g in,j =. We shall say that a player k is essential for i and j if player k lies on every path that joins i and j in the network. Let E(j, k; g) stand for the set of players who are essential to connect j and k in network g, where we shall often write simply E(j, k) thus dispensing with the reference to g when there is no risk of ambiguity. Correspondingly, we denote by e(j, k; g) = E(j, k; g) the cardinality of this set. To illustrate matters, note that the number of essential players between j and k is zero if the players have a direct link or, alternatively, if players are located around a circle in the latter case, for every pair of players 6

7 j and k the and every other player i, there is always a path joining j and k that does not include i. On the other hand, in a star every pair of peripheral players has a single and common essential player, namely the center of the star. In our framework, we posit that there are two types of advantages that players get from being part of a network. First, there is the direct advantage from being able to access others; second, there is the indirect advantage that arises from being an essential intermediary between others. We assume that the precise determination of these direct and indirect advantages is settled, through bargaining among all agents involved (independently across different surplus units). Specifically, consider any pair of players j and k that are connected by some network path and may thus can generate a unit of surplus. In general, this unit may be divided among all agents in N according to some vector of non-negative shares z jk = (z jk i ) i N. In the language of Cooperative Game Theory, such a vector defines an imputation for the coalition game associated to the surplus produced by j and k. We shall be interested in the imputation ẑ jk that represents the unique selection of the Core of such coalition game induced by the Kernel (Davis and Maschler (965)). Implicitly, this outcome reflects a bargaining process among the agents in which, eventually, only the players j and k as well as all those in E(j, k) who are essential in establishing the desired connection obtain a positive share of the surplus. In fact, this share happens to be equal for all since every agent l E(j, k) {j, k} has an equal bargaining power, i.e. any of them can similarly block the generation of the unit of surplus. The formal details of the approach are included in the Appendix. When traslated into payoffs, it gives rise to a payoff function that, for every strategy profile s = (s, s 2,..., s n ), defines the (net) payoffs to player i as follows: Π i (s i, s i ) = j N i (g) e(i, j; g) j,k N where c is the fixed cost payed per link by each player. I {i;j,k} e(j, k; g) + 2 η i(g)c, () 7

8 2.2 Equilibrium analysis We study the architecture of networks that are strategically stable and assess their efficiency. Our notion of strategic stability is a refinement of Nash equilibrium that allows for coordinated two-person deviations. Definition A strategy profile s is a Bilateral Equilibrium (BE) if the following conditions hold: for any i N, and every s i S i, Π i (s ) Π i (s i, s i ); for any pair of players i, j N, and every strategy pair (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 j, s j, s i j). We apply the term bilateral equilibrium to a strategy profile where no player or pair of players can deviate (unilaterally or bilaterally, respectively) and benefit from the deviation (at least one of them strictly, for bilateral deviations). This notion refines the original formulation of pair-wise stability due to Jackson and Wolinsky (996) by allowing pairs of players to form and delete links simultaneously. Our analysis will focus on a refinement of BE that we call strict. It rules out the existence of deviations (unilateral or bilateral) that have some consequence (i.e. affect the network) but nevertheless are payoff indifferent for the agents involved. In part, the motivation of this equilibrium concept is dynamic: a gradual adjustment process that allows pairs of players to revise the network in sequence will (only) be absorbed by equilibria of this sort see below for an elaboration. Definition 2 A strategy profile s is a Strict Bilateral Equilibrium (SBE) if the following conditions hold: for any i N, and every s i S i such that g(s i, s i ) g(s ), Π i (s ) > Π i (s i, s i ); 8

