Here we focus on nonspecific networking, meaning that an agent cannot select a specific subset of feasible links which he wants to establish or strengthen. Rather, each agent chooses an effort level or intensity of networking. In the simplest case, the agent faces a binary choice: to network or not to network. If an agent increases his networking effort, all direct links to other agents are strengthened to various degrees. We assume that benefits accrue only from direct links. The set of agents or players is finite. Each agent has a finite strategy set consisting of the networking levels to choose from. For any pair of agents, their networking levels determine the individual benefits which they obtain from interacting with each other. An agent derives an aggregate benefit from the pairwise interactions with all others. This aggregate benefit is a function of the chosen profile of networking levels. In addition, the agent incurs networking costs, which are a function of the agent's own networking level. The agent's payoff is his aggregate benefit minus his cost. The set of agents together with the individual strategy sets and payoff functions constitute a game in strategic form. Equilibrium means Nash equilibrium. 1

Despite its apparent simplicity, our hitherto unexplored model of nonspecific networking covers a broad spectrum of applications. It allows for social networking where some persons are more attractive than others, and some even possess negative attraction. Attraction or repulsion can be mutual or not. Certain individuals can have greater advantages from networking or smaller costs of networking than others and, therefore, may be considered natural networkers. To the extent that benefits are positive, under-investment in links can occur in equilibrium. When one allows for the possibility that benefits from interactions with certain agents are negative, a player prefers not to have links and interactions with such “bad neighbors”. Therefore, agents may refrain from networking even when link formation is costless. But an agent cannot prevent bad neighbors from networking and, consequently, may suffer from their efforts. Thus, there can be over-investment in the sense that less investment would increase aggregate welfare. The above 4-player game among friends and enemies demonstrates that under-investment is a possibility as well. In Section 6, we shall present an example where under-investment by one group of agents and over-investment by a second group coexist in equilibrium. A person's social networking activities may consist in joining social or sports clubs. Our general framework encompasses such instances of social networking, like the example following Fact 1 in subsection 3.3. In addition to social interaction and networking, the model is applicable in economics, in particular in the context of industrial organization. We mention in Section 8 that the model encompasses specific cases of user network formation.

The model comprises a large class of games which are both potential and supermodular (strategic complements) games. Finite potential games and finite supermodular games have in common that a Nash equilibrium in pure strategies exists. The literature on games which share both properties is scarce. Dubey, Haimanko, and Zapechelnyuk [3] show that games of strategic substitutes or complements with aggregation are “pseudo-potential” games.2 As a consequence, they obtain existence of a Nash equilibrium and convergence to Nash equilibrium of certain deterministic best response processes. Brânzei, Mallozzi and Tijs [5] investigate the relationship between the class of potential games and the class of supermodular games. They essentially focus on two-person zero-sum games (and a special case of Cournot duopoly). Their main result is that two-player zero-sum supermodular games are potential games and conversely that two-player zero-sum potential games can be transformed into supermodular games. In our model, suitable assumptions on the benefits from pairwise interaction give rise to a novel class of games which are both potential and supermodular games. A different set of assumptions generates an equally rich family of networking games which have both a potential and the strategic substitutes property.

Potential maximization, if applicable, has several strong implications. First of all, the set of potential maximizers is a subset of the set of Nash equilibria. Hence potential maximization constitutes a refinement of Nash equilibrium in potential games.3 If in addition, the game is supermodular, the set of potential maximizers forms a sublattice of the lattice of Nash equilibria. Consequently, if the game is also a symmetric game, then it has at least one symmetric potential maximizer. Finally, again in the case of supermodular potential games, one obtains comparative statics results for the smallest and the largest potential maximizer.

In Section 5, we obtain comparative statics results for Nash equilibria with respect to networking costs for either class of networking games, those with strategic complements and those with strategic substitutes. For networking games which are both potential and supermodular games, we obtain comparative statics results for the smallest and the largest potential maximizer.

