Communication Networks with Endogenous Link Strength

Transcription

1 Communication Networks with Endogenous Link Strength Francis Bloch, Bhaskar Dutta October 14, 2005 Abstract This paper analyzes the formation of communication networks when players choose how much to invest in each relationship. Players have a fixed endowment that they can allocate across links. We consider alternative technologies relating individual investments to link quality. Furthermore, we suppose that, in indirect connections, players use the path which maximizes the product of link strengths. When investments are perfect substitutes, the efficient network architecture is a symmetric star, which is stable with respect to individual deviations, but may be unstable when pairs of agents can jointly deviate. When investments are perfect complements, the efficient network is never a tree. For small numbers of players, the symmetric circle is the efficient network architecture, and is immune to individual deviations and deviations by pairs of players. When the value of an indirect connection only depends on the quality of the weakest link, the efficient network must be a tree. For perfect substitutes, the symmetric star is efficient and stable; for perfect complements, the symmetric line is efficient and stable. JEL Classification Numbers: D85, C70 Keywords: communication networks, network reliability We thank C.Blackorby and M. Jackson for helpful discussions. We have benefited from the comments of participants in seminars and conferences at Guanajuato, Paris and Montreal. Bloch is at GREQAM, Universite d Aix-Marseille, 2 rue de Charite, Marseille, France.Heisalsoaffiliated with the University of Warwick. Dutta is in the Department of Economics, University of Warwick, Coventry CV4 7AL, England. 1

2 1 Introduction Following a long tradition in sociology, economists have recently focussed attention on the role of social networks in economic activities. One of the main contributions of this emerging literature has been to propose strategic models of network formation, where self-interested agents establish bilateral links in order to maximize their utility. Following the pioneering work of Bala and Goyal (2000a) and Jackson and Wolinsky (1996), most of the literature assumes that all links are of the same quality, and that agents choose the set of agents with whom they link. In this paper, our objective is to allow for variations in the strength of bilateral links, and analyze a model where agents not only choose with whom they link, but also how much they invest in each relationship. 1 Our analysis is centered around communication networks networks where agents derive positive benefits from the agents with whom they are connected. Communication networks can either be formal networks (like the phone or internet), or informal networksofsocialrelations. 2 Both in formal and informal networks, the quality of communication links varies widely across the network. In formal communication networks, the reliability of a communication link depends on the physical characteristics of the connection, which is never constant across the network. In informal social networks, the strength of a social link depends on the frequency and the length of social interactions which also display a wide variation across a given network. Communication networks thus represent an obvious testing ground for a theory of network formation with endogenous link strength. The modeling of network formation with endogenous link strength poses new conceptual difficulties, which were absent in the literature where links have a fixed, exogenous value. First, the technology transforming individual investments into the quality of a bilateral link needs to be specified. In this exploratory analysis of the formation of links with endogenous quality, we focus attention on two polar cases: perfect substitutes and perfect complements. 3 When investments are perfect substitutes, as in Bala and Goyal 1 See Goyal(2004), which emphasizes the importance of studying networks where the srength of links can be chosen endogenously. 2 The role of social networks as communication devices has long been acknowledged. The use of social networks in job referrals has been studied,a mong others, by Granovetter (1974), Boorman (1975), Montgomery (1991), Calvo-Armengol and Jackson (2004). A classic reference on the role of social networks in the diffusion of innovation is Coleman, Katz and Menzel (1966). 3 We will show that the analysis of the perfect substitutes case can be extended to a larger family of technologies which are separable and convex functions of individual 2

3 (2000a), the formation of a link does not require consent from the other agent. In the case of perfect substitutes, as in Jackson and Wolinsky (1996), links can only be formed if both agents agree. The second difficulty stems from the modeling of benefits from indirect connections. In the existing literature, it is assumed that agents always choose to connect through the shortest path, and that indirect benefits are a decreasing function of the length of the connection. 4 When link strength is endogenous, it is natural to suppose that agents choose to connect through the path with the highest reliability (measured by the product of link strengths normalized to belong to [0, 1]) which is not necessarily the shortest path. However, this is not the only way to model the reliability of an indirect connection. At the end of the paper, we consider an alternative, simpler model, where the value of an indirect connection depends on the weakest link along the path. Throughout the paper, we consider the following problem. We suppose that agents are endowed with a fixed endowment X that they allocate across different connections. In formal communication networks, X should be interpreted as a total budget that the agent can invest on communication links ;ininformalsocialnetworks,x represents the fixedamountoftimethatan agent can spend on different social links. Our first task is to compute the strongly efficient network architecture, which maximizes the sum of benefits of all agents. In a second step, we ask whether this efficient architecture can be decentralized, namely whether it is an equilibrium of a noncooperative game where agents independently choose how to allocate their endowment over different links. 5 The main results of our study can be summarized as follows. When investments are perfect substitutes, the efficient network is a symmetric star. This star is stable if one only considers individual deviations, but for some values of the endowment level X, pairs of agents have an incentive to deviate and reallocate their investments. This incentive disappears when the number of players increases. The fact that stars are efficient when investments are perfect substitutes is easy to explain. With perfect substitutes, the sum of direct benefits is independent of the network architecture. The star is the investments. 4 Jackson and Wolinsky (1996) specify a decay factor δ [0, 1] and assume that indirect benefits are given by δ t v where t is the length of the shortest path connecting two agents and v a fixed parameter. For most of their analysis, Bala and Goyal (2000a) suppose that the value of a connection is independent of the distance between two agents. 5 Because of the well-known coordination problem in noncooperative games of bilateral link formation, we consider two equilibrium concepts. We say that a network is Nash stable if it is immune to individual deviations, and strongly pairwise stable if it is immune to deviations by pairs of agents. 3

