A connections model with decreasing returns link-formation technology

We study a connections model where the strength of a link depends on the amount invested in it and is determined by an increasing strictly concave function. The revenue from investments in links is the value (information, contacts, friendship) that the nodes receive through the network. First, assuming that links are the result of investments by the node-players involved, there is the question of stability. We introduce and characterize a notion of marginal equilibrium, where all nodes play locally best responses, and identify different marginally stable structures. This notion is based on weak assumptions about node-players’ information and is necessary for Nash equilibrium and for pairwise stability. Second, efficient networks in absolute terms are characterized. Efficiency and stability are shown to be incompatible, but partial subsidizing is shown to be able to bridge the gap.


Introduction
This paper seeks to contribute to the literature on economic models of strategic network formation.In this line of work, an increasing ‡ow of research has been produced by game-theorists and economists in general since Myerson (1977) and Aumann and Myerson (1988). 1 In the wake of these pioneering papers in the …eld, two seminal in ‡uential models of network formation are Jackson and Wolinsky's (1996) connections model and Bala and Goyal's (2000) non-cooperative two-way ‡ow model.In both models, networks are the result of creating links between pairs of individuals, by bilateral agreements in the former and unilateral decisions in the second, enabling information to ‡ow through the resulting network.In both models, the cost of a link and its strength or quality (i.e. its decay factor) are exogenously given, giving rise to two-parameter models.The simplicity of these basic models imposes some rigidity: Necessarily bilateral formation and compulsory equal share of the …xed cost of each link in Jackson and Wolinsky's model; and unilateral formation requiring full-covering of that …xed cost by its creator in Bala and Goyal's model, and a …xed level of quality for the resulting link in both.The point of this paper is to provide and develop a more ‡exible model in both link-formation and link-performance.
We develop a model of network formation where links are the result of investments and the quality or strength of a link, i.e. the …delity level of transmission through it, is never perfect and depends on the amount invested in it.A decreasing returns link-formation technology determines the quality of the resulting link as a function of the investment and is the only exogenous ingredient in the model.Formally, a decreasing returns link-formation technology is a di¤erentiable, increasing, strictly concave function whose range is [0; 1), i.e. however much is invested in a link, transmission is never perfect.The revenue from investment in links is, as in the seminal models, the information that the nodes receive through the network that results.
The question of e¢ciency is addressed …rst.It is established that the only possible non-empty e¢cient architectures for a decreasing returns link-formation technology are the complete network and the all-encompassing star, whose precise structures are also established.The family of decreasing returns link-formation technologies which have one of these non-empty structures as e¢cient is also characterized.Conditions for optimal investments constrained to supporting a given network are also obtained.
We then consider a decentralized context where links are formed according to a decreasing returns technology available to all players, and each link is the result of investments by the node-players that it connects, whose investments are assumed to be perfect substitutes.In this game-theoretic scenario the question of stability in the underlying network-formation game arises.We …rst examine a notion of marginal equilibrium of a classical ‡avor which is natural in this marginalist model but new in networks literature to the best of our knowledge.In a marginal equilibrium every player is playing a locally best response.More precisely, an investment pro…le is a marginal equilibrium if the investment vector of every player in the links in which he/she is involved is locally optimal, in the sense that su¢ciently small changes of these investments do not increase his/her payo¤.Necessary and su¢cient conditions for marginal stability are established by imposing that the marginal bene…t of the investment of any player in each of his/her links must be zero.The characterizing conditions that result from this classical economic principle have a clear intuitive interpretation which permits us to identify a variety of marginally stable architectures and their precise structures.At the same time, given that marginal stability is weaker than Nash-stability, these conditions are necessary for Nash equilibrium.
A comparison of the results on e¢ciency and stability yields the conclusion that non-empty e¢cient structures are not stable, not even marginally, and vice versa.Nevertheless, it is proven that subsidizing up to half the cost of each link bridges the gap between e¢ciency and marginal stability.
The paper is organized as follows.Section 2 introduces basic notation and terminology.Section 3 introduces the model.Section 4 addresses the question of e¢ciency, …rst in general, characterizing e¢cient networks (4.1), and then the question of the e¢cient support of a given "infrastructure" speci…ed by a set of feasible links (4.2).Section 5 is devoted to stability: Marginal stability (5.1) and Nash stability (5.2).Section 6 examines the incompatibility of e¢ciency and stability, and shows how a partial subsidy can bridge the gap.Section 7 brie ‡y reviews some related literature.Finally, Section 8 summarizes the results and suggests some possible extensions of the model.All proofs are relegated to an Appendix.

Preliminaries
An undirected weighted network (shortened in what follows to a network) is a pair (N; g) where N = f1; 2; :::; ng with n 3 is a set of nodes and g is a set of links speci…ed by a symmetric adjacency matrix g = (g ij ) i;j2N of real numbers g ij 2 [0; 1), with g ii = 0 for all i.Alternatively, g can be speci…ed as a map g : N 2 ![0; 1), where N 2 denotes the set of all subsets of N with cardinality 2. When no ambiguity arises we omit N and refer to g as a network.In what follows ij stands for fi; jg and g ij for g(fi; jg) for any fi; jg 2 N 2 . 2 When g ij > 0 it is said that a link of weight g ij connects i and j.N d (i; g) := fj 2 N : g ij > 0g denotes the set of neighbors of node i, and its cardinality is the degree of i.A path connecting nodes i and j is a sequence of distinct nodes of which the …rst is i, the last is j, and every two consecutive nodes are connected by a link.If i and j are two consecutive nodes in a path p, we write ij 2 p or ij 2 p. P ij (g) denotes the set of paths in g connecting i and j.N (i; g) denotes the set of nodes connected to i by a path.A network is connected if any two nodes are connected by a path.A subnetwork of a network (N; g) is a network (N 0 ; g 0 ) s.t.N 0 N and g 0 g.A component of a network (N; g) is a maximal connected subnetwork.An isolated node (i.e.not connected to any other) is a trivial component.A network has a cycle if there are two nodes connected by a link and also by a path of length 2 or more (the length of a path is the number of links that it contains, i.e. the number of nodes minus 1).
When the codomain of g is f0; 1g instead of [0; 1), i.e. g ij only takes the values 0 or 1, we say that g is a graph and it can be speci…ed as a set of links S N 2 .In particular, the non-weighted underlying graph S g of a weighted network g is S g := fij 2 N 2 : g ij > 0g.When a given graph S N 2 constrains the construction of a network which must have it as its underlying graph, we call S an infrastructure.
The empty network/graph is the one for which g ij = 0 for all ij 2 N 2 .A complete network/graph is one where g ij > 0 for all ij 2 N 2 .A subcomplete network/graph has only one non-trivial component which is a complete subnetwork, i.e. g ij > 0 if and only if ij 2 M 2 for some M N .A star network/graph is one with only one non-trivial component with k nodes (3 k n) and k 1 links in which one node (the center) is connected by a link with each of the other k 1 nodes.A tree network/graph is one with only one non-trivial component and no cycles.A circle network/graph has only one non-trivial component with k nodes (3 k n) and k links, each of them connecting one node with the next one and the last one with the …rst one for a given ordering of the k nodes.A tree, a star or a circle network/graph is said to be all-encompassing if k = n.