If a finite strategic game and specifically a networking game is a potential game, then perturbed best response dynamics with logit trembles yield the maximizers of the potential as the stochastically stable states, as shown by Blume [7,8], Young [9], Baron et al. [10], among others. Hence in this case, all results for potential maximizers apply to stochastically stable states as well. Two qualifications are warranted. First the coincidence of the set of potential maximizers and the set of stochastically stable states need not hold if the potential is not exact or updating is not asynchronous (like in the above papers), as Alos-Ferrer and Netzer (2010) have shown. Second, this is not to say that the study of stochastically stable states under logit perturbations has to be confined to games with exact potentials. See Alós-Ferrer and Netzer [11], Baron et al. [12], and Section 7 of Baron et al. [10].

Nonspecific networking admits a differential impact of an agent's networking efforts on the strength of links to various other agents. In particular, undirected graphs serve as a descriptive tool throughout the paper to distinguish between pairs of agents which can form links among themselves and those pairs which cannot reach each other. Such a graph represents a “social structure” in the sense of Chwe [13]. Chwe investigates which social structures are conducive to coordination in a “local information game”. In contrast to Chwe's, our model falls under the rubric of “local interaction games”. Our concern is not whether people coordinate, but who networks and how much, e.g., whether natural networkers invest more in networking than others.

The model of Bramoullé and Kranton [14] is similar to ours in many respects, with one important exception: Interaction is not pairwise but rather with the entire group of one's neighbors. For further details, see Remark (c) at the end of Section 4. The model of Cabrales, Calvó-Armengol and Zenou [15] constitutes an instance of nonspecific networking, both with respect to network formation (socialization) and with respect to network utilization. In both respects, the model exhibits strategic complements and quadratic costs. The investments in networks (socialization) give rise to a weighted graph or network. Given the network, productive investments, say a parent's time spent on homework with their child not only affects their own child's scholarly achievement, but also the achievement of other children with whose parents the parent is linked via the network. Under certain conditions, the model has three Nash equilibria, an unstable one where nobody invests in networks and two stable ones with positive investments in networks, one with low levels of networks and production (resulting in under-investment relative to the efficient outcome) and one with high levels (resulting in over-investment). The model shares several traits with ours: Nash equilibrium, pairwise interactions, comparative statics, among others. Their model differs in that investment is in two dimensions, networks and production, payoffs are of a special functional form, and results obtain asymptotically, for replica games. Galeotti and Merlino [16] adopt the network formation (socialization) part of the Cabrales et al. [15] model, with linear costs and link weight or strength replaced by link reliability or probability. Then the investments in networks yield a random graph. In the realized network, a worker with a “needless job offer” can pass on the offer to an adjacent job seeker. The authors find that investment in the network is high and the resulting networks are well connected when the job destruction rate is at intermediate levels, whereas investment is low and the emerging networks are not well connected when the job destruction rate is either low or high. Goyal and Moraga-González [17] consider a finite number of quantity-setting firms. In the first stage of a three-stage game, costless specific (directed, earmarked) network formation occurs, with pairwise stability à la Jackson and Wolinsky as the equilibrium concept. At the second stage, each makes a costly investment in R&D which reduces its marginal cost of production in the third stage. There are non-specific networking effects in that the firm's investment not only reduces its own marginal costs but also those of other firms and more so the costs of its direct neighbors. If at the third stage, the firms operate in independent markets, the complete network is stable and serves them best. If at the third stage, all firms compete in the same market, then each faces a trade-off: To the extent it benefits from the marginal cost reduction efforts of its competitors, its own investment also reduces the marginal costs of the competitors. Hence, with more connectivity, firms invest less, which increases marginal costs and reduces equilibrium outputs. Therefore, while stable, the complete network is undesirable both in terms of industry profits and total surplus.