4 structure which maximizes the value of indirect benefits,as itallowsfora large number of indirect connections, for a high value of link quality, and for short indirect connections. We also note that the star remains the efficient network architecture when link quality is a separable, convex function of individual investments. However, in that case, the hub of the star may concentrate its investment on a small number of links, and the star may fail to be stable, as the hub may benefit from reallocating his investment on a single link. With perfect complements, the efficient network architecture is very different. Trees cannot be optimal, because end players of the network waste part of their endowments, and have an incentive to invest what remains from their endowment on a new link. Regular graphs (where all agents have the same number of links) are likely candidates for efficient network architectures, and we show that for small numbers of players, the circle (the only regular graph of degree 2) isinfacttheefficient network architecture. However, when the number of player increases, the circle can be dominated by other network architectures, and we have been unable to obtain a full characterization of efficient networks. Finally, we note that the circle is always stable (immune to both individual deviations and deviations by pairs of players), so that the efficient network can be decentralized when the number ofplayersissmall. In the last Section of the paper, we consider a model of weakest-link reliability, where the value of an indirect connection is independent of the distance, and is only affected by the strength of the weakest link along the path. In this case, it is easy to see that the efficient network architecture must be a tree. (In any cycle, the weakest link will never be used, and the value of the network increases by reallocating the investments to other links). With perfect substitutes, the symmetric star again emerges as an efficient network architecture ; with perfect complements, the line is an optimal structure. 6 Both the symmetric line and symmetric star are stable, as any deviation from a network where all links have equal strength is bound to decrease the value of the weakest link. Related Work Given the obvious importance of networks with links of varying strength, a number of recent papers have proposed models where agents choose how much to invest in a relationship. In most of these models, agents choose their investment after the network has been established. In a specific model 6 We show however, that these are not the only efficient network architectures, and that other networks which generate the same distribution of link strengths are equally efficient. 4

5 of strategic alliances among firms, Goyal and Moraga Gonzales (2001) consider a two-stage model where firms first form links and then decide their R&D investment in every bilateral relationship. The strength of a link (measured by the investments in R&D by both partners in the alliance) is thus determined endogenously. The analysis is conducted with a specific cost and demand formulation, and the main focus of the analysis is on regular networks (where all firms have the same number of links), and the effect of the number of links on the R&D investments. Durieu, Haller and Solal (2004) construct a model of nonspecific networking. Agents choose a single investment, which applies to the links with all other agents. This formulation seems adequate for settings where agents cannot discriminate among other agents in the society, but does not capture situations where agents choose to form bilateral links. Finally, Brueckner (2003) considers a model of friendship networks which is closely related to ours. Agents choose to invest in relationships, and the value of indirect benefits isgivenbythe product of the strength of links, as in our model of min reliability. For most of his analysis, Brueckner (2003) concentrates on three player networks, and studies the effect of the network structure on the investment choices in the complete and star networks. In this paper, we consider an arbitrary number of agents, and simultaneously solve for the formation of the network and the choice of investments. The rest of the paper is organized as follows. We introduce the model and notations in Section 2. In Section 3, we compute the value of connected networks among four players, in order to illustrate the results and arguments of our analysis. Section 4 is devoted to the study of networks with perfect substitutes, and Section 5 to networks with perfect complements. We analyze weakest-link reliability in Section 6, and conclude in Section 7. All the proofs are collected in an Appendix. 2 Model and Notations Investments and link strength Let N = {1, 2,...,n} be a set of individuals. Individuals derive benefits from links to other individuals. These benefits may be the pleasure from friendship, or the utility from (non-rival) information possessed by other individuals, and so on. In order to fix ideas, we will henceforth interpret benefits as coming from information possessed by other individuals. Each individual has a total resource (time, money) of X>0, and has to decide on how to allocate X in establishing links with others. 5

6 Let x j i denote the amount of resource invested by player i in the relationship with j. Then, the strength of the relationship between i and j is a function of x j i and x i j. Let s ij denote this strength as a function of the amounts x j i and xi j. Let this functional dependence be denoted as s ij = f(x j i,xi j) with s ij [0, 1]. Alternative assumptions can be made about the functional form of f. In the paper, we will focus on the following two special cases. (Perfect Substitutes) :Foreachi, j N, f(x j i,xi j )=xj i + xi j (Perfect Complements): Foreachi, j N, f(x j i,xi j )=min(xj i,xi j ). When investments are perfect substitutes, agents can form links without the consent of the other agent; when investments are perfect complements, consent is needed to guarantee that the link is formed. In our view, these assumptions correspond to two polar cases on the technology of link formation. Throughout the paper, we emphasize when our results can be extended beyond these polar cases. Link strength and reliability We say that individuals i and j are linked if and only if s ij > 0. Each pattern of allocations of X, that is the vector x (x j i ) {i,j N,i6=j} results in a weighted graph, which we denote by g(x). 7 We say that ij g(x) if x j i + xi j > 0. Given any g, a path between individuals i and j is a sequence i 0 = i, i 1,...,i m,...,i M = j such that i m 1 i m g for all m. Two individuals are connected if there exists a path between them. Connectedness defines an equivalence relation, and we can partition the set of individuals according to this relation. Blocks of that partition are called components, and we let N (g) denote the set of components of the graph g. Suppose i and j are connected. Then, the benefit thati derives from j depends on the reliability with which i can access j s information. There are different ways to model how the strength of links affects the reliability of the communication channel. In our view, the most natural interpretation is that the strength of a link is an index of the quality of the transmission, so that messages sent along stronger links have a higher probability of being 7 To simplify notation, we will sometimes ignore the dependence of g on the specific pattern of allocations. 6

