## 1 Introduction

Local externalities are a phenomenon of great significance in a wide range of different contexts and the performance of an economic system hinges upon how agents respond to them. Local externalities are important, for example, in problems of human capital accumulation, learning and search, crime, productivity and growth, technological adoption and R&D collaboration.^{1} By their very nature, local externalities greatly depend on the pattern of interaction; that is, the social network. It is essential, therefore, to understand in detail the interplay between the topology of the network and agents’ incentives.

There are, however, few papers that explore this issue systematically and in some generality (see below for a summary). The main objective of the present paper is to suggest an approach to the study of local effects in a context where the social network is complex, volatile, and agents have only local information about it. We say that the network is *complex* because its architecture displays substantial heterogeneities and no clear patterns. We posit that agents have only *local information* about the network because they are taken to know how many first neighbors they have but ignore the number of second neighbors (i.e. neighbors of neighbors). Finally, the network is best conceived as *volatile* (i.e. with links being short-lived), so players can only use probabilistic information on ex-ante regularities; in particular, they have no time to learn the type or behavior of their neighbors.

To accommodate these considerations in a strategic game, we use the framework provided by the theory of random networks.^{2} In essence, a random network is to be conceived as a stochastic ensemble; that is, a probability measure (typically uniform) defined on a given family of possible networks. This family is usually characterized in terms of certain overall properties, such as a particular degree distribution, degree correlations or clustering. The basic postulate is then that, while all eligible networks satisfy the properties required, the specific network realized is uncertain.

In our framework, players are connected through a social network, whose statistical properties are solely characterized by a degree distribution. Each player knows this distribution and his or her own degree (which can be conceived as her “type”). With this information at hand, every player has to choose his or her costly effort. The equilibrium decision so taken by each player must depend on a number of factors. First, it has to reflect the intensity of interaction of the player in question (i.e. his or her degree). Second, it ought to hinge upon the overall distribution of types prevailing in the population. Finally, it must be shaped by the precise nature of local externalities. Although our analysis is fully general in terms of the underlying degree distribution, concerning payoffs we focus on a paradigmatic case where an agent’s gross payoffs are given by a Cobb-Douglas function of all efforts (or investments) displayed by the agent and the agent’s neighbors and individual costs are quadratic. An interesting feature of this formulation is that the nature of the network externalities becomes an endogenous outcome of the model. Therefore, whether the externalities induced by neighbors are positive or negative depends on equilibrium play: in particular, on whether their effort is high or low.

To better understand this feature of our model, let us fix ideas and conceive *interaction* as the mechanism through which agents accumulate human capital: an agent’s level of skills and education is the result of combining his or her own effort with that of his or her partner’s. Our payoff formulation implies that, for a given set of those partners, the investment in human capital displays strategic complementarities. However, the effect of changes in the number of partners crucially depends on whether players exert high or low effort. In the, former case, an increased number of partners yields positive externalities, whereas it induces negative externalities in the latter case. In this light, the following questions naturally arise. Under what conditions does a player’s interaction induce high effort and, therefore, positive spillover effects on one’s own (investment in) education? Do more connected societies support a higher level of human capital, or quite the opposite? Are more homogeneous societies (in player connectivity) better positioned to induce individuals to invest more in human capital accumulation? Alternatively, do more polarized societies generate higher incentives for individuals to do so?

It is easy to see that the model always allows for a trivial noninterior equilibrium where every agent, independently of his or her degree, exerts zero effort. Naturally, our interest is in equilibria where agents play nontrivially as a function of their degree. To guarantee that any such equilibrium is interior (and, therefore, can be characterized through marginal conditions), it is convenient to posit that the range of possible effort/investment is unbounded. We can show that there always exists a unique nontrivial equilibrium. In this equilibrium, there is a clear relation between effort profile, expected utilities, players’ network degree and investment costs. When the cost of effort is low (respectively, high), equilibrium efforts and equilibrium utilities are declining (respectively, increasing) in the degree. In this case, agents impose negative (respectively, positive) externalities on their neighbors. The intuitive basis for this conclusion can be explained as follows. When investing in effort is relatively cheap, any effort profile in which neighbors generate positive externalities always induces players to exert additional effort. This positive “social-multiplier effect” is inconsistent with an interior equilibrium. So, in this case, neighbors must generate negative externalities at equilibrium and, consequently, the more connected a player is, the lower his or her incentive to invest.