In Section 2, we introduce concepts which are of interest not only for networking games. We set the stage in Section 3, where we develop the general model and some of the main results about Nash equilibria, potentials, and potential maximizers. In Section 4, we examine the question of networkers and networking in a class of games with pairwise symmetry. Section 5 is devoted to comparative statics. In Section 6, we present two classes of games with linear benefits and costs. In Section 7, we elaborate on stochastic stability under logit perturbations. Section 8 contains conclusions and extensions.

Let X be a partially ordered set, with partial order ≥. That is, ≥ is a reflexive, transitive and antisymmetric binary relation on X. Antisymmetric means that for any x, y ∈ X, if x ≥ y and y ≥ x then x = y. Given elements x and z in X, denote by x ∨ z or sup{x, z} the least upper bound or join of x and z in X, provided it exists, and by x Λ z or inf {x, z} the greatest lower bound or meet of x and z in X, provided it exists. A partially ordered set X that contains the join and the meet of each pair of its elements is called a lattice. A lattice in which each nonempty subset has a supremum and an infimum is complete. In particular, a finite lattice is complete. If Y is a subset of a lattice X and Y contains the join and the meet with respect to X of each pair of elements of Y, then is Y is a sublattice of X.

Let X and Y be two partially ordered sets and U : X × Y → ℝ.

When appropriate, we shall employ the concept of a potential P for a game G = (I, (S_i)_i∈I, (u_i) _i∈I) pioneered by Monderer and Shapley [21], i.e., a function P : S → ℝ such that u i ( s i , s − i ) − u i ( s i ′ , s − i ) = P ( s i , s − i ) − P ( s i ′ , s − i )for all i ∈ I, s_i, s′_i ∈ S_i, s_−i ∈ S_−i. The game G is called a potential game if it has a potential. For a game G with potential P, S * = arg max s ∈ S P ( s )denotes the set of potential maximizers. Notice that S* is a subset of the set of Nash equilibria and independent of the particular choice of the potential P.

As a first result, we obtain

Obviously, every finite potential game has a Nash equilibrium. Moreover, for a networking game that has a potential and satisfies assumption (A) of Proposition 1, the set S* of potential maximizers forms a nonempty sublattice of the equilibrium set S^NE:

To formulate sufficient conditions on benefit functions for the existence of a potential of G, let us consider for any pair of distinct players ij, the two-player game G_ij with:

player set I_ij = {i, j};

Suppose β_ij is a potential for G_ij. We say that β_ij is symmetric, if β_ij(s_i, s_j) = β_ij(s_j, s_i) for all (s_i, s_j) K × K. Existence of a symmetric potential for all pairwise interactions is sufficient for the existence of a potential of the entire networking game:

Suppose the game G_ij is zero-sum. Then the following properties are equivalent:

(α) G_ij has a potential.

(β) There exist functions f_ij: K → ℝand g_ij: K → ℝ such that b_ij(s_i, s_j) = f_ij(s_i) − g_ij(s_j), b_ji(s_j, s_i) = g_ij(s_j) − f_ij(s_i) for all (s_i, s_j) ∈ K × K.

(γ) G_ij is supermodular.

We consider a population of N = 2M players with M ≥ 2. The players form the circular undirected graph Γ = (I, E⁰). The set of available networking levels is K = {h/2: h = 0,1,…, 10}. The pairwise benefit functions constitute a special case of (3.2): b i j ( s i , s j ) = s i + s j ⋅ υ i j for { i , j } ∈ E 0 ; b i j ( s i , s j ) = 0 for { i , j } ∉ E 0 .

Costs are of the linear form c_i(s_i) = C_i · s_i with C_i > 0.

Let G be a networking game satisfying (B)-(D) and let G′ be a second networking game that differs from G only in the marginal networking costs, which are C 1 ′ > 0 , … , C N ′ > 0 in G′. Further, let s ∈ S be an equilibrium of G and s′ ∈ S be an equilibrium G′. Suppose C i ′ ≤ C i for all i and s′ ≠ s. Then s i ′ > s i for some i.

