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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

A new model of strategic networking is developed and analyzed, where an agent's investment in links is nonspecific. The model comprises a large class of games which are both potential and super- or submodular games. We obtain comparative statics results for Nash equilibria with respect to investment costs for supermodular as well as submodular networking games. We also study supermodular games with potentials. We find that the set of potential maximizers forms a sublattice of the lattice of Nash equilibria and derive comparative statics results for the smallest and the largest potential maximizer. Finally, we provide a broad spectrum of applications from social interaction to industrial organization.

Models of strategic network formation typically assume that each agent selects his direct links to other agents in which to invest. Yet in practice, a person's networking efforts may not only establish or strengthen desirable links to specific agents, but also create or reinforce links to many if not all other individuals. Beneficial links may come along with detrimental ones. For example, being better connected and more accessible implies potentially more calls from phone banks, more “spam”, more encounters with annoying or hostile people.

To illustrate the latter, consider a population of four persons where each has two friends and one enemy. Each individual _{i}_{i}_{i}_{j}_{i}_{j}_{i}_{j}_{i}_{i}_{i}_{i}

Nonspecific networking does not mean that an individual's networking effort affects everybody else. As a practical matter, networking may be possible between certain persons and not between others. We employ graphs to model restrictions on networking. Then formally, networking takes place within a given network or graph. Not only may different persons be affected differently by an individual's networking effort; but individuals may also differ in their networking efforts, even if the same range of effort levels is available to them. Both in traditional and in electronic interactions, some agents are much more active in networking than others and might be called “networkers”. Some might be considered designated networkers because they have higher benefits or lower costs from networking than others. To fix ideas, consider a population of four individuals. Their networking possibilities are described by a circular graph as follows where ↔ means that networking between the two persons is possible. ■ stands for a high cost person and □ stands for a low cost person or “natural networker”. Individuals are identical ex ante in all other respects.

If benefits and costs of networking are positive, one would expect low cost □-persons are networking more (or at least not less) than high cost individuals. Indeed, this is the case for a wide range of model parameters. But it is not necessarily the case when low costs and high costs are very close. High cost individuals may be networking in equilibrium while low cost individuals are not. This intriguing result is driven by strategic substitutes in networking: In that case, if a □-person has two neighbors whose networking efforts are high, no effort may be the best response, and if a ■-person has two neighbors choosing zero effort, a significant effort may be the best response. For details, we refer to Example 2. A related yet different question is whether more networking occurs when

To recapitulate, we develop and analyze a new model of strategic network formation—or rather network utilization in many instances—where

an agent's effort or investment in links is nonspecific;

the intensity and impact of links can differ, possibly with a negative impact of certain links;

networking may take place within a given network (graph).

Our model holds much promise for several reasons:

A broad spectrum of applications.

A rich class of games which are both potential and supermodular games.

Possibility of comparative statics with respect to networking costs.

Possibility of stochastic stability analysis.

Possibility to explore networking within a “social structure”.

In the remainder of the section, we shall elaborate on these points.

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.

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 [

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.

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 [

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 [

The model of Bramoullé and Kranton [

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.

Here we collect definitions and results that are of interest beyond the investigation of nonspecific networking. Throughout, we consider finite games in strategic or normal form
_{i}_{i}_{j} S_{j}_{−i} = ∏_{j}_{≠}_{i} S_{j}

Let

Let

The function U satisfies

The function

Let

Let Euclidean spaces ℝ^{l}_{i}_{1} × … × _{n}^{N}_{1},…, _{N}_{i}_{−i}). Similarly, we shall write _{i}_{j}_{−ij}) in case _{−ij} instead of ∏_{k}_{≠}_{i,j} S_{k}

A function u: _{−ij}): _{i}_{j}_{−ij} ∈ _{−ij}.

Since _{1} × … × _{N}_{i}

For a finite _{i}_{i∈I}, (_{i}_{i}_{∈}_{I}_{i}

The game _{i}_{i}_{−i}) ∈ _{i}_{−i}.