The model
As in the seminal connections models of Jackson and Wolinsky (1996) and Bala and Goyal (2000), we consider a set of nodes or players, each of them endowed with an information of value v > 0 to any other node that receives it intact.The main di¤erence between our model, brie ‡y sketched in the introduction and to be formalized in detail now, and the seminal models concerns link-formation.In Olaizola and Valenciano (2020), a link-formation technology is a non-decreasing map : R + ![0; 1) s.t.(0) = 0: If c is the amount invested in a link to connect two nodes, (c) is the level of …delity of the transmission of information through the link.More precisely, (c) is the fraction of information ‡owing through the link that remains intact. 3Flow occurs only through links invested in ( (0) = 0), but perfect …delity in transmission between di¤erent nodes is never reached (0 (c) < 1).In this paper we assume a decreasing returns link-formation technology.
2) It is strictly concave.
Assuming smoothness of makes it possible to use di¤erential calculus, which allows for a relatively simple formal marginal analysis without getting involved in more sophisticated technical issues.C.2 amounts to assuming technology to be decreasing returns.
We consider the following model based on this basic ingredient.A set N = f1; 2; :::; ng of nodes or players can be connected by links formed according to a given decreasing returns link-formation technology .Players can invest in links with other nodes.An investment pro…le is speci…ed by a matrix c = (c ij ) i;j2N , where c ij 0 (with c ii = 0) is the investment of player i in the link connecting players i and j, and determines a link-investment vector c : which in turn, through the link-formation technology available, , yields a weighted network denoted by g c or by g c , where Thus players' e¤orts are perfect substitutes.Let P ik (g c ) denote the set of paths in g c connecting i and k.For a path p 2 P ik (g c ), let (p) denote the product of the …delity levels through each link in that path, i.e. if p = ii 2 i 3 :::i m k, then (p) = (c ii 2 ) (c i 2 i 3 )::: (c imk ).Thus, player i values information originating from k that arrives via p by v (p): As in Jackson and Wolinsky (1996) and Bala and Goyal (2000), we assume that player i's valuation of the information originating from k 6 = i, denoted by I ik (g c ); is that which is routed via the best possible route from k, that is where p ik is an optimal path connecting i and k, i.e. p ik 2 arg max p2P ik (g c ) (p) (if no path connects i and k we set (p ik ) = 0).Then i's overall revenue from g c is Thus, i's payo¤ is the value of the information received by i minus i's investment: and the net value of the network resulting is the aggregate payo¤, i.e. the total value of the information received by the nodes minus the total cost of the network: In this setting two main issues arise.A game in strategic form, where a strategy of a player i is a vector of investments (c i = (c ij ) j2N , with c ii = 0) and the payo¤ function is given by (1), is implicitly de…ned.Thus the question of stability arises: What structures are stable and under what conditions?The notion usually applied in a context such as this is Nash equilibrium: An investment pro…le is Nash-stable if no player has an incentive to change his/her investment vector.Nevertheless, we …rst devote particular attention to a weaker notion of stability new in this context: Marginal equilibrium.A second issue is the question of e¢ciency: What structures are e¢cient in the sense of maximizing the net value given by (2) and under what conditions?
We address the question of e¢ciency …rst, and then look at stability.Thus we deal with a model with two parameters, the number of nodes/players n and the value v of the information at each node.A third "parameter" is the link-formation technology represented by function : 44 E¢ciency

E¢cient networks
In the model just described, the net value of a network g c , given by (2), that results from an investment pro…le c = (c ij ) i;j2N , depends entirely on c = (c ij ) ij2N 2 , where c ij := c ij + c ji .In other words, given that players' e¤orts are perfect substitutes, the question of e¢ciency depends entirely on the investments in every link, but it is immaterial who pays for them.Thus the answer to the question of e¢ciency is the same, regardless of whether the investments are made by node-players in a decentralized way or by a central planner.For this reason we give preference in this section to expressing results in terms of investment vectors and c = (c ij ) ij2N 2 and the resulting network g c .In Olaizola and Valenciano (2020) it is proved that for any link-formation technology , i.e. any non-decreasing and s.t.(0) = 0, the only possibly e¢cient nonempty networks are the all-encompassing star, the complete network and, under certain conditions, also a whole range of intermediate particular nested split graph structures. 5his conclusion thus also applies to DR-technologies.We …rst show the necessary conditions for a star and a complete network to be e¢cient for a DR-technology, based on the conditions obtained in Olaizola and Valenciano (2020), which will enable us to re…ne these conclusions for DR-technologies.
Proposition 1 For a complete network g c to be e¢cient under a DR-technology , the following conditions are necessary: or, equivalently, Therefore, in an e¢cient complete network all links are of the same strength, b c ef s.t.(3) or, which is equivalent for a DR-technology, s.t.(4).Note that there is certain to be a unique b c ef > 0 s.t. ( 3) and ( 4) if and only if 0 (0) > 1=2v.
In order to establish the structure of an e¢cient all-encompassing star, we …rst prove the necessary symmetry of an optimal star, i.e. a star with the highest net value, for any technology.
Lemma 1 For any link-formation technology for which some star yields a positive net value, the optimal star is all-encompassing and all its links have the same strength.
Proposition 2 For an all-encompassing star g c to be e¢cient under a DR-technology , the following conditions are necessary: (i) All links receive the same investment c ef s.t.
for which a necessary condition is for all c > 0.
The following result shows that the existence of an optimal symmetric star is guaranteed unless the technology is "too bad" in a precise sense, but whatever the DRtechnology if n is big enough.
Proposition 3 For a DR-technology , there is an optimal all-encompassing star unless (c) ' n (c) for all c 0, where For every DR-technology, there is an optimal all-encompassing star if n is big enough.
Therefore function ' n (c); de…ned by (8), sets a precise bound below which a technology is poor enough to make the formation of any star non-pro…table.Notice that, as can easily be checked, ' n (0) = 0; ' 0 n (c) > 0; ' n 00 (c) < 0, and consequently function ' n meets all but one of the conditions for a DR-technology as per De…nition 1: for a big enough c (for c > nv, in fact) ' n (c) > 1.In other words, constraint (c) ' n (c) is actually active as far as ' n (c) < 1; i.e. for c 2 (0; nv) (note that ' n (nv) = 1 for all n).
Then we have a characterizing result.
Theorem 1 Under a DR-technology : (i) The only non-empty possibly e¢cient networks are the complete network described in Proposition 1 and an all-encompassing star (as described in Proposition 2).(ii) The empty network is e¢cient if and only if (c) ' n (c) for all c, with ' n given by (8).Otherwise, either the complete network or an all-encompassing star is e¢cient.
Namely, for any DR-technology worse than ' n , i.e. whose graph is below that of ' n , and only for such DR-technologies, no all-encompassing star and no complete network yields a positive net value.Figure 1 illustrates this, showing the graph of function ' n for v = 1 and di¤erent numbers of nodes: n = 5; 12; 22 and 42.The greater the number of nodes, the lower the graph of this function is, i.e. the worse the technology must be to make any star unpro…table.Two dashed lines represent the graphs of two DR-technologies: Obviously, technology 2 is worse than 1 .Thus, for instance, for n = 5 no symmetric star or complete network yields a positive net value under 2 , while under 1 there are sure to be both optimal complete and star networks.For n = 22 there are sure to exist both optimal complete and star networks under both technologies, 1 and 2 .