7 delivered without delay or distortion. With this interpretation, the reliability of any path between i and j is given by the product of the link strengths along the path. For any path p(i, j) =i, i 1,...,i M 1,j,wethusdefine r(p(i, j)) = s ii 1...s i m 1 i m...s i M 1 j. We assume that agents always choose to transmit information along the path with the highest reliability, and for any connected pair of agents i, j, we let p (i, j) denote the most reliable path in the set of all paths P (i, j) p (i, j) = arg max p(i,j) P (i,j) r(p(i, j)). The benefit of the connection from i to j is then given by R(i, j) =r(p (i, j)) = max r(p(i, j)) p(i,j) P (i,j) The total utility that agent i obtains in the weighted graph g can then be computed as: U i (g) = X j6=i R(i, j), and the total value of the graph is given by V (g) = X i U i (g). Before we proceed to define efficient and stable graphs, we would like to mention that alternative definitions of reliability could be used. In the last Section of the paper, we consider a different notion of reliability, where agents only care about the speed of transmission which depends on the bottlenecks of the network. In this model, the reliability of a path is given by the weakest link along the path, br(p(i, j)) = min s i m 1 i m p(i,j) s i m 1 i m Agents choose to use the paths with the highest reliability, and we define the benefit of a connection from i to j in this setting as: br(i, j) = max min s i p(i,j) P (i,j) s i m 1 i m p(i,j) m 1 i m Finally, we note that there is another definition of reliability, which has been used by Bala and Goyal (2000b). They suppose that, for any nonweighted graph g, agents obtain a utility u i (g).the strength of the link s ij 7

8 is interpreted as the probability that the link between i and j forms. In that case, the communication network is a random graph, and players evaluate the graph according to their expected utility, using the link strengths to compute the probability distribution over all possible graph realizations. Efficient and stable graphs We now define efficient and stable graphs. The notion of efficiency that we use is the strong efficiency notion introduced by Jackson and Wolinsky (1996). Definition 2.1 A graph g is efficient if V (g) V (g 0 ) for all g 0. Jackson and Wolinsky (1996) initiated the analysis of the potential conflict between efficiency and stability" of endogenously formed networks. We now describe the concepts of stability that will be used in this paper. Given any pattern of investments x, and individual i, (x i,x 0 i ) denotes the vector where i deviates from x i to x 0 i. Similarly, (x i,j,x 0 i,j ) denotes the vector where i and j have jointly deviated from (x i,x j ) to (x 0 i,x0 j ). Definition 2.2 A graph g(x) is Nash stable if there is no individual i and x 0 i such that U i(g(x i,x 0 i )) >U i(g(x)). So, a graph g induced by a vector x is Nash stable if no individual can change her pattern of investment in the different links and obtain a higher utility. Definition 2.3 A graph g(x) is Strongly Pairwise Stable if there is no pair of individuals (i, j) and joint deviation (x 0 i,x0 j ) such that U i (g(x i,j,x 0 i,j)) + U j (g(x i,j,x 0 i,j)) >U i (g(x)) + U j (g(x)) So, a graph is strongly pairwise stable if no pair of individuals can be jointly better off by changing their pattern of investment. Notice that we define a pair to be better off if the sum of their utilities is higher after the deviation. This leads to a stronger definition of stability than a corresponding definition with the requirement that both individuals be strictly better off after the deviation. The current definition implicitly assumes that individuals can make side payments to one another. The availability of side 8

9 payments is consistent with our definition of efficiency - a graph is defined to be efficient if it maximizes the sum of utilities of all individuals. 8 Jackson and Wolinsky (1996) define a weaker notion of stability - pairwise stability. They basically restrict deviations by assuming that only one link at a time can be changed. 9 Specific networks Some specific network architectures will be important in subsequent sections. We define these below. Definition 2.4 A graph g isastarifthereissomei N such that g = {ik k N,k 6= i}. The distinguished individual i figuring in the definition will be referred to as the hub". Thedegreeofanodei in graph g is #{j ij g}. Thatis,thedegreeof a node equals the number of its neighbors. Definition 2.5 A k-regular graph is a graph where every node has degree k. A circle is the unique 2-regular graph. Definition 2.6 A graph g is a line if there is a labelling of individuals i 1,...,i N such that g = {i k i k+1 k =1,...,N 1}. 3 Efficient and stable networks with four players In this Section, we introduce our analysis by characterizing efficient and stable networks with a small number of players four players is the minimal number for which different interesting network architectures can be compared. Both in the case of perfect substitutes and perfect complements, we 8 While we allow for side-payments among agents, we also assume that initial endowments are not transferable across players. If X is a fixed allocation of time, this is clearly understood. If X is a fixed budget, we assume that individuals can only make sidepayments ex post, once the communication network has formed, but cannot trade their initial endowments. 9 Our current definition corresponds to the definition of pairwise stability used by Dutta and Mutuswami (1997). See also Gilles and Sarangi (2004) and Bloch and Jackson (2005) for a comparison of different stability concepts. 9