Pairwise supermodularity is a strategic complements condition when reaction functions exist and equivalent to ^{2}_{i}_{j}_{i}

When appropriate, we shall employ the concept of a _{i}_{i∈}_{I}_{i}) _{i}_{∈}_{I}_{i}_{i}_{i}_{−i} ∈ _{−i}. The game

Our model of nonspecific networking constitutes a game in strategic form. There is a finite player set _{0}_{1},…,_{T}_{1},…, _{T}

Players receive _{ij}_{i}_{j}_{i}_{i}_{j}_{j}_{ij}_{i}_{i}_{i}_{i}_{i}_{i}_{1}), …, _{i}_{T}_{0}_{1},…, _{T}_{i}_{1},…, _{N}

For specific interpretations, it proves advantageous to decompose benefit functions as follows:
_{ij}_{ij}_{ij}_{ij}_{ij}_{ij}_{ij}_{ij}_{ij}_{ij}

The list _{i}_{i}_{∈}_{I}_{i}_{i}_{∈}_{I}^{NE} denote the set of Nash equilibria of

We adopt the standard notion of efficiency in the literature on networks. Let

It proves convenient and instructive to distinguish the pairs (_{ij}

In the sequel, we frequently assume a graph (_{ij}

Several of the subsequent examples will be based on the ^{0}) with
^{0} a one-dimensional interaction structure in contrast to two-dimensional interaction structures.

We are going to explore the implications of two opposite conditions, (A) and (B), on the benefits from networking. We will further consider condition (C) on networking costs and condition (D) on best responses:

There exists an undirected graph (without isolated nodes) (_{ij}_{ij}_{i}_{j}_{i}_{j}

There exists an undirected graph (without isolated nodes) (_{ij} = 0 for {_{ij}_{i}_{j}_{i}_{j}

There exist _{1} > 0,…, _{N}_{i}_{i}_{i}_{i}_{i}_{i}

For _{−i} ∈ _{−i}.

Let us consider Igor, player

As a first result, we obtain

Let ^{NE} ⊆ ^{NE} is a lattice.

The proof consists in verifying that the hypothesis of _{i∈I} _{i}_{i}_{i}_{j}_{−ij}) satisfies increasing differences in (_{i}_{j}_{i}_{j}_{−ij}∈_{−ij} because of the functional form _{i}_{i}_{−i}) has increasing differences in (_{i}_{−i}) on _{i} × _{−i}. Hence

Since

In general, a networking game need not have a Nash equilibrium in pure strategies:

We consider a population of ^{0}). _{i}_{i}_{i}

In this example, ^{NE} is empty. Namely, if at least one of the even numbered players plays 0, then the best response of both odd numbered players is to play 0. Against the latter, the best response of both even numbered players is 1. In turn the best response of both odd numbered players is 1. Against the latter, the best response of both even numbered players is 0, and we have reached a cycle where players alternate their choices. If none of the even numbered players plays 1, we also reach a cycle where players alternate their choices.

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 ^{NE}:

Suppose

(

(^{NE} and of

(_{i} is supermodular on

(

The two equalities follow from the definition of a potential _{i}_{j}_{−ij} ∈ _{−ij}. As this property holds for all

(^{NE} is a lattice with respect to the partial order induced by the partial order of ^{NE}. Thus we have that ^{NE} ⊆ ^{NE}.

(_{i}^{∞} = arg max_{s}_{∈}_{S} W^{∞} ∩ ^{∞} is a sublattice of ^{∞} ∩

Observe that if in addition,

The result that the set of potential maximizers forms a nonempty sublattice of _{1},…,_{N}^{NE}. For the conclusion of Proposition 1 that the set of Nash equilibria ^{NE} is a nonempty lattice can be hardly replaced by the stronger assertion that ^{NE} is a sublattice of the set of strategy profiles

Part (^{∞} ∩

The results contained in Propositions 1 and 2 do not depend on the particular form of the payoff functions _{i}_{i}, s_{−i}) ∈ _{i}_{−i}.

In general, a networking game satisfies neither condition (A) nor condition (B) as Example 1 demonstrates. A networking game need not be a potential game either. But which restrictions on benefit functions would yield a potential game?