E¢cient support of an infrastructure
Now consider the situation where a given infrastructure speci…ed by a set of feasible links S N 2 is to be supported in the most e¢cient way.We say that an investment De…nition 2 Given S N 2 , an investment pro…le c =(c ij ) i;j2N supports S e¢ciently if it supports S and v(g c ) v(g c ); for all c =(c ij ) i;j2N which support S: That is, investments are constrained to be made in all links in S and only in them, and the function to be maximized is The following result establishes necessary conditions for an investment vector to support an infrastructure S e¢ciently using the following notation: if p kl is an optimal path connecting nodes k and l s.t.ij 2 p kl ; de…ne: In other words, p ij kl can be seen as a path from k to l which results from replacing link ij in path p kl by a "perfect" link with no decay.In particular, if fk; lg = fi; jg, (p ij ij ) := 1: Proposition 4 Let be a DR-technology.For a link-investment vector c = (c ij ) ij2N 2 that supports an infrastructure S N 2 to do so e¢ciently the following conditions are necessary.For any two connected nodes in g c there must be a unique optimal path connecting them, and for each ij 2 S, Two comments are worth making here.First, note that ( 11) has a clear interpretation.The denominator of its right-hand side is the total amount of information that crosses link ij (subject to a decay (c ij )), i.e. between all pairs of nodes whose optimal connecting path contains link ij.The greater this amount the greater the denominator is and the smaller the quotient, i.e. the smallest 0 (c ij ) and consequently the greater its strength (c ij ): Second, as the investments in an e¢cient complete network and in an e¢cient all-encompassing star both e¢ciently support the infrastructure of their underlying graph, ( 4) and ( 6) are particular cases of (11).

Stability
Whether investments are made by a planner or in a decentralized way by node-players is immaterial in addressing the question of e¢ciency, but now we consider the situation where nodes are players who form links by investing in them and using an available DR-technology.An investment pro…le c = (c ij ) i;j2N (an n n matrix with zeros in the main diagonal) where c ij 0 is the investment of player i in the link connecting players i and j, actually represents a strategy pro…le, where its i-row, c i = (c ij ) j2N with c ii = 0, is the strategy of player i, whose payo¤ is given by6 This situation raises the question of stability.We …rst consider a weak form of stability which is, as far as we know, new in network literature, but quite natural in the context of this "marginalist" model.Moreover, in addition to its interest per se, its characterization provides necessary conditions for stronger notions of stability.