Our second set of results compare equilibria across different networks. We start by comparing degree distributions that are ranked according to the criterion of first-order stochastic dominance (FOSD). This amounts to comparing networks whose respective levels of connectivity can be unambiguously ordered. Then, as network connectivity is increased, we find that individual efforts uniformly adjust upward when costs are low, while they uniformly adjust downward otherwise. To understand the intuition, note that higher network connectivity in this sense simply implies that each player increases his or her probability of interacting with more connected players. This strengthens the negative (positive) externalities imposed on players by their neighbors when costs are low (high), thus leading players to uniformly increase (decrease) effort to offset such an effect.

Finally, to understand the effects of network polarization, we compare networks that have the same average connectivity but differ in the way the links (always the same number of them) are allocated across players. Specifically, we consider degree distributions that can be ordered according to the mean preserving spread criterion. We show that efforts are systematically lower in networks displaying broader degree distributions. The intuition here relies on the curvature of the equilibrium strategies. Because equilibrium effort happens to be given by a convex function of the degree, whenever the degree distribution becomes broader (keeping the same mean) each link becomes, on average, more valuable. At equilibrium, therefore, players exert lower effort in order to compensate for this effect.

Local knowledge, network complexity and network volatility are the three aspects that distinguish our paper from the existing published literature on local externalities. In the literature, there are two polar approaches that have been used to model local effects. One approach posits that agents interact with their neighbors in a fixed (or relatively stable) socioeconomic network, whose architecture is common knowledge.^{3} Hence, players know who their neighbors are, and the neighbors of their neighbors etc., and can fully analyze the situation as a game of complete information. In the second approach, the specific pattern of interaction is not known beforehand but each agent anticipates that he or she will be interacting with some fixed number of individuals randomly sampled from the whole population. Therefore, even though players do not enjoy a precise knowledge of the realized network, social interaction has an extremely simple structure and players confront, ex-ante, a fully symmetric social environment.^{4}

As compared to these two branches of literature, our approach introduces an incomplete-information scenario that seems better suited to understanding strategic behavior in large and typically complex social networks in the real world. Furthermore, it delivers predictions that are markedly different from those obtained in models where the network architecture is common knowledge.

The present approach was first explored in our original version of this paper, Galeotti and Vega-Redondo (2005), where we focused on scenarios given by three paradigmatic families of (parametrized) degree distributions: Poisson, geometric or scale-free. In that paper we characterized the equilibria for these three scenarios and also conducted some basic exercises of comparative statics. A development of this approach has been undertaken by Galeotti *et al.* (2009). They carry out a general study of how the network topology impinges on strategic behavior in games where payoffs display what they call either degree complements or degree substitutes.^{5} Their payoff formulation, however, is crucially different from the present one. For example, in their model, whether there is degree complementarity or substitutability is an exogenous assumption, whereas, in our case, it is (as advanced above) an endogenous outcome of the equilibrium. We borrow, however, from Galeotti *et al.* (2009) the tools used to compare different degree distributions: essentially, first-order and second-order dominance criteria. More specifically, our comparative-static analysis applies that methodology to a setup that generalizes to any random network the analysis we originally undertook for specific contexts in Galeotti and Vega-Redondo (2005).

The rest of the paper is organized as follows. Section 2 presents the general theoretical framework. Section 3 specializes this general framework to a context with strategic complementarities. Section 4 concludes. The detailed proof of the results is relegated to the Appendix.