9 for any pair of players, i, j N and every strategy pair (s i, s j ) with g(s i, s j, s i j ) g(s ), 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 j, s j, s i j). Finally, we introduce the notion of efficiency considered. In line with the assumption that the setup involves transferable utility, different networks g are assessed in terms of the total surplus generated, W (g) i N Π i(g). Definition 3 A network g is efficient if W ( g) W (g) for all g G. 2.3 Networks A network is connected if there exists a path between any pair i, j N. A network, g g, is a component of g if for all i, j g, i j, there exists a path in g connecting i and j, and for all i g and k g, g i,k = implies k g. A component g g is complete if g i,j = for all i, j g. Two networks g and g are said to have the same architecture if one network can be obtained from the other by a permutation of the players labels. We shall say that a network is symmetric if every player has the same number of links, i.e., η i (g) = η(g) i N. Then we refer to η as the degree of the network. It should be noted that if the number of players is even, then a symmetric network of every degree is possible; this is not true when the number of players is odd. In some parts of the analysis, we explore symmetric networks and then we will be (implicitly) assuming that the number of players in the game is even. The complete network, g c, is a symmetric network in which η = n, i N, while the empty network, g e, is a symmetric network in which η = 0, i N. Another example of a symmetric network is a cycle where η = 2 and the whole set of nodes can be ordered in a list i, i 2,...i n with g i,i 2 = g i2,i 3 =... = g in,i =. We shall say that a network is asymmetric if there is at least one pair of players who have different number of links. One important example is the star 9

10 where there is a single node, i c, with η(i c ) = n while η(i) = for all other i i c. An interesting mixture is what we shall call the hybrid cycle-star network where there is a subset of k nodes C = {i, i 2,...i k } arranged in a cycle and some particular player i x C such that g j,ix = for all j N\C. 3 Results We start by characterizing efficient networks. Proposition If c < n/4 then an efficient network is minimally connected, while if c > n/4 then an efficient network is empty. Proof: It is easy to see that an efficient network cannot be partially connected and that it must be minimal. The aggregate payoff from a minimally (fully) connected network is (n )[n/2 2c], while the payoff from the empty network is 0. It then follows that a minimal connected network is efficient if and only if c < n/4, while an empty network is efficient if c > n/4. We now turn to the study of equilibrium networks, i.e. the networks induced by (S)BE which are simply labelled (S)BE networks. Our first finding is that an equilibrium network is either connected or empty. Proposition 2 A BE network is either empty or connected. The proof of this result makes use of the following technical result which is of independent interest. For any link g i,j = in g, define M i (g i,j ; g) = Π i (g) Π i (g g i,j ). Lemma Consider any network g. If g i,j = and the link is critical then M i (g i,j ; g) = M j (g i,j ; g). Proof: By hypothesis g i,j is critical, and so it follows that i and j lie in different components in the network g g i,j. Let C i (g), C j (g) be the components that contain i and j, respectively, where we shall usually dispense with an explicit 0

11 account of the dependence and simply write C i and C j. The marginal payoff of the link ij for player i, is given by: M i (g i,j ; g) = 2 + k C j \{j} e(i, k) l C i \{i} k C j \{j} e(l, k) l C i \{i} e(l, j) + 2 c where the first two terms refer to access benefits while the latter two terms refer to essentiality benefits. player j form link g i,j as: Similarly, we can write the marginal payoffs of M j (g i,j ; g) = 2 + l C i \{i} e(j, l) l C i \{i} k C j \{j} e(l, k) k C j \{j} e(i, k) + 2 c It follows then that M i (g i,j ; g) = M j (g i,j ; g) and the proof is complete. Equipped with this lemma we can now complete the proof of the connectedness result. Proof of Proposition 2: Let g be a non-empty BE network, and suppose it is not connected. Let Ĉ be the largest component in g. If Ĉ contains 2 players then it is immediate that both players in this component have a strict incentive to form a link with any other player not in the component. Given Lemma it follows that every player outside Ĉ in g also has an incentive to form a link with these players. Thus in the rest of the proof we concentrate on the case where Ĉ contains more than 2 players. This also means that there is at least one player (say) i Ĉ such that η i 2. Let Ni m (g) = {j C i : e(i, j) = m)} be the players whom i accesses via m essential players and let η m i payoffs of this player i in network g are then given by: η 0 i (g) 2 + η i (g) 3 + η2 i (g) 4 m (g) = Ni (g). The ηr i (g) L + 2 η i(g)c (2)