Let G be a networking game that satisfies (B) and (C) and let G′ be a second networking game that differs from G only in the marginal networking costs, which are C 1 ′ > 0 , … , C N ′ > 0 in G′. Further, let s ∈ S be an equilibrium of G and s′ ∈ S be an equilibrium of G′. Suppose C i ′ < C i for all i and s′ ≠ s. Then s i ′ > s i for some i.

Consider a family of networking games G^τ satisfying (A) and (C) which differ in the marginal cost parameters τ = ( C 1 , … , C N ) ∈ ℝ + + N. Then the smallest and the largest equilibrium of G^τ are non-increasing functions of τ.

Suppose that G θ = ( I , ( S i ) i ∈ I , ( u i θ ) i ∈ I ) , θ ∈ Θ, is a collection of finite potential games with respective potentials P^θ, θ ∈ Θ. Further suppose that:

For each i ∈ I, S_i is a finite subset of ℝ.

Then the largest (smallest) potential maximizer for each game G^θ is weakly increasing in θ on Θ.

Sympathy or antipathy among people need not be mutual, but often they are and here we assume that they are. We consider the special case of (3.2) with b_ij(s_i, s_j) = (s_i + s_j)·υ_ij. Let (I, E) be any undirected graph on I. If (iv) holds, then (i)–(iv) and, therefore, (I)–(III) hold and G is a potential game. Moreover, (A) and (B) hold. Hence with (iv), the assumptions of Propositions 1 and 2 are met.

Let us specialize further and postulate the mutual affinity condition (iv) and linear networking cost functions satisfying (C). Finally, we assume that players make binary choices, to network, s_i = 1, or not to network, s_i = 0. Accordingly, K = {0,1}.

To analyze the specific game G, let N_i be the set of player i′s neighbors and define W_i = Σ_j∈Ni υ_ij for i ∈ I. Each player i has weakly dominant strategies. Namely, the player's best responses are 1 if W_i − C_i > 0; 0 if W_i − C_i < 0; 0 and 1 if W_i = C_i.

A player's decision creates own payoff (W_i − C_i)s_i and the surplus (2W_i−C_i)s_i. Hence a player's best response is inefficient in two instances, if W_i = C_i and the player chooses s_i = 0 and if W_i < C_i < 2W_i. Therefore, inefficiencies always constitute under-investments. The aggregate functions P and W assume correspondingly simple forms: P ( s ) = ∑ i ( W i − C i ) s i ; W ( s ) = ∑ i ( 2 W i − C i ) s i

In particular, all equilibria are potential maximizers. Depending on model parameters and tie-breaking, equilibria may be efficient or inefficient.

Now mutual affinity allows for mutual lack of interest, υ_ij = υ_ji = 0 and mutual dislike, disadvantage, animosity, antipathy, enmity, or hostility, υ_ij = υ_ji < 0. In the beginning of the Introduction, we have presented an example of four players where each has two friends and one enemy. Obviously, affinities and adversities can give rise to a host of interesting social spill-overs, where a player is affected by affinities between other players. We confine ourselves to one more instructive example.

“The enemy of my enemy is my friend” usually means that if j is i′s enemy and k is j′s enemy, then i and k might form an alliance against j. Yet in the present situation, i may benefit from hostility between j and k in a different way: If i and j are enemies, υ_ij < 0, then i prefers that j is not networking. This is certainly the case if υ_jk < 0 for all k, that is if j has only enemies. For instance, let N = 3, υ₁₂ = υ₂₁ < 0, and υ₁₃ = 0. Then 1 prefers that 2 is not networking. This is guaranteed if W₂ = υ₂₃ + υ₂₁ < C₂. Since υ₂₁ < 0 the latter holds if 2 and 3 are enemies, υ₂₃ < 0, or not too close friends, 0 ≤ υ ₂₃ < |υ₂₁| + C₂.