Marginal stability
If c = (c ij ) i;j2N is an investment pro…le and c 0 i = (c 0 ij ) j2N an investment vector of player i, let (c i ; c 0 i ) denote the investment pro…le that results from replacing row i in c by c 0 i .De…nition 3 An investment pro…le c = (c ij ) i;j2N is marginally stable (or a marginal equilibrium) if for some " > 0 the following holds: for all i 2 N and all In other words, an investment pro…le is marginally stable if the investments of every node in its links are locally optimal, in the sense that su¢ciently small changes in its investments in the links in which it is involved do not increase its payo¤.
It is worth emphasizing the interest of this weak notion of equilibrium per se.In this model, Nash equilibrium poses computational and informational di¢culties.Apart from the computational di¢culties of calculating best responses in a complex network, it requires a huge amount of information.Moreover, if the network is the means of transmission of information, how do players know about the revenue from links with players with whom they are not directly or even indirectly connected?If players are only aware of the marginal contribution of their investments in links in which they are actually involved (a much weaker assumption about their information) a marginal equilibrium means that no player receives signals inducing him/her to change his/her investments and the situation will remain unchanged.Note that the clause "s.t.c 0 ik > 0 only if c ik > 0" restricts responses to existing links.In other words, the creation of new links is not a response w.r.t. which a marginal equilibrium must be immune.A stronger variant of De…nition 3 closer to Nash equilibrium, but still weaker, is obtained by eliminating this clause.
De…nition 4 An investment pro…le c = (c ij ) i;j2N is strongly marginally stable (or a strong marginal equilibrium) if for some " > 0 the following holds: for all i 2 N and all Although strictly speaking one should refer to stability of investment pro…les, we often express our results in terms of the resulting networks.Thus a "(strongly) marginally stable network" should be read as a weighted network that results from a (strongly) marginally stable investment pro…le.The following lemma shows that the two notions are equivalent for connected networks.
Lemma 2 Let c = (c ij ) i;j2N be an investment pro…le.If g c is connected, then c is strongly marginally stable if and only if it is a marginal equilibrium.
The following result establishes a necessary and su¢cient condition for the empty network to be (strongly) marginally stable.
Proposition 5 Let be a DR-technology.The empty network is marginally stable whatever the technology, and is strongly marginally stable if and only if 0 (0) 1=v: It is convenient to introduce some notation in order to formulate and prove the following characterization establishing necessary and su¢cient conditions for an investment pro…le to be marginally stable.Note that expression (12) of the payo¤ of a player i, involves the choice of an optimal path p ik for each k 2 N (i; g c ).We denote by p i = fp ik : k 2 N (i; g c )g any particular choice of such optimal paths.We make use of a special case of notation (10): (p ij ik ) := (p ik )= (c ij ) whenever p ik is an optimal path that contains link ij.If C g i;j is the set of nodes that are connected with i in g through optimal paths that contain link ij, i.e.C g i;j := fk 2 N : 9p ik s:t: ij 2 p ik g; then, choose for each k 2 C g i;j an optimal path p ik s.t.ij 2 p ik and de…ne Note that K g i;j does not depend on the choice of the p ik 's such that ij 2 p ik because if p ik and q ik are two di¤erent optimal paths containing ij, then (p ik ) = (q ik ) and consequently (p ij ik ) = (q ij ik ).We have then the following result making use of Kuhn-Tucker's conditions: Theorem 2 Under a DR-technology , for an investment pro…le c = (c ij ) i;j2N to be marginally stable the following conditions are necessary and su¢cient.For all i; j 2 N (i 6 = j) s.t.c ij > 0; (i) If c ij > 0 any optimal path connecting i and k that contains link ij is the only path connecting them and 0 (c ij ) = 1 Part (i) establishes that, in a marginal equilibrium, if i sees k through an optimal path in which he/she invests it cannot be the case that i sees k also through another optimal path.In other words, in a marginally stable pro…le, the optimal paths in which a player invests form a unique tree rooted at that node.
As to (13), it is the result of requiring the marginal bene…t of the investment of any player in each of the links that he/she invests in to be zero.The resulting condition when c ij > 0 is : This has a clear interpretation: If player i invests in a link with j, the denominator of the fraction in formula ( 13) that yields 0 (c ij ) is v times the sum of the …delity levels through all subpaths up to j of optimal paths containing link ij through which player i receives information.In other words, vK g c i;j is the actual amount of information that reaches j on its optimal way to i. Thus this sum is a measure of the importance of link ij to player i: the greater this amount, the smaller 0 (c ij ), i.e. the greater c ij and (c ij ).Condition (ii) means a lack of incentives to invest in a link entirely supported by the other player.Condition (14) ensures that not investing in link ij is optimal for i because player j is investing in the link the amount that player i would be willing to invest for all the information that he/she can receive through link ij or even more. 7 Equation ( 13) and inequality (14), which are necessary conditions for an investment pro…le c to be marginally stable, involve only the resulting investment vector c , not directly the investment pro…le c .Nevertheless parts (i) and (ii) actually involve c, because which condition (( 13) or ( 14)), applies for a link ij, depends on whether c ij > 0 or c ij = 0.A direct consequence of these conditions is the following important conclusion.
Corollary 1 Under a DR-technology , if two players are connected by a link in the network resulting from a marginally stable investment pro…le but do not receive the same amount of information through that link, all the investment in that link is made by the player who receives more information through it.
As for strong marginal equilibrium we have Proposition 6 Under a DR-technology , an investment pro…le c = (c ij ) i;j2N is a strong marginal equilibrium if and only if in addition to conditions (i) and (ii) of Theorem 2, either 0 (0) 1 v+K , where K is the information received by the node that receives the greatest amount of information in g c , or g c is connected.
The rest of this section is devoted to show how Theorem 2 and Corollary 1 can be applied to establish that certain graph architectures, as subcomplete, complete, star, tree or circle graphs, are the result of investment pro…les marginally stable and characterize such pro…les.De…nition 5 Given a graph S N 2 , an investment pro…le c =(c ij ) i;j2N sustains S in marginal equilibrium if it supports S and c is a marginal equilibrium.When such c does exist we say that graph S is sustainable in marginal equilibrium.
The following two propositions refer to subcomplete and star networks. 7Alternatively, Theorem 2 can be reformulated like this: Theorem 2 (reformulated) Under a DR-technology , an investment pro…le c = (c ij ) i;j2N is marginally stable if and only if for all i; j 2 and whenever c ij > 0 any optimal path connecting i and k that contains link ij is the only optimal path connecting them.
Proposition 7 Let be a DR-technology and let c = (c ij ) i;j2N be an investment pro…le such that g c is subcomplete, then c is marginally stable if and only if 0 (0) > 1=v and all links receive the same joint investment b c eq > 0, such that If g c is complete these conditions are necessary and su¢cient also for c to be strongly marginally stable.
Proposition 8 Let be a DR-technology and let c = (c ij ) i;j2N be an investment pro…le such that g c is a star connecting p nodes (3 p n), then c is marginally stable if and only if g c is a periphery-sponsored star where all peripheral players invest the same amount c p;eq in the only link in which each of them is involved s.t.
If p = n (i.e. the star is all-encompassing), these conditions are necessary and su¢cient also for g c to be strongly marginally stable.
In particular, Propositions 7 and 8 establish necessary and su¢cient conditions for the only two non-empty architectures that can be e¢cient, i.e. the complete and the all-encompassing star networks (case p = n), to be sustainable in marginal equilibrium.In the case of the complete graph the existence of an investment pro…le satisfying these conditions is guaranteed if the technology satis…es the condition 0 (0) > 1=v.Before addressing the question of existence of c p;eq s.t.(16) for a star to be marginally stable we establish a result relative to any tree based on a …xed point argument.
Proposition 9 Under a DR-technology continuously di¤erentiable s.t. 0(0) > 1 v , any tree is sustainable in marginal equilibrium, and any all-encompassing tree graph is sustainable in strong marginal equilibrium.
As mentioned in the proof, by Corollary 1, in the marginally stable pro…le c s.t.g c is a tree, peripheral or terminal nodes must pay the full cost of their links, and the cost of any link where the nodes that it connects do not receive the same amount of information through it must be paid for fully by the node who receives more information through it.
Thus even if condition 0 (0) > 1 v does not hold, a star graph continues to be sustainable in marginal equilibrium for a su¢ciently big p.Compared with this result for the star, condition 0 (0) > 1 v seems rather strong relative to trees.In fact, this condition enables the results of Proposition 9 (…rst part) and Proposition 10 to be proved for any number of nodes 8 .The symmetry of the star enables the precise smaller bound 0 (0) to be calculated, while a similar re…nement of this bound for an arbitrary tree would require speci…c study.
The following proposition shows that a circle graph also can be sustained in marginal equilibrium.
Proposition 11 Under a DR-technology continuously di¤erentiable s.t. 0(0) > 1 v , any circle graph of k nodes (3 k n) can be sustained in marginal equilibrium with all links of the same strength, (c); if k is odd, given by ; and with links alternating two levels of strength, (c) and (c), if k is even, given by If k = n (i.e. an all-encompassing circle graph) it can be sustained in strong marginal equilibrium.
Remarks: (i) Note that even though the preceding results refer to graphs with only one non-trivial component, be it complete or subcomplete, a tree, a circle or a star graph, it follows immediately that any graph which has trees, circles, stars and subcomplete graphs as non-trivial components also can be sustained in marginal equilibrium if the conditions of Propositions 7-11 hold. 9ii) A property of marginal equilibria worth noting is its resilience in response to shocks such as deletion of nodes under certain conditions.For instance, a marginally stable star network ceases to be so if a spoke node vanishes.Nevertheless, by diminishing the investments of the remaining spoke nodes a new marginal equilibrium sustaining the new star with one arm less can be obtained surely if 0 (0) > 1 v or, otherwise, if the number of nodes is big enough.A similar situation occurs by the elimination of a node in a tree network.This yields a network with a number of components equal to the degree of the node eliminated: namely, in general, some isolated nodes and some tree networks of smaller diameters.In a circle network the elimination of a node yields a line (particular case of a tree).In all these cases some of the resulting components can possibly be sustained in marginal equilibrium by readjusting the investments of the nodes.In the case of a marginally stable subcomplete network the elimination of a node yields a new marginally stable subcomplete network with a node less.
(iii) The variety of graph architectures that have been shown to be sustainable in marginal equilibrium can be misleading, conveying the impression that every graph is sustainable.This is not so as the following example shows.Consider a star with su¢cient number, p, of nodes and a technology such that (b c eq ), s.t.(15), and (c p 1;eq ), s.t. ( 16), verify (b c eq ) < (c p 1;eq ) 2 : Then, the graph that results from adding to a star graph of p nodes a link connecting two spoke nodes is not sustainable in marginal equilibrium.The reason is clear: whatever the investments of those two spoke nodes in the link connects them, both nodes would have an incentive to diminish their investment in that link.