12 for some r n 2. Since g is an equilibrium, it follows then [ η 0 i (g) + η i (g) + η2 i (g) ηr i (g) ] c. (3) η i (g) r + 2 Now let us examine marginal returns for player j / Ĉ from forming a link with some player i. Suppose, for simplicity, that player j is a singleton component. Then the marginal returns to j are given as follows: η 0 j (g + g i,j) 2 + η j (g + g i,j) 3 + η2 i (g + g i,j) ηr i (g + g i,j) r + 2 c. (4) Note now that for every k Ĉ\{i}, e(j, k; g + g i,j) = e(i, k; g) +. Thus ηj m(g + g i,j) = η m i (g) for every m and ηj 0(g + g i,j) =. Using these facts we can write the marginal returns of player j from the link with j as follows: Now we argue that 2 + η0 i (g) + η i (g) ηr i (g) c. (5) 3 4 L + 3 c 2 + η0 i (g) + η i (g) 3 4 > [ η 0 i (g) 2 2 η i (g) + η i (g) 3 [ η 0 i (g) ηr i (g) L η i (g) 3 + η2 i (g) 4 + η2 i (g) ηr i (g) ] r ηr i (g) r + 2 The first inequality is immediate, while we use η i (g) 2 in deriving the second inequality and equation (3), in deriving the final inequality. We now apply Lemma to conclude that player i also has a strict incentive to form a link with j, given that all existing links are retained. But note that given that link g i,j is formed, player i has no incentive to delete any of his erstwhile links since (roughly speaking) the marginal returns from each of these links ] 2

13 has actually increased. Thus players i and j have a strict incentive to form an additional link and this link is a credible two-person deviation. These arguments extend directly to cover the case where j belongs to a non-singleton component. Thus g is not a BE network, a contradiction that completes the proof. We now look more closely at the architecture of equilibrium networks. The following result gives examples of two simple networks that arise in equilibrium. Proposition 3 If /6 < c < /2 + (n 2)/6 then a star is a BE network, while if c > /2 then the empty network is a BE network. Proof: We first prove that the star is a BE network for the parameter range specified. First consider the incentives of the center. The payoffs of the center are given by n 2 + (n )(n 2) (n )c. (6) 3 2 Note that the payoffs from a subset of links k are given by k 2 + (k)(k ) kc. (7) 3 2 It can be verified that the marginal payoffs are increasing with respect to k, and so the optimal number of links is either zero or n. It follows then that center maintains all links if c < /2+(n 2)/6. We next examine the incentives of peripheral players. Their payoff in the star is given by /2 + (n 2)/3 c. The payoffs from having two links are given by /2 + /2 + (n 3)/3 2c. It then follows that the peripheral player will choose to maintain one link with the center if /6 < c < /2 + (n 2)/3. This proves that the star is a BE network if /6 < c < /2 + (n 2)/6. It is straightforward to see that the empty network can be supported by a BE for c > /2. The star is a BE network for a wide range of parameters and this is due to the fact that centrality generates large payoffs from essentialness. However, a star also exhibits an extreme form of essentialness, with only one player being 3

14 essential for all pairs of players. This raises the question: are there other perhaps more egalitarian network architectures that can arise in BE? The following results provide a complete answer to this question, for large societies. Theorem Suppose that n is large. If /6 < c < /2 then any SBE network is a star while if /6 < c < /2 + (n 2)/6 then any non-empty SBE network is a star. Finally, if c > /2 + (n 2)/6 then the unique BE network is empty. This result provides a complete characterization of SBE networks for large societies. The proof proceeds through five steps. The first step shows that any minimally connected BE network is a star, for large n. The second step shows that there cannot be two or more cycles in an SBE network. The third step shows that there cannot be a single cycle containing all players in equilibrium. The fourth step shows that a non-empty SBE network with a cycle must be a hybrid cycle-star network. The final step shows that a hybrid cycle-star network cannot be SBE for large n. This means that for large n the only candidates for SBE networks are the star and the empty network. Lemma 2 Given c, there exist some ˆn(c) such that if n ˆn(c), any BE network that is minimally connected is a star. Proof: Consider a g that is minimally connected (thus, a tree) but not a star. Then, there are two players, say i and j, such that (a) E(i, j; g) 2; (b) they are end players, i.e. η i (g) = η j (g) =. Then, it readily follows that for at least one of them, say i, the following holds: There is a player x with e(i, x; g) = (thus, x is two steps away from i in g) and the set of players k for whom x is essential in connecting i and k has a cardinality that is at least (n 4)/2, i.e. {k N : x E(i, k; g) (n 4)/2. We argue that such a network g cannot be induced by an equilibrium. To see this, first note that, given the linking cost c, the number of essential players that can be supported in equilibrium between any two players has some (finite) upper bound ê(c), independent of n. Consider then the possibility that i and x 4

