Cooperative networks with robust private monitoring☆
Introduction
Individuals cooperate in a range of social and economic settings even without formal institutions to enforce cooperative behavior. The sharing of information between innovative firms, the exchange of favors among colleagues, or the informal risk-sharing arrangements in developing economies, are examples of cooperative behavior that rely on informal enforcement of cooperative norms. While cooperation can sometimes be sustained bilaterally through infinitely repeated interaction, when there is heterogeneity in preferences, information, or enforcement opportunities, it can be more efficient to pool incentive constraints across groups, communities, or markets (Bernheim and Whinston, 1990). Community enforcement leverages contagion strategies, triggering a spread of non-cooperative behavior following deviations (Kandori, 1992; Ellison, 1994). A recent strand of research seeks to formalize the structure of community interactions in terms of networks (Jackson, 2009; Goyal, 2012), enhancing the empirical relevance of the theory by generating predictions about observable network structure.
In this paper, we develop a theory of community enforcement that generates predictions about networks that support cooperative behavior under private monitoring. We model network formation and continuation play in a unified framework, with a simple trade-off between investments in the formation of ties and the ongoing benefits of cooperation. In the formation phase, players can make heterogeneous investments in bilateral ties. In the continuation phase, players have the opportunity to cooperate on their bilateral ties, receiving a stream of cooperative payoffs, or permanently ending a bilateral interaction to receive a one-off premium. In a given period, the stage-game on a bilateral tie therefore has the structure of a Prisoner’s Dilemma, providing a stylized model of the tension between individual incentives and social efficiency. Investment in a tie during the formation phase has an inverse relationship to the subsequent cost of cooperation, reflected in a “formation technology” that can be different in each bilateral relationship. This generates a trade-off between the effort cost in forming ties, and the effort cost required to cooperate. For a range of parameters, this trade-off can imply that strategically isolated bilateral ties are never incentive compatible, and cooperative behavior can emerge only when ties are embedded in a network that pools incentive constraints.
With perfect monitoring virtually any network structure can emerge under the right configuration of parameters, even when strategically isolated ties are not incentive compatible. In particular, full cooperation can be supported by equilibria that dictate defection on every active relationship for any observed deviation, thereby pooling incentive constraints across the network. However, in a range of network applications, such rich monitoring structures seem implausible. For example, in settings such as rural villages in developing countries, farmers may have a disincentive to broadly disclose their history of wealth shocks. In a network of innovative firms, firms may be reluctant to disclose the content of their relationships in order to protect trade and technological secrets. Our primary analysis therefore focuses on an environment with private monitoring. We introduce monitoring restrictions via a radius of information ρ, which specifies the longest network path on which a player can observe actions. When , players can only observe actions on their own ties; when , players can also observe actions on their neighbor's ties; when , players can observe their neighbors' neighbors' ties, and so on.
With private monitoring expected payoffs depend on players' beliefs about actions outside their radius of information. As a result, equilibrium predictions can be sensitive to the choice of equilibrium refinement, which imposes structure on beliefs. Our approach to this problem is to provide upper and lower bounds on the set of equilibrium network outcomes by studying arguably the weakest and strongest equilibrium refinements that accommodate monitoring restrictions. To provide an upper bound, we consider Perfect Bayesian Equilibrium, which requires only that, at each information set, every player best-responds given some system of beliefs that is consistent with the equilibrium strategy profile. Perfect Bayesian Equilibrium (PBE) is therefore a weak equilibrium refinement that ensures sequential rationality given Bayesian beliefs. To provide a lower bound, we consider Belief-Free Equilibrium (Piccione, 2002; Ely and Välimäki, 2002; Ely et al., 2005; Hörner and Lovo, 2009; Hörner et al., 2011), which requires that play is sequentially rational for all beliefs that are consistent with the equilibrium strategy. Belief-Free Equilibrium (BFE) is therefore a strong refinement that requires strategic network interactions to be robust to all off-path beliefs.
Our first result shows that, for any radius of information , the same set of network outcomes can emerge under both PBE and BFE. In particular, regardless of the refinements imposed on off-path beliefs, private monitoring does not restrict the set of equilibrium networks relative to perfect information because, when a player deviates against one neighbor, all other neighbors can observe this deviation and punish immediately.