Nash-stability
An investment pro…le is Nash-stable if no player is interested in changing his/her investments unilaterally.Formally: Obviously the notion of marginal equilibrium (strong or not) is weaker than that of Nash equilibrium.Consequently, the characterizing conditions for marginal stability established in Theorem 2 are necessary conditions for Nash stability.Thus we have the following results in part as a corollary of the results in the previous section.
Proposition 12 Let be a DR-technology.The empty network is a Nash network if and only if 0 (0) 1=v: As established in Section 4, the only non-empty possibly e¢cient networks are the all-encompassing star and the complete network.This raises the question of Nashstability conditions for these architectures.We …rst examine the Nash-stability of a complete network focusing on the symmetric one, i.e. a complete network where the cost of each link is equally shared.
Proposition 13 Let be a DR-technology and let b c n be an investment pro…le such that g b cn is complete, symmetric and marginally stable, i.e. c ij = b c eq and c ij = b ceq 2 , with 0 (b c eq ) = 1=v.Then b c n is Nash-stable, if and only if for all k = 2; :::; n 1 : where b c eq is s.t.(15) and c k is s.t.
Note that the structure of a Nash complete network is unique (all its links are b c eq -links), but there may exist di¤erent investment pro…les that support it depending on how the cost of each link is shared. 10The conditions for Nash equilibrium for a non-symmetric investment pro…le c = (c ij ) i;j2N , s.t.c ij = b c eq for all i 6 = j are much more complicated.Proposition 14 Let be a DR-technology and let c n be an investment pro…le such that g c n is an all-encompassing periphery-sponsored star which is marginally stable, i.e. the investment in each of its links is c eq = c n;eq s.t.(16).Then if 0 (0) 1=v, c n is Nashstable, while if 0 (0) > 1=v, c n is Nash-stable if and only if for all k = 1; 2; :::; n 2 : where b c eq is s.t.(15) and c k is s.t.
The complexity of conditions ( 20) and ( 22) for Nash-stability, even for so simple structures as that of the symmetric complete network and that of a symmetric allencompassing star, may seem somewhat disappointing compared with the simplicity of the conditions for their marginal stability established in Propositions 7 and 8.This corroborates the computational di¢culties that the notion of Nash equilibrium involves, particularly in a richer model like this, and enhances the tractability of marginal equilibrium.

E¢ciency vs. equilibrium
In view of the results on e¢ciency and on stability we have the following.
Proposition 15 Under a DR-technology, e¢ciency and Nash stability or even marginal stability are incompatible, unless0 (0) 1 2v , in which case the empty network is both e¢cient and Nash-stable.Similarly, e¢cient support and stable support of an infrastructure are incompatible.
The reason is clear: From the results in Sections 4 and 5 it follows that b c eq < b c ef and c eq < c ef , and consequently a non-empty e¢cient network requires link-investments which are not stable because they give players the opportunity of free riding by taking advantage of externalities, even if responses are restricted to being pro…table only marginally.The same occurs in the seminal discrete models of Jackson and Wolinsky (1996) and Bala and Goyal (2000).The robustness of this incompatibility, now in a much more ‡exible model, may seem somewhat surprising.Nevertheless, the reason is clear.Similarity and di¤erence between ( 4) and (15), and between ( 6) and ( 16), both stem from the same source.Conditions for optimality and marginal stability are based on the same economic principle: Imposing zero marginal bene…t, but social (i.e.aggregate) bene…t for e¢ciency, and individual bene…t for stability.From the point of view of e¢ciency the strength of a link must maximize its contribution to the aggregate payo¤, while from the point of view of either player involved in its support it must maximize his/her payo¤.Hence the incompatibility.
Nevertheless, in a mixed environment, if a central planner o¤ers to pay for half the investment to every player, or, more precisely, subsidizing each dollar invested by a player with another dollar, e¢ciency can be sustained in marginal equilibrium.This can be shown as follows.In this situation, if the actual investment of each player in which by an argument identical to that which that leads to ( 13) and ( 14), leads to the necessary conditions for equilibrium: (whenever c ij = 0 and c ij > 0): In particular, if g c is a complete network only the …rst one applies and becomes while for an all-encompassing star whose center is player 1, it yields, (for all i 6 = 1): That is, under this subsidy e¢ciency can be sustained in marginal equilibrium.
Notice that from the point of view of players the e¤ect of this subsidy is like replacing the actual technology by a better technology ; s.t.(c) = (2c), also a DR-technology.In fact, more generally, subsidies of the form (c) = ((1 + )c), i.e. of dollars per dollar invested, with ranging from 0 to 1, bridge the gulf between equilibrium and e¢ciency.Figure 2 shows the graphs of technology (c) = c 2c+2 and that of (c) = (2c) = c 2c+1 , superimposed over those of ' n for n = 5; 12; 22 and 42 (as in Figure 1).7 Related literature In this brief review we concentrate mainly on papers published after the seminal connections models of Jackson and Wolinsky (1996) and Bala and Goyal (2000), where agents derive utility from their direct and indirect connections, and focus on those most closely related to the model studied in this paper. 11Apart from other di¤erences between our model and those commented below, there is one that applies to all of them: in our approach to stability the central concept is that of marginal equilibrium, a weaker notion than Nash equilibrium.Bloch and Dutta (2009) introduce endogenous link strength in a connections model by replacing Jackson and Wolinsky's discrete technology by an additively separable convex function of players' investments in a link that determines its strength, i.e. they assume non-decreasing returns.They also assume that players' investments are limited by a unit of resources.We instead assume technology to be a concave function of the joint investments of the players (i.e.we assume decreasing returns), whose e¤orts are assumed to be perfect substitutes, and have no budget constraint, hence the di¤erent results.Bloch and Dutta prove that in their model the star is the only Nash-stable architecture and the only e¢cient one.Deroian (2009) studies a similar model, but with directed communication, i.e.where links are directed, and proves that, as in Bloch and Dutta (2009), in equilibrium agents concentrate their investment on a single link and the complete wheel is the only e¢cient architecture and the unique Nashstable architecture.Also in the wake of Bloch and Dutta (2009), So (2016) assumes that technology is an additively separable function of players' investments, which are limited by a budget.But unlike Bloch and Dutta, So assumes that the strength of a link connecting i and j where i invests x j i and j invests x i j is (x j i ) + (x i j ), with increasing and strictly concave; while in our model the strength is a function of x j i + x i j , that is, players's e¤orts (i.e.investments) are perfect substitutes.She obtains su¢cient conditions for the symmetric complete network to dominate all star networks and for the symmetric star and the complete network to be Nash-stable, but no characterization is provided.
Other models with endogenous link strength less closely related to ours are the following.In Cabrales, Calvo and Zenou (2011) players choose a level of socialization e¤ort which is distributed across all possible bilateral interactions in proportion to the partner's socialization e¤ort.In Feri and Meléndez-Jiménez's (2013) dynamic model the choice of whom to link to and a coordination game determine the strength of the links.In Harmsen-van Hout, Herings and Dellaert's (2013) model individuals derive social value from direct connections and informational value from direct and indirect connections, but the more links an individual sustains the weaker they are.Boucher (2015) considers a model where individuals with a limited budget derive utility from self-investment and from direct connections, assuming the utility of a direct link to be a convex function of the investments of the two players involved, whose distance also enters as an argument in their utility.In Salonen (2015), Baumann (2019) and Gri¢th (2019) individuals with limited resources derive utility from self-investment and from direct connections, but assuming that the utility of a link is a strictly concave function of the investments of the two players.Ding (2019) considers a constant elasticity of substitution link-formation technology that nests unilateral and bilateral network formation.