15 were to form a link. The gross gains, π i and π x, induced by that change for i and x (if all other links were to remain in place) is bounded below as follows: min{ π i, π x } (n 4)/2 ê(c) (n 4)/2 ê(c) = n 4 2ê(c)(ê(c) ). This expression is larger than c if n is large enough, which implies that both i and x benefit from a deviation that creates a link between them and keeps all other links. [[Argument for bilateral-proofness: Consider now the possibility that either i or x would want to delete some of their other links, simultaneously with the creation of the link between them. Would that alter their incentives to create that link? On the one hand, it is clear that x cannot profit from deleting any link, in particular the link to the single player y that was required to connect x to i in the network g. (Note that this link generates gross benefits to x that are again proportional to n.) Instead, it could be optimal for player i to remove the link to y, as she establishes the new link to x. This removal, however, can only decrease gross payoffs of x by /6, that of course cannot affect the optimality of creating the link for i if n is large.]] We now consider non-minimal networks. The following result shows that there cannot be more than one cycle in a strict equilibrium. Lemma 3 There can be at most one cycle in a SBE network. Proof: Suppose g is an equilibrium network and there are two or more cycles in it. Let χ = (i, i 2,...i n ) be ordered set of players in one cycle, and let χ 2 = (j, j 2,...j m ) be those in the other cycle. Since g is connected it follows that there are two possibilities: () cycles have common players and (2) cycles have no common players. We take these up in turn. (). Cycles have players in common: If there is a single common player i in the two cycles then it is easy to see that the partners of i (say) i 2 χ and j 2 χ 2 have a strict incentive to delete their links with i and instead form a link with 5

16 each other. (Throughout, we shall abuse notation and write i χ if i is one of the nodes in the ordered collection of nodes specifying χ.) Consider next the case with two or more players in common. Let (i, i 2,...i k ) be the players in common. Suppose that k 3; the case of k = 2 is simple and omitted. Then there exist players i, i x, and j y with the following properties: i x χ but i x / χ 2, while j y χ 2 and j y / χ and g i,i x = g i,j y = g i,i 2 =... = g ik,i k =. Note also that like player i, i k must again have links with a player who belongs to one of χ and χ 2 only. It then follows that i k and j y have at least a weak incentive to delete their current link with i k and i respectively and instead form a link with each other. It then follows that g cannot be sustained by a SBE. (2). Cycles have no common players. Since g is a SBE network it is connected and so there exists a path between the two cycles. Let (i, i 2,..., i k ) be members of such a path with i χ while i k χ 2. Suppose g i,i x = and g ik,j y =, where i x χ and j y χ 2. Now it is easy to use a variant of the earlier argument for case above to show that players i x and j y have a strict incentive to delete their link with i and i k and instead form a link with each other. The proof of the lemma is complete. We now turn to the networks with a single cycle. Lemma 4 Suppose n 4. 2 in a BE. A cycle containing all players cannot be sustained Proof: Consider two player i and j who are furthest apart in terms of geodesic distance in the cycle. Now consider the deviation in which each of the players deletes one link and they form a link with each other in such a way that they create a line. Assume, for simplicity, that n is even, so that there are (n 2)/2 players to one side of player i and (n 2)/2 players to the other side of player j in the line created. We now show that players i and j will strictly increase their payoff with this coordinated deviation. 2 We note that for n = 3 a complete network can be sustained in equilibrium for c < /6. 6