We consider networking games of the form (3.2) with (ii) which differ from games with pairwise symmetry. We postulate numbers numbers V₁,…, V_N such that ( v ) υ i j = V j

If the V_i differ, then (iv) is violated and, as a rule, the pairwise interaction games G_ij do not have symmetric potentials. Consequently, Propositions 1 and 2 need not apply. In the sequel, we focus on a linear model which allows a systematic inquiry. This linear model is essentially identical with the one developed and analyzed in the previous subsection, with the crucial exception of condition (v):

In Propositions 2 and 7 and elsewhere in Sections 3 to 6, S*, potential maximizer(s) and potential maximizing, respectively, can be replaced by S̃, stochastically stable state(s) and stochastically stable, respectively.

For instance, Lemma 1, Fact 1 and (7.2) apply to Example 3. There, it turns out that with a slight logit perturbation, the best response dynamics would stay most of the time in the equilibrium s^●, which is the unique stochastically state of the evolutionary model based on G or G′. In Example 2, case 1, all three equilibria are maximizers of the potential of P and, therefore, stochastically stable states. Consequently, under very small random perturbations, asymmetric outcomes are more likely (since they outnumber the symmetric one) than the symmetric equilibrium. Hence very likely, one observes that some players network more than others, although none of the players are distinguished as natural networkers.

Logit trembles have the appealing feature that mistake probabilities are state-dependent and the probability of making a specific mistake, that is of playing a specific non-best response, is inversely related to the opportunity cost of making the mistake.8 Furthermore, Mattsson and Weibull [31] and Baron et al. [10,12] derive a logit rule as the solution of a maximization problem involving a trade-off between the magnitude of trembles and control costs.

The investigation of logit perturbed best response dynamics for supermodular games with potentials and the associated set of stochastically stable states is one of the original contributions of the current paper. Dubey, Haimanko, and Zapechelnyuk [3] do not consider stochastic perturbations or “noise” and stochastic stability. To our knowledge only two earlier papers, Kandori and Rob [32] and Kaarboe and Tieman [33], combine stochastic stability and supermodularity in a general setting.9 These two papers focus on a class of global interaction games based on two-player and symmetric strict supermodular games. Players gradually adjust their behavior in taking a summary statistic into account. The adjustment process is perturbed by Bernoulli or uniform trembles or slight generalizations thereof. All authors obtain monotonicity results of best responses over the set of states and show that the limit sets of the unperturbed process correspond one-to-one with the set of (strict) Nash equilibria of the recurrent game. Consequently, the set of stochastically stable states is contained in the set of Nash equilibria. Hence supermodular games exposed to uniform trembles and potential games exposed to logit trembles both induce perturbed dynamics under which the stochastically stable states form a subset of the set of Nash equilibria. Unlike the present paper, the earlier literature does not examine the structure of the set of stochastically states and its variation in response to parametric changes.

Nonspecific networking means that an individual's networking effort establishes or strengthens links to a multitude of people. The individual cannot single out specific persons with whom she is going to form links. In the simplest case, the individual has a binary choice, to network or not to network. This particular case covers already a variety of interesting scenarios and phenomena. It encompasses scenarios with differential benefits across pairs of individuals, mutual versus non-mutual (positive or negative) affinities, leading for instance to second-order externalities such as the impact of an enemy of an enemy or to the co-existence of under-investment and over-investment in networking as exemplified in Section 6. Often, however, networking efforts are gradual and our model accommodates this possibility as well. Beyond expanding the descriptive scope of the model, the availability of several levels of networking effort makes the question of Section 5—how networking efforts respond to a change in networking costs—much more interesting. One conceivable generalization of our analysis, including the comparative statics, would assume multi-dimensional effort choices, like choosing software-hardware combinations.

The model also encompasses the formation of user networks, not dealt with in this paper. In that particular application, the player set I is interpreted as a finite population of users or adopters. Each player has to adopt exactly one technology or network good from the list K. The list K may consist of computer systems, word processors, internet providers, etc. The adopters of the same good constitute a user network. Baron et al. ([12], p. 574) consider the case of partial but imperfect compatibility of different technologies. Furthermore, two prominent classes of spatial games, both analyzed in detail in Baron et al. ([10], pp. 555-557) permit a novel interpretation as user network formation games. The first class consists of coordination games which can be reinterpreted as network formation games with perfect incompatibility of different technologies. The second class consists of minimum effort coordination games which allow an interpretation of network formation games with downward compatibility of technologies.