10 compute, for every possible connected network architecture, the optimal allocation of investments. This allows us to assign a value to every connected network, and to characterize the efficient network architecture. In a second step of the analysis, we investigate whether efficient network architectures are stable. The following picture shows the six architectures of connected networks among four players. t t Q QQ t t Star (S) t t t t Line (L) t t t t t Circle (C) t Q QQ t t Triangle (T) t t t t Almost Complete t t Full (F) 3.1 Perfect Substitutes When investments are perfect substitutes, the values of the six networks can easily be computed. Consider for example the star, S. Every peripheral player invests all his endowment, X, on the link with the hub. Let (x 1,x 2,x 3 ) denote the investments of the hub to each of the peripheral agents. The value of the network is given by: V (g) = 2(X + x 1 )+2(X + x 2 )+2(X + x 3 )+2(X + x 1 )(X + x 2 ) +2(X + x 1 )(X + x 3 )+2(X + x 2 )(X + x 3 ) 10

11 In order to compute the value of the star, we thus need to solve the problem: max x 1,x 2,x 3 V (g) subject to: x 1 + x 2 + x 3 = X. It is easy to see that the value is maximized when x 1 = x 2 =3x = X/3, so that the value of the star is given by: V (g) =8X +6(4X/3) 2. Similar computations enable us to determine the value of the six connected network architectures, which are reported in the following table 10 : Network V (g) V (g) for X =1/2 S 8X +6(4X/3) L 8X +4(X + x )(X x )(2 + X + x ) 5.22 where x = 2 1+X+X 2 2 X 3 C 8X +4X 2 5 T 8X +2(X + x ) where x = X X 2 /2 A 8X +2(X/2+x ) where x = 2X 2 +16X+16 X 4 2 F 8X 4 Table 1: Network values with perfect substitutes The last column of Table 1 suggests that the star is the optimal network architecture when investments are perfect substitutes. In fact, the value of the graph can be decomposed into two components: the value of direct links, which is identically equal to 8X in all networks, and the value of indirect links. The value of indirect links is maximized in the star. The complete graph has the lowest value, because there are no indirect connections. Comparing the star with the triangle (or the almost complete graph with the circle), we discover that the addition of links lowers the value of the graph, because it reduces the value of indirect connections. Turning to stability, we investigate whether the efficient graph S is indeed Nash stable. We note that the hub of the star obtains the highest 10 Notice that the value of the networks is computed assuming that all links are used. Hence, we impose that agents always prefer to use direct links when they exist. 11

12 possible benefits, 4X and has no incentive to deviate. Consider a peripheral agent. Clearly, this agent has no incentive to form links with the two other peripheral agents, since this would at most give him the value of direct benefits 4X/3. Suppose that she chooses to shift a fraction x of her endowment to another peripheral agent. This deviation only makes sense if x (4X/3 x)(4x/3). Two cases can arise: either the peripheral agent connects directly to the hub of the star (if 4X/3 x x(4x/3)) or she connects to the hub through the peripheral agent (if x(4x/3) > 4X/3 x). In the first case, the benefits after deviation are given by v 0 =4X/3+(4X/3 x)(4x/3) ; in the second case by v 0 = x + x(4x/3) + x(4x/3) 2. It is easy to check that both those benefits are lower than the value of a peripheral agent in the star, v =4X/3 +2(4X/3) 2, so that the star is stable. However, there are values of the endowment X for which the star is not strongly pairwise stable. By jointly deviating, and investing 2X on a link, two peripheral agents can each obtain a value v 0 =2X + X/3+(X/3)(4X/3). As long as X<9/28, v 0 >vand the deviation is profitable for both agents. 3.2 Perfect Complements We now compute the values of the six connected networks when investments are perfect complements. The following table reports the results. Network V (g) V (g) for X =1 S 2X +6(X/3) L 2[x + X +2x (X x )+x 2 (X x )] 4.54 X 2 +2X+7 (2 X) where x = 3 C 4X +4(X/2) 2 5 T 4X 2ε +2ε(X ε) with ε 0. 4 A 4X 2x +2(X/2 x ) with x = 3 9 X 2 2 F 4X 4 Table 2: Network values with perfect complements The last column of Table 2 suggests that the circle is the efficient network architecture when investments are perfect complements. The lowest value is obtained in the star, where the hub invests very little in each relationship, resulting in very weak links. The line appears to be the second 12