To formulate sufficient conditions on benefit functions for the existence of a potential of _{ij}

player set _{ij}

strategy sets _{i}_{j}

payoffs _{ij}_{i}_{j}_{ji}_{j}_{i}_{i}_{j}

Suppose _{ij}_{ij}_{ij}_{ij}_{i}_{j}_{ij}_{j}_{i}_{i}_{j}

If for each distinct pair _{ij}_{ij}

Analogous to proof of Proposition 1 in Baron

Suppose that each player _{i}_{i}_{i}_{1} > 0 and basic benefit _{1}. _{i}_{2} > c_{1} and basic benefit b_{2}. _{i}_{1} + c_{2} and basic benefit b_{3}. Moreover, player _{ij}_{i}_{j}_{i}_{i}_{1} − b_{1}, _{i}_{2} − b_{2}, _{i}_{1} + c_{2} − b_{3}. Then each game _{ij}_{ij}_{ij}

Next we impose directly certain restrictions on the pairwise benefit functions and discuss how they relate to the existence of symmetric potentials. For any distinct pair of players

Identical Benefits:_{ij}_{i}_{j}_{ji}_{j}_{i}_{i}_{j}

Symmetric Benefits: _{ij}_{i}_{j}_{ji}_{i}_{j}_{i}_{j}

Interchangeable Actions: _{ij}_{i}_{j}_{ij}_{j}_{i}_{ji}_{i}_{j}_{ji}_{j}_{i}_{i}_{j}

Condition (I) is tantamount to _{ij}_{ij}_{ji}_{ji}_{ij}_{ij}_{ji}_{ij}

If the conditions (I)–(III) hold, then the game _{ij}_{ij}

While obviously restrictive, existence of a symmetric potential for _{ij}_{ij}

Identity: _{ij}_{i}_{j}_{ji}_{j}_{i}_{i}_{j}

Symmetry: _{ij}_{i}_{j}_{ji}_{i}_{j}_{i}_{j}

Interchangeability: _{ij}_{i}_{j}_{ij}_{j}_{i}_{ji}_{i}_{j}_{ji}_{j}_{i}_{i}_{j}

We also consider a symmetry condition for the valuations _{ij}

Mutual Affinity: _{ij}_{ji}

Mutual affinity can result, e.g., from similarity (kindred spirits) or from complementarity (attraction of opposites). There can be mutual lack of interest, _{ij}_{ji}_{ij}_{ji}

We repeatedly consider games exhibiting pairwise symmetry of the form

In certain pairwise interactions, one party gains when the other loses and vice versa. One can think of chess matches, instances of gambling, or mutual industrial espionage. This means that for such a pair of players _{ij}_{ij}_{i}_{j}_{ji}_{j}_{i}_{i}_{j}_{ij}_{ji}_{ij}_{ij} and supermodularity of _{ij}

Suppose the game _{ij}

(_{ij}

(_{ij}_{ij}_{ij}_{i}_{j}_{ij}_{i}_{ij}_{j}_{ji}_{j}_{i}_{ij}_{j}_{ij}_{i}_{i}_{j}

(_{ij}

By Theorem 1 of Brânzei

If a zero-sum game _{ij}_{ij}_{i}, s_{j}_{ij}_{i}_{ij}_{j}_{ij}_{ij}_{ij}_{ij}_{ij}_{i}^{NE} = _{1} × … × _{N}. Essentially the same conclusions hold if each basic game _{ij}

Both in traditional and in electronic interactions, some agents are much more active in networking than others and might be called “networkers”. Some might be considered designated or natural networkers because they have higher benefits or lower costs from networking than others. In the Introduction, we already raised the question whether natural networkers would necessarily network more. To address this question, we examine the following example.

We consider a population of ^{0}). The set of available networking levels is

Costs are of the linear form _{i}_{i}_{i}_{i}_{i}

_{ij}_{i}

Then the networking game _{i}_{i}_{∈}_{I}_{i}_{i}_{∈}_{I}^{Δ} = (0,1, 0,1,…, 0,1) and s^{∇} = (1,0,1,0,…,1,0).

All three equilibria are inefficient, with the same value _{i}_{j}^{0}. The game has a potential

_{ij}_{i}_{i}

Then the odd numbered players have a cost advantage and are the “natural networkers”.

If the cost advantage is rather small, e.g., ^{Δ}, and ^{∇} are still Nash equilibria. In ^{∇}, the natural networkers are not networking while the other players are. However, the cost difference does have an impact: The Nash equilibrium ^{Δ}—where the natural networkers are networking and others are not — is the only potential maximizer.