Concluding remarks
We have developed a marginalist decreasing returns connections model which is a natural extension of the seminal discrete connections models of Jackson and Wolinsky (1996) and Bala and Goyal (2000).The basic logic is the same, payo¤ = information investment, but it is based on a non-discrete, smooth decreasing returns link-formation technology, which is the only exogenous ingredient in the model.
The characterization of e¢cient networks for DR-technologies is solved by Theorem 1, which establishes that the only possible non-empty e¢cient structures are symmetric all-encompassing stars and complete networks, and characterizes the family of DRtechnologies which admit one of these non-empty structures as e¢cient.This result shows the somewhat surprising robustness of the result on e¢ciency in the seminal discreet two-parameter connections model of Jackson and Wolinsky (1996).
As to stability, we introduce a notion of marginal equilibrium, natural in this marginalist model and new in the networks literature to the best of our knowledge, and obtain necessary and su¢cient characterizing conditions for this weak notion of stability (Theorem 2).In a marginal equilibrium, the optimal paths or channels for information which each player pays for form a well-de…ned tree, i.e. a multiplicity of such optimal paths is incompatible with marginal stability.Moreover, this along with the other characterizing conditions (Theorem 2 and Corollary 1) enables a variety of graph architectures sustainable in marginal equilibria to be identi…ed, such as subcomplete graphs, stars, trees and circles, and determine the investment pro…les that sustain them in marginal equilibrium (Propositions 7-11).A feature worth noting of marginal equilibrium is its resilience in response to shocks, such as deletion of nodes.Although no dynamic model has been provided, it is clear that nodes sensitive to the marginal revenue of its links can readjust to a new marginal equilibrium after a node vanishes by responding to such changes in many cases.
As to Nash-stability, no characterization has been obtained, only for a symmetric complete network and all-encompassing stars.Nevertheless, given that marginal stability is necessary for it, Proposition 12 gives necessary and su¢cient conditions for the empty network to be Nash-stable, Proposition 13 gives necessary and su¢cient conditions for a complete network to be Nash-stable and Proposition 14 for an allencompassing star to be Nash-stable.In a decentralized context, the comparison from Bala and Goyal (2000) on stability issues is pertinent here.In this respect, a salient di¤erence with Bala and Goyal (2000) is that, under a DR-technology, a Nash-stable (marginally stable) all-encompassing star is necessarily periphery-sponsored. 12Finally, the conditions for e¢ciency (Theorem 1) and stability, even if only marginal (Theorem 2), lead to the conclusion that they are incompatible and make it transparent why.Conditions for e¢ciency and for marginal stability are based on the same economic principle: imposing zero marginal bene…t, but social (i.e.aggregate) bene…t for e¢ciency, and individual bene…t for stability.Nevertheless, it is shown that subsidizing up to a dollar per dollar invested by each player would bridge the gap between e¢ciency and marginal stability.
There are three lines of further research that might be of particular interest.First, although Theorem 2 gives necessary and su¢cient conditions for marginal equilibrium, no complete characterization of the architectures sustainable in marginal equilibrium Assume w.l.o.g.c i c j ; then (c i ) (c j ); and consequently which implies that all three expressions must have the same value, from which it follows that which implies (c i ) = (c j ).Therefore all links in g necessarily have the same strength.Finally, if a star all of whose links have the same strength yields a positive net value but is not all-encompassing, its net value would increase by connecting any other node with a link of the same strength to the center.Proposition 2 Proof.(i) By Lemma 1, all links in an e¢cient all-encompassing star must have the same investment, c , which must maximize its net value, i.e. such that (5).Condition (6) stems from the …rst order condition for an extreme of (n 1) 2v (c) + (n 2)v (c) 2 c ; which is the net value of an all-encompassing star with n nodes and n 1 links of strength (c).
(ii) Otherwise, if 2v (c ef ) 2 < 2v (c) c for some c, connecting two spoke nodes would increase the net value of the network.Proposition 3 Proof.As (c) < 1 for all c 0, the net value of a symmetric all-encompassing star of c-links, denoted by g c-star , is and (n 1)(nv c) > 0 if and only if c < nv.In other words, an all-encompassing star of c-links yields a positive net value only if c < nv.Therefore, arg max v(g c-star ) [0; nv]; and such a maximum exists because v(g c-star ) is continuous on c.Moreover, that maximum is > 0 (i.e. an optimal symmetric star actually does exist) unless v(g c-star ) 0 for all c > 0, i.e. unless 2v (c) + (n 2)v (c) 2 c 0 (8c 0); and yield a positive net value, and consequently there is sure to be an e¢cient star or complete network.

Proposition 4
Proof.Let c = (c ij ) ij2N 2 be a link-investment vector s.t.c ij > 0 if and only if ij 2 S. Assume that c = (c ij ) ij2N 2 e¢ciently supports S and ij 2 S. Link ij is thus a necessary part of at least one optimal path in g c , the one connecting i and j, because otherwise c would not be e¢cient.The contribution of link ij, i.e. of investment c ij , to the net value v(g c ) given by ( 9) for a choice of optimal paths (p kl ) kl2N 2 is 2v X Thus for investment c ij to be optimal it must maximize for which it is necessary that which yields (11).A non-null derivative w.r.t.c at c ij means that by slightly increasing (if it is > 0) or decreasing (if it is < 0) the investment in link ij the aggregate payo¤ through those paths would surely increase, which contradicts the e¢ciency of c. Assume now that two nodes r and s are connected by two di¤erent optimal paths in g c .Then there is at least one link, say ij, that is part of one of these paths but not of the other.Then the right-hand side of (9) admits at least two di¤erent expressions: One where the optimal path between any pair of nodes k; l is p kl , and another where it is q kl , and such that for any pair k; l di¤erent from pair r; s, p kl = q kl , while for r and s the optimal path is di¤erent, i.e. p rs 6 = q sr ; and only the …rst one contains ij.In that case, which leads to a contradiction because (11) yields two di¤erent values for 0 (c ij ).