13 best architecture, followed by the almost complete graph. It is easy to see that the circle is also Nash stable and strongly pairwise stable. Given that investments are symmetric across agents on every link, any individual deviation can only lower the strength of links, and result in a loss. By the same argument, pairs of players have no incentive to decrease the investment they make on an existing link to shift it to other links. By creating a new link, a pair of players will not increase the value of direct benefits (which are equal to their maximal value in the circle), but would lose the benefit fromtheir indirect connection. Hence, pairs of players have no incentive to form new links, and the circle is strongly pairwise stable. Our analysis of efficient and stable networks among four players shows that the computation of optimal weighted graphs is a complex exercise and that general results are unlikely to be found. The architecture of efficient networks depends crucially on the specification of the relation between individual investments and link strength. When investments are perfect substitutes, the star is efficient ; when investments are perfect complements, the circle is efficient. Furthermore, whether the efficient network is stable or not also depends on the specification of the model. With perfect substitutes, the star is Nash stable but not necessarily strongly pairwise stable ; with perfect complements, the circle is strongly pairwise stable. In the next two Sections, we build upon the four player example to generalize our results to an arbitrary number of players. 4 Perfect Substitutes and Convex, Separable Investments In this Section, we characterize the efficient and stable networks when investments are perfect substitutes. Our results are then generalized to a larger class of models, where the strength of a link is a separable, convex function of individual investments. 4.1 Perfect substitutes When investments are perfect substitutes, as we noted in our discussion of the four-player example, the total value of direct benefits is independent of the network structure: the sum of link strengths is always equal to nx, and the value of direct benefits to 2nX. The architecture of the network 13

14 only affects the value of indirect links. Our first result shows that this value is maximized in a star where the hub uniformly distributes his endowment over all links to peripheral players. Proposition 4.1 When investments are perfect substitutes, the efficient network is a star where the hub invests X/(n 1) in all links with peripheral agents. Proposition 4.1 is a special case of Theorem 4.1, so that we defer the formal proof of the result to the end of the Section. However, we can delineate the different steps of the argument to explain the intuition underlying the result. First, we start from any connected network with link strengths (z 1,z 2,...,z M ), where link strengths are ordered so that z 1 z 2... z M. We construct a star, with (n 1) links of strength z 1,z 2,...,z n 2, P m t=n 1 z t. The value of this star necessarily dominates the value of the initial graph: the length of indirect connections has been reduced, and the strength of links has been weakly increased. One problem however is that the star may not be feasible in the sense that the hub may have to invest more than his initial endowment X. In order to solve this problem, in the second stepoftheproof,wereallocateinvestmentsacrosslinks,insuchawaythat links become feasible. This reallocation tends to equalize the strength of all links, and given that the value of indirect connections is the product of link strengths, this convexification will always increase the total value of the network. Hence, this argument shows that any connected network is dominatedbyastarwithequalstrengthoneverylink. Inthethirdstepofthe proof, we show that a single connected star always dominates a collection of stars, establishing the final result. Heuristically, it is easy to understand why the star is the efficient network architecture. First, the star is a tree, and so minimizes the number of direct connections amongst all connected graphs. This increases the average strength of direct links. Also, the fewer the number of direct links, the greater is the scope for indirect benefits. Second, it also minimizes the distance between all nodes which do not have a direct connection. Given the product form of the reliability function, the latter also increases the scope of higher indirect benefits. We now show that the star is always a stable network, but may fail to be strongly pairwise stable. The only agents who have an incentive to deviate unilaterally are the peripheral players, who may choose to reallocate 14

15 their investments over m links, (x 1,x 2,...,x m ). Two cases can arise: either the player still connects to the (n 1 m) players with whom she has indirect links through the hub of the star, or through another player i. In the first case, the payoff after the deviation is given by v 0 =4X/3 +(n 1 m)(x/3)(4x/3) ; in the second case by v 0 = X +max{x/3,x i 4X/3} + (n 1 m)x i (4X/3) 2. It is easy to check that both payoffs arelowerthan the payoff of a peripheral agent in the star, v =4X/3+(n 1)(4X/3) 2. Turning to pairwise strong stability, we consider a deviation by two peripheral agents, labeled 2 and 3. (We let 1 label the hub of the star). Suppose that both agents shift away some of their investment from the link to the hub of the star to form a new link, and let x 3 2 = δ 2,x 1 2 = X δ 2 and x 2 3 = δ 3,x 1 3 = X δ 3.Without loss of generality, let δ 2 δ 3. Individual 3 gains an additional direct benefit ofδ 2 from 2, but loses the indirect benefit of(x + n 1 X )2 from 2. In addition, 2 also suffers a loss in indirect benefit from each of the other peripheral nodes. The exact loss depends on whether p (3,i) for i>3 includes 31 or (32, 21). 11 Hence, the loss in indirect benefit from each of the other nodes 4,...,n is at least δ 2 (X + n 1 X ). So, the deviation is profitable if δ 2 >δ 2 (n 3)(X + X n 1 )+(X + X n 1 )2 If this inequality holds for some δ 2 <X,itmustalsoholdforδ 2 = X. Substituting this value of δ 2, and simplifying, we get (n 1) 2 X< n(n 2 3n +3) Summarizing, we have proved the following Proposition. Proposition 4.2 When investments are perfect substitutes, the symmetric star is Nash stable. Furthermore, the symmetric star is strongly pairwise stable if and only if X (n 1)2 n(n 2 3n+3). Proposition 4.2 thus illustrates the classic conflict between efficiency and (n 1) stability in our model. We note that, for n 3, 2 n(n 2 3n+3) is decreasing in n so that, as n increases, the lower bound on X decreases. This is easily interpreted. If two nodes divert investment from their links with the hub, they lose indirect benefits from other peripheral nodes. This loss increases with the number of peripheral nodes. As a consequence, it is easier to sustain the efficient network as a strongly pairwise stable network when the number of agents in the society increases. 11 Thelatterisapossibilitysincewehaveassumedδ 3 δ 2. 15