17 We proceed in two steps: the first step is to show that individual payoffs are strictly increasing as we move toward the center in the line. The payoffs of an individual player consist of two components, the returns from accessing others and the returns from being essential on paths between pairs of other players. Number the players on a line as, 2,...n. We show that the payoffs of player l + are greater than the payoffs of player l, where 2 l < n/2. The access returns to player l are given by l + l n l + while the access returns to player l + are given by (8) l + + l n l It now follows that access returns for player l + are larger than access returns for player l if l < n/2. We now turn to the returns from being essential. The returns to player l from being essential consist of two parts: the returns from being essential to pairs of players i and j where i < l and j > l +, and a second part which consist of returns from being essential for pairs i < l and player l +. The essentialness payoff to player l can be written as follows: (9) l n i= j=l+2 l e(i, j) i= e(i, l + ) + 2 (0) Similarly, the essentialness payoffs to player l + consist of two parts: those that arise out of linking players i < l with players j > l + and those that arise out of linking player l with players j > l +. These payoffs can be written as follows: l n i= j=l+2 e(i, j) n j=l+2 e(l, j) + 2 () 7

18 The first part of the essentialness payoffs to the two players are equal, while the second part of the payoffs are greater for player l + if l < n/2. Note that players < l < n all have two links in a line network and hence have the same costs. Combining the payoffs from access and essentialness we can then conclude that payoffs to player l + are greater than payoffs to player l, so long as < l < n/2. A similar argument shows that payoffs to player k are larger than payoffs of player k + if n/2 k < n. This completes our argument that payoffs are increasing as we move toward the center of the line. Let g C and g L denote the networks prevailing before the contemplated deviation and after it (i.e. the cycle and the line with i and j at the center, respectively). To show that i and j indeed obtain higher payoffs under g L, note that the aggregate gross payoffs obtained in both cases are the same. Let H W (g C ) + nc = W (g L ) + (n )c denote this common value. The above argument implies that i and j enjoy a higher share of H under g L than under g C. Thus, they also obtain higher gross payoffs under g L. Since their linking cost is the same in both cases (i.e. 2c), it follows that they obtain higher net payoffs as well, which completes the proof. We now move to the fourth step of the proof and show that any equilibrium network with a cycle must be a hybrid star-cycle network. Lemma 5 Consider an equilibrium network which has a cycle. It must then have the hybrid star-cycle architecture for large enough n. Proof: Suppose that g is a non-empty network sustained by a SBE. Then we know from Proposition 2 and Lemma 3 that it has at most one cycle and the rest of players belong to (possibly different) trees with a root at some node in the cycle or all belonging to a single tree if there is no cycle. Let x be the number of players who are outside the cycle, while y = n x is the number of players in it. Note that y 4 since y = 3 cannot be sustained in equilibrium 8

19 given that c > /6. Next we argue that, for any such c, there is a y(c) such that y y(c) in equilibrium. To prove this claim, first we note that, in view of Lemma 4, there must be at least one nondegenerate tree in the network g, with the root given by some player i and this player having a link to some other player j who does not belong to the cycle. Suppose now that the cycle is non-degenerate and let k be one of the nodes in the cycle that are connected to i. Clearly, player k always profits from shifting to j the link that connected it to i. On the other hand, j also benefits from such an adjustment if (y )/6 > c. So given a c, in order for the cycle to be stable, we must have the converse inequality, which implies that y y(c) 6c +. Now we argue that all nodes that do not belong to the cycle must be connected to a single node in the cycle. The initial observation is that, if n is large enough, there cannot be two distinct trees with different roots lying in the cycle. To see this, consider one of those subtrees that has a number of nodes at least equal to [n y(c)] /y(c) = (n/y(c)). At least one such tree must exist since there are at most y(c) nodes in the cycle. Let i be the root of this tree and i 2 be the root of any other cycle. Now consider any node j in the second tree different from its root, i 2. A straightforward adaptation of the arguments used in the proof of Lemma 2 lead to the conclusion that, if n is large, j and i can both profit from forming a link, whether or not j maintains its link with i 2. The reason is that, since the number of essential players e(l, l ) for any pair of players, l and l, is bounded by some ê(c), independently of n, the gross gains from the link between j and i grow linearly with n for both players, independendently of whether player j maintains the link to i 2. (Clearly, i cannot profit from destroying any of its links.) Once established that there can be at most one tree connected to the cycle through its root, again relying on the arguments introduced in Lemma 2 we arrive at the conclusion that, for any two nodes in the tree, u and v, the number of essential players e(u, v). This still leaves open the possibility that the tree consists of a star with a center at some node i c that is not part of the cycle but 9