Lemma 2
Proof.It is obvious that strong marginal stability implies marginal stability.Assume that g c is connected and c is marginally stable.Let c 0 i = (c 0 ik ) k2N , and let c 00 i be given by and c 00 = (c i ; c 00 i ).For c 0 i su¢ciently close to c i , the underlying graphs of g c and g c 00 are the same and s. t. i (c) i (c i ; c 00 i ) because c is marginally stable.And for all j s.t.c ij = 0 and c 0 ij 6 = 0, as i and j are indirectly connected in g c and g c 00 , i and j receive an amount of information from each other through a path in g c 00 .Thus, a su¢ciently small investment c 0 ij in link ij (namely, as far as (c 0 ij ) is smaller than the decay along that path) is sure to be unpro…table.Therefore, for c 0 i su¢ciently close to c i , Proposition 5 Proof.The empty network has no actual links, so it satis…es trivially marginal stability.Assume now that the empty network, i.e. c ij = 0 for all i; j 2 N , is strongly marginally stable.Then for some " > 0, for all c s.t.0 < c < ", it holds that v (c) c 0: Or, equivalently, (c) c 1=v for all c < ": Then taking limits, lim c!0 (c) c = 0 (0) 1=v: Assume now that the empty network is not strongly marginally stable,i.e. for every " > 0 there exists c < " s.t.v (c) c > 0: But then 0 (0) > (c)  c > 1=v: Theorem 2 Proof.(Necessity) Let be a DR-technology and c = (c ij ) i;j2N a marginally stable investment pro…le.
(i) Assume c ij > 0, then at least one of them, i or j, say i, invests in that link, i.e. c ij > 0. Then link ij is part of at least one optimal path in g c for i's information, the one connecting i and j, otherwise i would increase payo¤ by diminishing investment in it.Fix one of the, in principle, possible di¤erent but equivalent expressions on the right-hand side of (12).Then i's payo¤ for this particular choice of optimal paths p i is given by the right-hand side of (12), which can be rewritten like this: From the point of view of player i, with the investments by the other players j 6 = i taken as given, the right-hand side of (27) depends on i's admissible strategy c i , and it is a di¤erentiable function of as many variables as i has neighbors, (c ij ) j2N d (i;g c ) .Namely, Thus the terms in (28) where c ij enters are Therefore, for the strategy of player i, (c ij ) j2N , to be marginally stable given the investments made by the other players (which fully determine (p ij ik ) for all j 2 N d (i; g c ) and all k 2 N (i; g c ) s.t.ij 2 p ik ) the following must hold A non-null partial derivative w.r.t.c ij of (27) at c means that slightly increasing (if it is > 0) or decreasing (if it is < 0) investment by i in link ij would increase i's payo¤ (through the same available paths), which contradicts the marginal stability of c .Therefore 13 @ @c ij i (c i ; By construction, K i;j (p i ) K g c i;j , but note that it must be K i;j (p i ) = K g c i;j .Otherwise, a di¤erent choice of optimal paths p 0 i = (p 0 ik ) k2N (i;g c ) would yield K i;j (p i ) 6 = K i;j (p 0 i ) and then (29) would lead to a contradiction.Therefore (13) must hold.
This means that any optimal path p ik containing a link ij s.t.c ij > 0 has a positive impact on its cost, because (p ij ik ) is a summand in K g c i;j , the denominator in (13), so that 0 (c ij ) decreases and c ij increases.Then, if an optimal path p ik contains ij and (p ik ) = (q ik ) for some other optimal path q ik in g c , the optimality of p ik would be super ‡uous because its marginal revenue for i is v (p ik ) = v (q ik ) at a cost that can be spared given that it is also received through q ik , i.e. a small decrease in c ij would increase i's payo¤, contradicting the marginal stability of c .Thus, every optimal path that contains link ij and connects node i with another node must be the only optimal 13 Just note that by the chain rule path connecting them.In other words, the optimal paths connecting one node with other nodes in which a node invests form a well-de…ned tree rooted at that node.
(ii) Assume now that c ij > 0 and c ij = 0; i.e. the link ij is entirely supported by j.A similar argument to the one used to prove part (i) leads in this case to the conclusion that v 0 (c ij )K i;j (p i ) 1 0 must hold at c i , whatever the choice of optimal paths p i = (p ik ) k2N (i;g c ) .Otherwise player i's payo¤ increases by investing in link ij, which yields 0 (c ij ) . And choosing p i s.t.K i;j (p i ) is maximal, i.e.K i;j (p i ) = K g i;j , we have 0 (c ij ) Thus conditions (i) and (ii) are necessary for c to be marginally stable.(Su¢ciency) Assume conditions (i) and (ii) hold for an investment pro…le c = (c ij ) i;j2N .Then i's payo¤ for a particular choice of optimal paths p i is given by ( 27), that is j2N is an alternative admissible strategy of player i s.t.c ij 6 = 0 only if c ij 6 = 0, then i's payo¤ through the same paths is given by (28), that is, which is a di¤erentiable concave function of (c ij ) j2N d (i;g c ) 14 .Moreover, the Kuhn-Tucker conditions for a maximum of i (c i ; c for all j 2 N d (i; g c ): Now if c ij > 0 and (13) holds, given that any optimal path containing ij and connecting i with any node k is necessarily the only path connecting them, it must be K i;j (p i ) = K g c i;j : Then whenever c ij > 0 condition (K-T.3)holds, condition (K-T.1)becomes v 0 (c ij )K g c i;j 1 + j = 0; which holds along with (K-T.2) and (K-T.4) with j = 0: Whereas if c ij = 0 and 0 (c ij ) < 1 vK g i;j ; (K-T.2) and (K-T.3)hold, while (K-T.1)and (K-T.4)hold with j = (v 0 (c ij )K g c i;j 1) > 0: Thus Kuhn-Tucker conditions for a maximum of i (c i ; c i ; p i ) constrained by c ij 0 hold at c i .Given that i (c i ; c i ; p i ) is concave, these conditions are also su¢cient for a maximum.In short, the necessary conditions guarantee that whatever the choice of optimal paths p i through which player i receives information, the investments of each player in his/her actual links are optimal to receive it trough them.Therefore, a su¢ciently small change of investments of any player is necessarily non-pro…table.

Corollary 1
Proof.Let c be a marginally stable investment pro…le and assume c ij > 0. Then if both invest in link ij condition (13) must hold for i and j, and j and i, i.e. i and j can interchange roles in (13), which yields two expressions for 0 (c ij ): : But this is possible only if the sums in both denominators are equal, in other words, only if both players, i and j; receive the same amount of information through link ij.Otherwise, the two conditions are incompatible and stability is possible only if the player who receives more information through link ij, say i, covers the whole investment, so that j;g c ) s:t: ji2p jk (p ji jk )

Proposition 6
Proof.Conditions (i) and (ii) of Theorem 2 are equivalent to marginal stability.Assume now that c is strongly marginally stable, but g c is not connected and 0 (0) > 1 v+K , where K is the information received by the node, say i, that receives the maximal amount of information.Then, if j is any node in a di¤erent component, any su¢ciently small investment of j in a link with i is sure to increase j's payo¤, contradicting c 's strong marginal stability.Reciprocally, if i and j are in di¤erent components and 0 (0) 1 v+K , where K is the information received by the node that receives the maximal amount of information, then (c)  c < 0 (0) 1 v+K for all c, i.e. (v + K) (c) c < 0, and also replacing K by the information received by the node that receives the maximal amount of information in the component of i or that of j.Consequently any investment in a link connecting them is not pro…table for either of them.Whereas if g c is connected, by Lemma 2 it is also strongly marginally stable.Proposition 7 Proof.Assume c = (c ij ) i;j2N is s.t.c ij 6 = 0 if and only if ij 2 M 2 for some M N .Then, for all ij 2 M 2 , p ij = ij, and by Theorem 2, ( 13) is necessary for marginal stability, i.e. 0 (c ij ) = 1=v for all i; j 2 M (i 6 = j).That is, for all i; j 2 M (i 6 = j) c ij + c ij = b c eq , s.t. 0(b c eq ) = 1=v; which is feasible only if 0 (0) > 1=v.Moreover, such a b c eq is unique by strict concavity of .But this is also su¢cient because then ( 14) is also satis…ed however the cost b c eq of each link is shared.
In particular, if M = N , then g c is connected and, by Lemma 2, also strongly marginally stable.