If the cost advantage is sufficiently large, then only natural networkers are networking in equilibrium. E.g., if

The equilibria ^{Δ}, ^{∇} and

If in CASE 2, the payoff function _{i}_{i}

Bramoullé and Kranton [_{i}^{2}-function _{+} → ℝ_{+} with _{i}_{i}_{i}_{i}^{Δ}, and ^{∇} above arise.

Intuitively, one would expect that networking activities intensify if networking costs decline. This conjecture proves at least partially true in the presence of strategic substitutes in pairwise interactions. To be precise, we consider conditions (B)–(D). Notice that condition (B) constitutes the antithesis of condition (A). It is satisfied in Example 2. Both (A) and (B) hold for the linear models of subsections 6.1 and 6.2.

Let

Let _{1},…, _{N}_{i}_{i}_{i}_{−i},

The assumption (D) of unique best responses can be disposed of if one postulates strict cost reductions instead:

Let

Let _{1},…, _{N}_{i}_{−i} in _{i}_{i}_{i}_{i}_{j}_{i}_{i}

Notice that the conclusion of Propositions 4 and 5 cannot be substantially strengthened for two reasons. For one, _{i}

Without a strategic substitutes assumption, a cost decline is consistent with a universal reduction of networking activities. Next we provide a numerical example with this property.

We consider a population of ^{0}). The set of available networking levels is ^{1/4} −1, e−1} where e = exp(1) is the Euler number. Put _{ij}_{i}, s_{j}

With _{i}^{−1} for all ^{0} = (0,…, 0) and ^{●} = (e − 1,…, e − 1).

Setting
^{0}, ^{●}, and ^{●●} = (^{1/4} − 1,…, ^{1/4} − 1).

Thus, the example has actually several interesting features. First, there exists the equilibrium ^{0}, an instance of mutual obstruction where nobody has an incentive to network if nobody else is networking. Next there exists the equilibrium ^{●} where everybody exerts maximum networking effort. Further, a cost reduction leads to the emergence of a third equilibrium, ^{●●} where everyone makes a positive but less than maximal effort. Regarding our original point, the conclusion of Propositions 4 and 5 obviously need not hold if the strategic substitutes assumption of the form (B) is violated.

The example satisfies assumptions (A) and (C). In addition, the games ^{0} is the smallest equilibrium and ^{●} is the largest equilibrium in both games. Thus, the smallest and the largest equilibrium prove immune to a cost reduction. This observation is consistent with the claim that in response to a cost decrease, the smallest and the largest equilibrium will never decrease. Formally, we obtain a weak monotonicity result by applying an earlier result of Milgrom and Roberts [

Consider a family of networking games ^{τ}^{τ}

Endow the parameter space
^{τ}^{τ}

By Proposition 2, if in addition to satisfying (A) and (C), a networking game is a potential game, then the set of potential maximizers forms a nonempty sublattice of the set of equilibrium points. As a consequence of this added structure, there exist a smallest and a largest potential maximizer. Interestingly enough, the comparative statics à la Milgrom and Roberts for supermodular games extend to the smallest and largest potential maximizer. We choose a more abstract formulation in this instance than before. Let Θ be a nonempty subset of some Euclidean space ℝ^{n},

Suppose that
^{θ}

For each _{i}

For each _{i}_{−i}) ∈ _{i}_{−i}.

For each _{−i} ∈ _{-i}_{i}, θ_{i}

Then the largest (smallest) potential maximizer for each game ^{θ}

Pick any _{i}_{i}^{θ}^{ϑ}_{i}

This means that ^{θ}^{θ}_{s}_{∈}_{S} P^{θ}

Now consider ^{p} S_{S}_{S}_{S} S_{S} _{S}_{S} S_{S} S_{S} S_{S} S_{S} S_{S} S_{S} S

Suppose that for some integer _{i}^{θ}_{1},…, _{N}

Note that the potential does not necessarily depend on _{i}_{+} and the payoff function is defined by
^{θ}_{i}_{∈}_{I}_{i}

Further note that the pairwise benefit function
_{i}_{j}_{i}_{j}_{i}_{j}

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 _{ij}_{i}, s_{j}_{i}_{j}_{ij}

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, _{i}_{i}

To analyze the specific game _{i}_{i}_{j}_{∈}_{Ni} υ_{ij}_{i}_{i}_{i}_{i}_{i}_{i}

A player's decision creates own payoff (_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{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, _{ij}_{ji} = 0 and mutual dislike, disadvantage, animosity, antipathy, enmity, or hostility, _{ij}_{ji}

“_{ij}_{jk}_{12} = _{21} < 0, and _{13} = 0. Then 1 prefers that 2 is not networking. This is guaranteed if _{2} = _{23} + _{21} < _{2}. Since _{21} < 0 the latter holds if 2 and 3 are enemies, _{23} < 0, or not too close friends, 0 ≤ _{23} < |_{21}| + _{2}.