16 4.2 Convex, separable investments The characterization of the efficient network does not rely on investments being perfect substitutes, and can be generalized to a larger class of functions relating individual investments to link strength. Consider the family of functions: s ij = φ(x j i )+φ(xi j) where φ is convex, increasing with φ(0) = 0 and φ(x) 1/2. This specification implicitly assumes the presence of increasing returns in individual investments: the more a player invests in one link, the larger the marginal return from his investment. Hence, with convex, separable investments, players have an incentive (at least in order to maximize direct benefits) to concentrate their investments on a small number of links. It turns out that this incentive is aligned with the objective of maximizing the value of indirect links, so that one can characterize efficient networks with convex, separable investments as stars. Theorem 4.1 Suppose that the strength of links is a convex, separable function of individual investments. Then the unique efficient graph is a star with every peripheral agent investing fully in the arc with the hub. Theorem 4.1 shows that with increasing returns the efficient architecture is a star where all peripheral nodes invest fully in the link with the hub, but does not characterize the allocation of investments by the hub. Except in the case where investments are perfect substitutes, there is a conflict between maximization of direct and indirect benefits. In order to maximize direct benefits, the hub should invest fully in one link with a peripheral node. This however lowers aggregate benefits for all other peripheral nodes, and may result in a lower total value of the network. The optimal distribution of investments of the hub thus depends on the curvature of the function φ. To illustrate this point, we consider in the following example a quadratic transformation function, and characterize the optimal allocation of investments in the hub as a function of the parameters of the utility function. Example 4.1 Let φ(x) = 1 2 [λx +(1 λ)x2 ],forsomeλ [0, 1] and assume X =1 For λ =1, the transformation function is linear, and for λ =0,the function is quadratic. Lower values of the parameter λ correspond to higher degrees of convexity of the transformation function. Consider a star where 16

17 the hub, denoted n, allocates his investments on the different peripheral nodes, x 1 n,...,x n 1 n and each peripheral node invests fully on the link to the hub. The value of the star is given by: V = n(n 1) +2n(1 λ)+(λ 2 2n(1 λ)) X X x i 2 nx j n i j +λ(1 λ) X X x i nx j n(x i n + x j n)+(1 λ) X X 2 (x i nx j n) 2. i j i j If x k n is given for all k 6= i, j, (x i n +x j n)=1 P k6=i,j xk n is fixed, and V can be written as a quadratic convex function of the product x i nx j n.asv is a convex function, it reaches its maximum either at x i nx j n =0or x i nx j n =(x i n +x j n) 2 /4. We thus conclude that, at the optimum, if the hub invests a positive amount on two nodes x i n and x j n,thenitmustinvestthesameamountonthesetwo nodes. In other words, at the optimum, the hub chooses the number k of links on which it invests, and invests an equal amount 1/k on each of those links. The following table lists, for n =5and different values of λ, the number of links on which the hub invests at the optimum. λ optimal number of links Table 3: Optimal allocation of investments of the hub Table 3 shows that for a large fraction of the parameter range, the hub optimally invests in a single link. As the degree of convexity of the transformation function goes down, the number of links on which the hub optimally invests increases, and when the function is linear, the hub invests equally on all links, as we noted in Proposition 4.1. Example 4.1 also shows why the efficient graph may not be stable when φ is strictly convex. For λ high enough, the efficient graph is a star where the hub does not specialize completely in one of the links. But, then such a 17

18 graph cannot be Nash stable because the agent at the hub is strictly better off by investing X in just one of the links - the hub does not derive any indirect benefits and so is better off by maximizing her own direct benefit. 5 Perfect Complements The analysis of efficient networks with four players when investments are perfect complements highlights two facts. First, when a player is at the end of the network, part of his endowment is wasted, because his neighbors always contribute a smaller part of their endowment to the relationship. This argument indicates that trees perform badly, and that agents must be connected to at least two other agents in the optimal network. Second, regular networks, where all agents have the same number of neighbors, are likely candidates for efficient networks, as they allow for a symmetric distribution of investments across links, which maximizes the value of direct benefits. The following results show that these two intuitions are (partially) confirmed in the general case. We first establish that trees cannot be optimal networks. Proposition 5.1 Suppose that investments are perfect complements. If g is efficient, it cannot have any component with three or more nodes which is a tree Proposition 5.1 relies on the following observation. In a tree, two agents located at the end of the network have an incentive to use what remains from their endowment to establish a new link. It is easily checked that the value of this direct link is higher than the value of the indirect connection between the two end agents, so that the addition of the link leads to an increase in the value of the network. Our second result shows that, for a small number of players, the circle, which is the only regular network of degree 2, is indeed the efficient network structure. Proposition 5.2 Suppose Assumption 2 holds. For 3 n 7, thecircle where every link has value X/2 is the unique efficient network. Unfortunately, we have been unable to extend Proposition 5.2 beyond seven players. With a larger number of players, the circle may be dominated 18