20 has a link to some other node k in the cycle. But, in that case, the node i c and any of the neighbors of k in the cycle, say k, can jointly profit from establishing a direct link, even if k destroys the link with k. The final step in the proof shows that the hybrid cycle-star network is not an equilibrium for large n. Lemma 6 Suppose that c > /6. The star is the unique non-empty equilibrium, for large n. Proof: Suppose that g is non-empty equilibrium network and it has the hybrid cycle-star structure. As in the proof of Lemma 5, let x denote the number of peripheral players in the star, while y = n x stands for the number of players in the cycle. As explained, it may be assumed that y y(c) for a function y(c) that is independent of n. Thus, fixing some c, consider the class of hybrid networks g in which y y(c). Let i be the player in the center of the star and suppose that j and k are the players in the cycle who are connected to i. We show that players j and k have a strict incentive to form a link if the set of peripheral players x n y(c) is sufficiently large. The payoffs of players j and k in the hybrid network g H are given by π j (g H ) = π k (g H ) = x 3 + y 2 2c (2) Now consider a deviation by players j and k in which player k deletes his link with player i and player j deletes his link with player m in the cycle and instead players j and k form a link with each other. The resulting network is a minimal network g in which there are x peripheral players and a line starting with player i which consists of y players. The payoffs of player j in g are given by π j (g ) = x 3 + y 2 + k=2 These payoffs are bounded below by y+ k + x k=4 y k + k=3 2c (3) k x 3 + xy 2 2c = M (4) y + 20

21 Next note that M > x/3 + (y )/2 2c if x > y y + 2 y 2 (5) Since y 4, the right hand side is increasing in y and bounded above by [y(c) ][y(c)+]/2[y(c) 2]. The final step is to note that x = n y n y(c) and so (5) applies for sufficiently large n. Thus player j has a strict incentive to switch links to player k for large n. We now turn to the incentives of player k. The payoffs of player k in g are given by π k (g ) = x y k=2 These payoffs are bounded below by Note that M > x/3 + (y )/2 2c if y+ k + x k=5 y k + k=4 2c (6) k x 4 + xy 3 y + 2c = M (7) x :>: 6(y )(y + ) y 37 (8) Since y 4, the right hand side is positive and increasing in y and so is bounded above by 6[y(c) ][y(c) + ]/[y(c) 37]. Note that x = n y n y(c) is larger than this term for sufficiently large n. Thus player k has a strict incentive to switch links to player j, for sufficiently large n. The proof is complete. 4 Extension: Two-person coalition proof networks In our analysis so far we have assumed that a deviation by two players is credible so long as it yields higher payoffs to both players. We have not looked at profitable deviations from the agreed upon deviation, in the spirit of the standard notion of coalition proofness. In this section we will examine the implications of 2