Proposition 8
Proof.Assume that c is marginally stable and g c is a star of p 1 links (3 p n). Assume w.l.o.g. that node 1 is the central player, connected with 2; 3; :::; p, and M = f1; 2; :::; pg.Obviously, in a star there is only one path connecting any two nodes.By Corollary 1, the star must be periphery-sponsored, that is, for any spoke player i, c i1 = c i1 .The payo¤ of spoke player i is then We show …rst that in marginal equilibrium all spoke nodes invest the same amount in the link that each of them supports and receive the same payo¤.Assume two spoke nodes, say i and j, invest di¤erent amounts in their links, and assume that Assume c i1 > c j1 : As is di¤erentiable and strictly concave, Which along with (30) yields ; which is a contradiction, given that, by (13), : Through the same steps, c j1 > c i1 leads to a similar contradiction for 0 (c i1 ).Therefore marginal equilibrium implies that the star must be entirely symmetric: All spoke that the optimal path connecting any two nodes is the shortest, and at least one of the two nodes that each link connects invests in it, through which the information that it receives is v( (c) + (c) 2 + (c) 3 + ::: 2 ): Thus condition ( 14) applies and becomes (17).If (17) holds, however players share the cost of each link all conditions of Theorem 2 hold and c is marginally stable.
Case k is even: Let c = (c ij ) i;j2N be an investment pro…le where c ij = c or c, alternating c-links and c-links so that c ij = 8 < : c; if ij 2 T and minfi; jg is odd, c; if ij 2 T and minfi; jg is even, 0 otherwise, i.e. c 12 = c; c 23 = c; c 34 = c, etc., with c < c.Note that if k=2 is even, the optimal path is the shortest when there is only one, while when there are two of the same length the information through the one containing the c-link that the player is involved in is A term to term comparison shows that the optimal path is the one containing the c-link that the player is involved in.Thus condition (14) applies to 0 (c) becoming (18).Then the actual information a node receives through the path containing the c-link that that node is involved in is Thus condition ( 14) applies to 0 (c) and becomes (19).Therefore if (18) and ( 19) hold, c is marginally stable.A similar argument leads to the same conclusion if k=2 is odd.Again a …xed-point argument proves that the existence of such c and such c is guaranteed if 0 (c) > 1=v.Finally, the result for k = n follows from Lemma 2.

Proposition 12
Proof.The necessity is a corollary of Proposition 5.As for su¢ciency, let be a DRtechnology and g 0 the empty network, i.e. c ij = 0 for all i; j 2 N .In these conditions a player has an incentive to invest c > 0 in a link with another (or any number of them) only if v (c) c > 0. But, by the assumptions on technology , if 0 (0) 1=v and c > 0; then (c) < c 0 (0) c=v; i.e. v (c) c 0 for all c.Proof.Let c n be an investment pro…le such that g c n is a marginally stable allencompassing star.By Proposition 8, g c n must be a periphery sponsored star of c eq -links.The central node is obviously playing its best response.If 0 (0) > 1=v the possible best responses of a spoke node i consist of connecting k spoke nodes (with 1 k n 2) by b c eq -links and replacing the link c eq -link that connects it with the center by a link that optimizes the bene…t of connecting with the center of the star formed by the remaining n k 2 c eq -links (see Figure 4), i.e. by a c k -link s.t.(23).Then, if b k denotes the response described, we have Then c n is Nash-stable if and only if for all k = 1; 2; :::; n 2 : i.e.
((n k 2) (c eq ) + 1)v( (c eq ) (c k )) (c eq c k ) + k(v (c eq ) 2 (v (b c eq ) b c eq )) 0; which yields (22).Note that if 0 (0) 1=v; the only best response of a spoke node is to keep the investment unchanged.Proof.In view of Theorem 1, the empty network is e¢cient if and only if (c) ' n (c) for all c, with ' n given by (8), and by Propositions 5 and 12 the empty network is strongly marginally stable and Nash-stable if and only if 0 (0) 1=v: But if (c) ' n (c) for all c, then 0 (0) 1=2v < 1=v.Thus whenever the empty network is e¢cient it is also strongly marginally and Nash-stable, but not reciprocally: whenever 1=2v < 0 (0) 1=v, the empty network is stable in both senses but not e¢cient.In any other case, Propositions 1 and 2 establish necessary conditions for e¢ciency under a DR-technology for the only non-empty structures that are proven in Theorem 1 to be possibly e¢cient for any DR-technology: all-encompassing stars and complete networks.Propositions 7 and 8 provide necessary and su¢cient conditions for these structures to be marginally stable.Comparing (4) and (15), with (6) and ( 16), makes it clear that e¢cient all-encompassing stars and e¢cient complete networks are not Nash-stable, or even marginally stable.This incompatibility extends to the case of an investment supporting an infrastructure e¢ciently and in marginal equilibrium according to De…nitions 2 and 6.This follows immediately from a comparison between (11) and (13).

Figure 2 :
Figure 2: Graph of ' n for v = 1 and n = 5; 12; 22; 42, and (c) and (2c) v( (c) + (c) (c) + (c) (c) (c) + (c) 2 (c) 2 + ::: + (c) while through the one containing the c-link that the player is involved in, is v( (c) + (c) (c) + (c) (c) (c) + (c) 2 (c) 2 + ::: + (c) v( (c) + (c) (c) + (c) (c) (c) + (c) 2 (c) 2 + ::: + (c) Proposition 13   Proof.Let b c n be an investment pro…le such that g b cn is complete, symmetric and marginally stable, i.e. c ij = b c eq and c ij = b ceq 2 , with 0 (b c eq ) = 1=v < 0 (0) by Proposition 7. The possible best responses of a node i consist of withdrawing support from k 1; links (with 2 k n 1) and replacing one of the remaining b c eq -links that connects it with a node connected with them by b c eq -links by a c k -link that optimizes the bene…t of this indirect connection with these k 1 nodes (see Figure315 ), i.e. by a c k -link s.t.(21) by investing c k b ceq 2 in it. 16Then, if a k denotes the response described, b c n is Nash-stable if and only if i (b c n ) i (a k ) 0; for all k = 3; :::; n; i.e., if and only if k(v (b c eq ) b c eq =2) (c k )(v + (k 1) (b c eq )v) (c k + b c eq =2) 0;which yields(20).

Figure 3 :
Figure 3: A possible best response to a complete network

Figure 4 :
Figure 4: A possible best response to the star