19 Figure 1: The Petersen graph by denser regular graphs, where the number of indirect connections is lower but the distance of indirect connections is lower as well. The following example shows that the circle can indeed be dominated by another network architecture (the Petersen graph") 12 for n =10. Remark 5.1 Let n =10. The circle may be dominated by the Petersen graph. The Petersen graph is a regular graph of degree 3 such that, for any node i, andanyneighborsj and k of i, the set of direct neighbors of j and k (other than i ) is disjoint. Consider the Petersen graph where every link has value X/3. The direct benefits are maximized, and each player has 6 indirect connections of length 2 and value (X/3) 2. Hence, the value of indirect connections for any player is 6X 2 /9=2X 2 /3. In the circle, each player has 2 indirect connections of length 2, 2 indirect connections of length 3, 2 indirect connections of length 4 and 1 indirect connection of length 5, so the total value of indirect connections is given by IC = X2 2 + X3 4 + X4 8 + X5 32. For small values of X, itisobviousthat 12 See Holton and Sheehan (1993). IC < 2X2 3 19

20 so that the Petersen graph dominates the circle. Turning to stability, we can show that the symmetric cycle is always strongly pairwise stable. Hence, for small numbers of players, the efficient graph is always stable ; for larger number of players, in spite of our best efforts, we have been unable to characterize the efficient graph and to study the tension between efficiency and stability. Proposition 5.3 Let investments be perfect complements. Then, the symmetric cycle is Nash stable and strongly pairwise stable. 6 Weakest link reliability In this Section, we modify the definition of path reliability, and suppose that the value of a path only depends on the weakest link along the path. With this notion of reliability, the distance of indirect connections becomes irrelevant, allowing for a simpler characterization of efficient and stable network architectures. We first show that, in the general case, efficient networks cannot contain cycles. Proposition 6.1 Suppose that the strength of a link is an increasing function of individual investments. Then the efficient network must be a tree. The intuition underlying Proposition 6.1 is easily grasped. If the graph contains a cycle, the weakest link in the cycle is never used to connect any agents. Hence, the value of the graph can be increased by reallocating the investments of the weakest link in the cycle to other links. Among trees, the star still plays a prominent role when investments are perfect complements, or when the strength of a link is a convex, separable function of individual investments. Theorem 6.1 Suppose that the strength of a link is a convex separable function of individual investments. Then any efficient network is equivalent to a star where all peripheral agents invest fully in their relation with the hub. Furthermore, if investments are perfect substitutes, then any efficient network is equivalent to a star where the hub distributes equally its endowment across all links with peripheral agents. 20

21 Theorem 6.1 shows that, as in the case where path reliability depends on the product of link strengths, the star is an efficient network architecture with convex, separable investments. However, when the reliability of a path only depends on its weakest link, the distance of connections is irrelevant, and other trees, which generate the same distribution of link strengths than the star, are equally efficient. In fact, as the following Proposition shows, in the case of perfect substitutes, one can generate links with equal strength for any tree, so that any tree can be supported as an efficient network architecture. Proposition 6.2 Suppose that investments are perfect substitutes. any tree can be supported as an efficient network. Then We now briefly examine the stability of efficient networks when the reliability of a path depends on its weakest link. When the function linking individual investments to link strength is strictly convex, the efficient network may force the hub of the star to invest equally among its links, whereas it would prefer to concentrate all his investment on a single link. Hence, one can easily construct examples where the star is not Nash stable. However, with perfect substitutes, the star is always Nash stable and even strongly pairwise stable. If two peripheral agents reallocate investments away from their link with the hub, the value of indirect connections through the hub must decrease, resulting in a lower payoff for the peripheral agents. When investments are perfect complements, the line emerges as the optimal network among trees, because it minimizes the degree of any node (and hence maximizes the strength of all links). Furthermore, when investments are perfect complements, the strength of a link is not increasing in individual investments, so that Proposition 6.1 does not hold. In fact, the circle is also efficient, as it generates exactly the same value as the line. Proposition 6.3 If investments are perfect complements, the efficient graphs are the symmetric line and the cycle. Because distances do not matter with weakest link reliability, in the symmetric line and cycle, each node derives a benefit of X 2 from every other node. It is trivial to check that no deviation by a pair (or coalition) can improve mutual payoffs. Hence, the symmetric line and the cycle are both Nash stable and strongly pairwise stable. 21