22 this further requirement, which of course can only enlarge the set of equilibria in general, only a subset of bilateral deviations qualify as valid. We start with a definition of bilateral proofness in our context, focusing directly on the strict version of this concept. Definition 4 A strategy profile s is a Strict Bilateral-Proof Equilibrium (SBPE) if the following conditions hold:. for any i N, and every s i S i such that g(s i, s i ) g(s ), Π i (s ) > Π i (s i, s i ); 2. For any pair of players i, j N, and every strategy pair (s i, s j ) with g(s i, s j, s i j ) g(s ) and Π i (s i, s j, s i j ) Π i(s i, s j, s i j ), one of the following two conditions hold: (a) Π j (s i, s j, s i j ) < Π j(s j, s j, s i j ) (b) k, l {i, j}, k l, and some s k S k, such that Π k ( s k, s l, s k l ) > Π k (s k, s l, s k l ). Most of the insights obtained in Section 3 through the SBE concept are maintained if we consider the the less demanding SBPE concept. (Note that SBPE requires valid bilateral deviations to be internally consistent, which can only enlarge the set of equilibria as compared with the SBE notion.) Indeed, reviewing the Propositions and Lemmata that underlie the proof of Theorem, it is straightforward to check that all of them continue to apply for the SBPE concept, except for Lemma 4 and Lemma 6. results in turn. We next address each of these First, Lemma 4 rules out that a single cycle involving all players may define a SBE network. This network architecture can no longer be ruled out as a SBPE network, as established by the next result. Proposition 4 Given any c > 0, there exists an n(c) such that for all n n(c), a cycle containing all players is a pcp-equilibria. 3 3 The stability of the cycle which is not minimal is an instance of the inefficiency discussed in Jackson (2003) where players will over-connect to avoid positional disadvantages. 22

23 Proof: In a cycle, all players are symmetrically located, so we need only consider the incentives for a typical player, say some player i. First note that the payoffs of player i in a cycle are (n )/2 2c. The payoffs from deleting both links are 0 and clearly for a given c, there is always a large enough n such that deleting both links is not optimal. Consider next the deviation of deleting one link. If player i deletes one link and keeps the other link as in the cycle then he becomes an end-player of the line network. In this network his payoffs is given by n k=2 The payoff from both links in cycle are higher if n 2 k c (9) n k=2 k > c (20) Clearly this inequality holds for large enough n. We next consider the deviation in which player i deletes both links but forms a new link (say) with the center of the line network that arises. Again, a variation of the above argument shows that for large n this reduces payoffs. We finally consider deviations by player i in which player i maintains one of the links as is but changes the partner of the other link. Suppose that in the cycle he has a link with i and i +. In the deviation, he maintains the link with i but deletes the link with i + and instead forms a link with some player k. If player k retains all his links as in the cycle, it is easy to see that the payoffs of the player go down strictly since the costs remain the same (he maintains two links) while the gross payoffs decline since player k is essential for accessing at least one player, namely i +. So this deviation is not profitable. If player k deletes both his links then the payoffs from the deviation are still lower and so it clearly not profitable. We turn to the final case, where players i and k coordinate and link up with each other but delete a link each so that the new network is a line. From Lemma 4 it follows that players such a deviation is profitable. However, now we need to check whether this deviation is credible: 23

24 do the players have an incentive to actually delete one of their links? We show that for large n, at least on the players has a strict incentive to retain their links in the cycle. Number the players in the line as, 2..., n, from left to right. So there are i players to the left of player i while there are n k players to the right of player k in the line. Suppose without loss of generality that i n k. Player k gets a payoff i+ l=2 /l in the line network from these players, 2..., i. The payoffs can be increased to i/2 if player k forms a link with player. Clearly, it is profitable for player k to deviate from the deviation and form a link with player, if n is large. It is similarly possible to show that player would have an incentive to form a link with player k, if n is large. Thus the deviation in which i and k deviate to create a line is not two-person coalition proof. Finally, we note that starting from a cycle it is not profitable for player i to form any additional links. The proof for cycle being a SBPE network is complete. Secondly, we note that the proof of Lemma 6 crucially depends on the fact that players are not allowed to deviate unilaterally from a bilateral deviation. (Specifically, observe that this proof contemplates a joint deviation by two players in the cycle that is not in the interest of any one of them, under the assumption that the other abides by it.) In fact, it is not difficult to construct examples where a full-fledged hybrid cycle-star network is SBPE for a large enough population. As a brief outline of matters, consider some such network where there are r nodes involved in the cycle. We know from the proof of Lemma 5 that, in order for such a configuration to be an equilibrium, it must be that r < 6c +. Under the latter inequality, it is clear that no (bilateral) deviation involving a player outside the cycle can be profitable for this player. Thus, we are left with deviations involving pairs of non-neighboring players in the the cycle, as those considered in the proof of Lemma 6. Let i and j be any such players. The only way in which they can gain is by attempting to become central. This must involve establishing a link between them and, simultaneously, destroying one of their respective links to the cycle. Now we argue that any such deviation is 24