We consider networking games of the form (3.2) with (ii) which differ from games with pairwise symmetry. We postulate numbers numbers _{1},…, _{N}

If the _{i}_{ij}

We assume an undirected graph (_{ij}_{i},s_{j}_{i}_{j}_{j}_{ij}_{i},s_{j}_{i}_{i}_{i}_{i}_{i}_{j}_{Ni} V_{j}_{i}_{i}_{i}_{i}_{i}_{i}_{i}(_{i}_{i}_{i}_{i}_{i}_{i}V_{i}_{i}_{i}

First, “bad neighbors” may not only harm “good neighbors”, but can also harm each other through their networking efforts. For example, let _{1} = _{2} = −1, _{3} = _{4} = 1, 0 < _{i}_{1}(_{1},_{2}(_{2} and _{3}(_{4}(^{0} = (0, 0,0, 0). But given any choices by 3 and 4, players 1 and 2 find themselves in a Prisoner's Dilemma. Incidentally, the efficient outcome would be _{3} + _{4}). Hence, the equilibrium

Second, the particular networking game _{ij}_{ij}_{ij}_{ij}_{ij}_{i}, β_{ij}_{j}, β_{ij}_{i}_{j}_{ij}_{i}_{j}

It turns out that potential maximizers in finite games are the stochastically stable states for a particular kind of stochastically perturbed best response dynamics. Therefore, any results obtained for potential maximizers also hold for those stochastically stable states. Our specific concept of stochastic stability of outcomes (joint strategies) in a finite N-player game
_{1},…, _{n}) ≫ 0 be an _{i}

The perturbed adaptive rule is a logit rule: Suppose the current state is _{j}_{j}_{∈}_{I}_{−i} = (_{j}_{j≠i}_{i}_{i}_{i}_{−i} is more likely to be chosen than a non-best reply. As _{s,s′}_{s,s′}_{i}_{−i}), then
_{ϵ}_{→0}

The profiles in

As an immediate consequence of (7.2), we obtain:

In Propositions 2 and 7 and elsewhere in Sections 3 to 6,

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 ^{●}, which is the unique stochastically state of the evolutionary model based on

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.

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 [

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

Supermodularity and increasing differences, utilized in some of our comparative statics, are cardinal properties. As Milgrom and Shannon [_{ij}_{i}_{j}

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 [^{N}^{N}^{NE}^{NE}^{N}^{N}

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 [

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 [

We would like to thank the two referees and the editor for helpful suggestions. Financial support by the French National Agency for Research (ANR)—research program “Models of Influence and Network Theory” ANR.09.BLANC-0321.03—is gratefully acknowledged.

The recent literature on network formation employs mainly two alternative equilibrium concepts—and combinations thereof. Jackson and Wolinsky [

The notion of pseudo-potential games is a generalization of the notion of best-response potential games introduced by Voorneveld [

Peleg, Potters and Tijs [

In case _{i}^{l}_{i}_{i}_{i}_{−i} ∈ _{−i}. In our case,

Or strategically equivalent to a zero-sum game.

For

For all ^{p} ^{p} is the strong set order. Precisely, ^{p} S_{S}_{S}

The most prominent alternative, Bernoulli or uniform trembles, does not have this feature. Both types of trembles often, but not always lead to the same set of stochastically stable states or long-run equilibria.

Other papers on stochastic stability and supermodularity (or submodularity) exist but they exclusively deal with symmetric aggregative games that are either submodular or supermodular [Alós-Ferrer and Ania [

After learning about our work, Sudipta Sarangi pointed out to us Brueckner's paper and a further common trait of the two papers: Brueckner presents two asymmetric examples, one with an agent who creates higher benefits than others and a second example with an agent who is more accessible than others. See Roy and Sarangi [