22 7 Conclusion In this paper, we analyze the formation of communication networks when players choose how much to invest in each relationship. We suppose that players have a fixed endowment that they can allocate across links, and consider alternative technologies relating individual investments to link quality. Furthermore, we suppose that, in indirect connections, players use the path which maximizes the product of link strengths. When investments are perfect substitutes, the efficient network architecture is a symmetric star, which is stable with respect to individual deviations, but may be unstable when pairs of agents can jointly deviate. When investments are perfect complements, the efficient network is never a tree. For small numbers of players, the symmetric circle is the efficient network architecture, and is immune to individual deviations and deviations by pairs of players. When the value of an indirect connection only depends on the quality of the weakest link, the efficient network must be a tree. For perfect substitutes, the symmetric star is efficient and stable; for perfect complements, the symmetric line is efficient and stable. In our view, this paper provides a first step in the study of networks where agents endogenously choose the quality of the links they form. An ambitious objective would be to revisit the recent literature on strategic network formation assuming that networks are represented by weighted graphs, but the complexity of the analysis in thesimplecaseofcommunication networks indicates that a general study of weighted networks may be intractable. Instead, we would like to suggest three possible directions for further research. First, we would like to extend the analysis to situations where agents face investment costs, rather than opportunity costs due to a fixed budget. Second, we plan to analyze network formation for other specifications of indirect benefits, assuming for example that information decays completely for paths of length greater than two. Finally, we need to understand better the relation between investments and network structures, and wish to study in more detail the pattern of link investments for fixed network architectures, and the comparative statics of investments with respect to changes in the network structure. 8 References Bala, V. and S. Goyal (2000a) A noncooperative model of network formation," Econometrica 68,

23 Bala, V. and S. Goyal (2000b) A strategic analysis of network reliability," Review of Economic Design 5, Bloch, F. and M. O. Jackson (2005) "Definitions of equilibrium in network formation games", mimeo, GREQAM and Cal Tech. Boorman, S. (1975) A Combinatorial Optimisation Model for Transmission of Job Information through Contact Networks", Bell Journal of Economics 6, Brueckner, J. (2003) Friendship networks", mimeo., University of Illinois at Urbana Champaign. Calvo-Armengol, A. and M.Jackson (2004) The Effects of Social Networks on Employment and Inequality" (2004) American Economic Review 94, Coleman, J., E. Katz and H. Menzel (1966) Medical Innovation: A Diffusion Study, BobbsHill,NewYork. Durieu, J., H. Haller and P. Solal (2004) Nonspecific networking", mimeo., Virginia Tech.and University of Saint-Etienne. Dutta, B. and S. Mutuswami (1997) Stable networks", Journal of Economic Theory 76, Gilles, R. and S. Sarangi (2004) "Social network formation with consent", mimeo., Virginia Tech and University of Louisiana. Goyal, S. (2004) Strong and Weak Links", mimeo., University of Essex. Goyal, S. and J. Moraga Gonzales (2001) R&D networks", RAND Journal of Economics 32, Granovetter, M. (1974) Gretting a job: A Study of Contacts and Careers, Harvard University Press, Cambridge. Holton, D.A. and J. Sheehan (1993) The Petersen Graph, Cambridge University Press. Jackson, M. and A. Wolinsky (1996) A strategic model of economic and social networks," Journal of Economic Theory 71, Montgomery, I. (1991) Social Networks and Labour Market Outcomes" (1991) American Economic Review 81, Appendix Proof of Theorem 4.1: We prove the Theorem in three steps. Consider any feasible component h of g of size m. Step 1: We construct a star Ŝ with higher aggregate utility than h. Step 2: If Ŝ is infeasible, then we construct a feasible star S which has higher aggregate utility than Ŝ. 23

24 Step 3: If the graph g contains different components, we construct a single connected star which has higher aggregate utility than the sum of the stars S ProofofStep1:Let h have K m 1 links. Welabelthestrengthofthese links z k and assume, without loss of generality, that z 1 z 2... z K We construct the star Ŝ with hub m as follows. For each i 6= m, define x m i =min(φ 1 (z i ),X), x i m = φ 1 (z i ) x m i If g is a tree, then K = m 1. Letˆx j i = xj i for all i, j, andŝ h(ˆx). If g is not a tree, then K>m 1, and the investments involved in {z m,...,z K } have not been distributed. Since z i z i+1, x m i x m i+1. If xm i = X for all i =1,...,m 1, thenfrom convexity of φ, m 1 P x i m <X.Inthiscase,let x m i =ˆx m i for all i =1,...,m 1, and i=1 ˆx 1 m = X m 1 P x k m > x 1 m,andˆx k m = x k m for k =2,...,m 1. LetŜ h(ˆx).13 k=2 Otherwise, let k 1 be the smallest integer such that x m k < X. Define ˆx m i = x m i = X for all i< k. Then, distribute {z m,...,z K } sequentially amongst { k,...,m 1} as follows: ˆx m k = min(x, x m k + for k> k, ˆx m k = min(x, x m k + Finally, if this procedure has not exhausted KX φ 1 (z j )) j=m KX k 1 X φ 1 (z j ) (ˆx m k 0 xm k 0)) j=m K P j=m k 0 = k φ 1 (z j ), then distribute the excess sequentially to ˆx 1 m, ˆx 2 m,..., while satisfying the constraint that none of them exceeds X. Again, define Ŝ h(ˆx). Suppose h is a tree. Then, from convexity of φ, we know that φ(ˆx m i )+φ(ˆx i m) z i for all i =1,...,m 1 (1) This implies that direct benefits are at least as high in Ŝ as in h. A similar argument ensures that direct benefits are at least as high in Ŝ as in h even when the latter is not a tree. 13 Notice that in this case Ŝ is a feasible graph since mp = X for all i. 24 x j i j=1,j6=i