25 not inmune to a unilateral omission by at least one of the players in her link deletion. By keeping her link, there is at least one of the players, say i, half of the players in the cycle without any intermediation costs. Thus, for large r, the gain Π i from retaining the link (assuming j abides by the deviation) can be approximated as follows: Π i r 2 2 ln r 2 c. Now suppose r = c/α for some α (/6, /4). Then, we have Π i > 0, while still maintaining the condition that r < 6c +. This confirms that, under high enough costs and a large population, there is a SBPE that supports a hybrid cycle-star network. The above discussion indicates that, under the requirement of bilateral proofness, a richer set of network architectures can be supported at equilibrium specifically, the cycle and a hybrid cycle-star network are also possible, in addition to full stars. Thus, in this sense, the force towards polarization that proved so acute under unqualified bilateral stability becomes mitigated when we insist that bilateral deviations be internally consistent. We now argue, however, that such alternative architectures (a cycle and a hybrid cycle-star) are, in effect, rather fragile. Specifically, they should not be expected to last under the pressure of occasional perturbations that affect some of the links. A natural way of formulating precisely this idea is to check the performance of some suitably modelled dynamics of adjustment under the pressure of ocassional random mutations. In what follows, we outline such a dynamic model and explain the gist of the argumen. While we dispense with the formal details, they are available from the authors upon request. Consider a dynamic process of bilateral adjustment in which, at every point in (discrete) time t =, 2,..., a pair of players is selected at random and are given the option to create a link between them (if this link is not already in place) and, simultaneously, destroy any subset of the existing links whose maintenance they control. In line with the considerations that underlie the SBPE concept, 25

26 let us suppose that, in evaluating any joint move by the two players involved in the adjustment, only those that are internally consistent are admitted as valid. That is, joint moves are only judged valid if they are inmune to a unilateral deviation (that, in this case, would always involve keeping a link that should otherwise be maintained under the contemplated move). In addition to this adjustment dynamics, suppose that there is also a slow dynamics of link decay. Specifically, postulate that, at (the end of) every t, each of the existing links vanishes with some probability ε, conceived as small. The dynamics just described may be parametrized by ε. If ε = 0, we simply have an adjustment dynamics, whose stationary points are SBPE networks these include cycle, hybrid cycle-star, or star networks, as well as the empty network if c > /2. Let us now ckeck the dynamic robustness of each of the non-empty architectures if one of the existing links, randomly selected, simply vanishes. The implicit assumption here is that such a perturbation is very infrequent so that no more than one instance of it may occur with significant probability before the adjustment dynamics restores the stationarity of the situation. Consider first the case of the cycle or a hybrid cycle-star. For concreteness, let us focus on the latter case, the (pure) cycle network, since the second one is analogous. Suppose that one of the links in the cycle vanishes. Then we argue that (if the population is large enough) there is chain of bilateral-proof adjustments that lead to the star, where the dynamics is eventually absorbed. After the deletion of the link, the resulting network is a line, and let players be indexed consecutively from one end to the other, i =, 2,..., n, with n odd for simplicity. Suppose that player n 3 2 (who is linked to player n 2 ) is given the option of linking to the central player in the line, n+ n 3 2. Player 2 is indifferent to do so (in which case she will delete the link to n n+ 2 ) but player 2 strictly benefits. This is the case for any c, provided n is large enough, and the entailed adjustment is bilateral-proof. Now we may proceed iteratively with players that are two steps apart from the central player in the current network (e.g. either player n 5 2 or n+3 2 at the next iteration) until a star centered at player n+ 2 is 26