Supermodularity and increasing differences, utilized in some of our comparative statics, are cardinal properties. As Milgrom and Shannon [36] point out, comparative statics questions are inherently ordinal questions, and the conditions on objective functions and constraints necessary for comparative statics conclusions should possibly be ordinal. Indeed, Milgrom and Shannon [36] find such ordinal conditions for monotone comparative statics. They introduce and study quasi-supermodular functions and functions with the single crossing property. These functions generalize supermodular functions and functions with increasing differences and preserve the monotonicity conclusion for parametric optimization problems. A list of a wide variety of problems in economics and in noncooperative games presented by Milgrom and Shannon [36] makes a convincing case for the value added of their ordinal extension of complementarity conditions. In view of these results, one might ask whether Proposition 2 can be extended further by invoking such ordinal conditions. Precisely, if we assume that each b_ij satisfies the single crossing property on S_i × S_j, are we then able to show that the potential P is quasi-supermodular on S? Unfortunately, one cannot draw such a conclusion. The reason is that the generality of the single crossing property has its drawbacks: Namely, in the proof of Proposition 2 we make use of Corollary 2.6.1 in Topkis [18] which states that for a function defined on a finite product of totally ordered sets, pairwise supermodularity implies supermodularity. This crucial auxiliary result no longer holds when the single crossing property is substituted for the pairwise supermodularity property Shannon ([37], p. 220) demonstrates that the single crossing property in each pair of variables does not imply quasi-supermodularity in all variables.

Proposition 7 establishes a weak monotonicity result on the set of potential maximizers. It states that the largest (smallest) potential at a lower parameter value is smaller than the largest (smallest) potential maximizer at a higher parameter value. But this result does not assert that a given potential maximizer at a lower parameter value is smaller than any other potential maximizer at a higher parameter value. Echenique and Sabarwal [38, p. 309] give a condition on a pair of parameters θ,θ′ ∈ Θ, θ ≤ θ′, which implies sup S^N(θ) ≤ inf S^N(θ′) for the two equilibrium sets S^NE(θ) and S^NE(θ′). Since S*(θ) ⊆ S^N(θ) and S*(θ′) ⊆ S^N(θ′), their condition also implies supS*(θ) ≤ inf S*(θ′).

A further alternative could make the set of available efforts a (one- or multi-dimensional) interval or convex set and assume sufficient differentiability of the cost and benefit functions. As Brueckner [39] demonstrates in the context of specific networking, one arrives at some conclusions very elegantly, if such a continuous model is highly symmetry, but does not get very far otherwise. Most of our subcases and examples can be easily embedded into a larger continuous model. But again, while this might produce some eloquence and quickness of derivations in some cases, it would only render the analysis more complicated in others. An added complication stems from the fact that the concept of stochastic stability developed in the literature so far (based on logit or other perturbations) and employed in the present paper relies on a finite state space.

The idea that the strength or reliability of a link might depend on the efforts of both agents involved, is also central to the model of Brueckner [39].10 Similarly, Haller and Sarangi [41] and Baron et al. [42] consider the possibility that the reliability of a link between two agents depends on the efforts of both agents. Bloch and Dutta [43] consider the possibility that the strength of a link between two agents depends on the efforts of both agents. In Cabrales et al. [15], link intensity depends on the socialization or networking efforts of both players constituting the link. Moreover, their model exhibits nonspecific networking and productive investments with spill-overs across the network. Since we allow for negative affinity or attraction, some agents might not only abstain from networking but might take counter-measures against the networking attempts of others and be willing to incur costs in order to weaken or sever links. This eventuality suggests a further extension of the formal model.