This strong equivalence breaks down when , which we call local monitoring. With local monitoring, players can deviate from cooperation on some ties without other neighbors observing this deviation. As a result, the network not only pools incentive constraints, but also plays a crucial role in the transmission of information. Players cannot monitor directly and so require contagion strategies that spread through the network to enforce cooperation. Since contagion takes time, the effectiveness of third-party punishment can depend on the distance between nodes, and tighter networks (with shorter paths between nodes) are needed to make cooperation incentive compatible. In such environments, we first characterize the network outcomes that can emerge in a PBE, and show how local monitoring restricts observable network outcomes relative to perfect monitoring by restricting network paths. Our characterization of PBE network outcomes thereby extends previous literature (e.g., Raub and Weesie, 1990; Lippert and Spagnolo, 2011) to a framework where incentives in the formation of ties are in tension with cooperative behavior on the resulting network.
Requiring belief-free best-responses generates much sharper predictions on network outcomes. In a BFE, we show that network ties must satisfy a “triadic closure” property, which requires that, in many cases, two neighbors of player i also need to be direct neighbors with each other. Triadic closure is both necessary and sufficient for belief-free enforcement mechanisms because it significantly reduces the time taken for contagion to spread through the network. At the network level, triadic closure implies a “clustering” of network relationships that is commonly observed in real-world networks. Our model provides a strategic explanation for such patterns. Overall, while BFE may be overly decisive in refining the set of equilibria in some applications, our results provide a useful boundary in distinguishing the network structures that can be rationalized under different assumptions about the extent of belief coordination.
Finally, our results address the relationship between heterogeneity in the formation technology, network structure and cooperative outcomes. In a setting with a homogeneous formation technology, a network can only facilitate cooperation when there are non-convexities in the technology. We provide a simple example of such non-convexities in Section 4.1, where a network of cooperative ties can emerge in equilibrium, and all heterogeneity in network ties are due to differential investments. When there are asymmetries in the formation technology, non-trivial networks can emerge in equilibrium even for convex formation technologies. We provide two such examples in Sections 4.2 and 4.3, where asymmetries in the formation technology seem natural, and relate them to applications in informal risk-sharing and information-sharing networks.
Our paper contributes to the literature by providing a novel economic rationale for the “small-world” property, which has been observed in a number of social and economic networks (Milgram, 1967; Watts, 1999). Small-world networks have two main characteristics: (1) small diameter and average path lengths, and (2) high clustering relative to networks generated by an independent random process (Jackson and Rogers, 2005). There is a long-standing interest in rationalizing such network structures: while an early literature uses mechanical network formation processes to generate small-worlds (see, e.g., Jackson and Rogers, 2007), there has been more recent interest in providing economic explanations. Jackson and Rogers (2005) provide a simple formation model that generates small-worlds based on heterogeneity in costs and benefits of forming ties, without modeling continuation play on the network explicitly. Jackson et al. (2012) show that, under perfect information with infrequent bilateral interactions, favor exchange is supported by renegotiation-proof equilibria through high clustering of relationships in “social quilts,” which generate a localized penalty to those who deviate. The clustering concept (“supported”) is a homogeneous relationships analogue of our version of triadic closure. Our model provides an alternative mechanism to rationalize the clustering observed by Jackson et al. (2012) based on the robust enforcement of cooperative norms under local monitoring.
Our paper also contributes to a literature on network enforcement under imperfect information, where players can choose heterogeneous actions on each tie.1 Raub and Weesie (1990) model the intuition that tightly-connected networks shorten contagion time, without making specific predictions about network structure. Lippert and Spagnolo (2011) study a fixed network of asymmetric, bilateral Prisoner's Dilemma games, and consider the role of key nodes in sustaining cooperation with limited Word-of-Mouth communication. Ali and Miller (2013) study the role of local monitoring in a setting with exponentially-distributed matching, where each player is limited to at most d partnerships, and show that the socially optimal network is composed of disconnected cliques of players. Balmaceda and Escobar (2017) study an optimal network design problem, and characterize networks under different assumptions about the flow of information through the network. Under limited information, cohesive networks tend to be efficient as they facilitate coordination and, similarly to Ali and Miller (2013), optimal networks have cliques of equally-sized sub-components.
We contribute to this literature in at least two ways. First, we do not start with an exogenous network, but explicitly model the formation and continuation of heterogeneous ties in a unified framework.2 Our model therefore generates predictions about a different observable. In models with a fixed network, equilibrium predictions address the degree of cooperative behavior, which is often difficult to measure in practice. Our results make predictions directly about network structure, which has been a growing topic of empirical research (see, e.g., Newman et al., 2006; Jackson, 2009; Latora et al., 2017). Extending the analysis to incorporate network formation explicitly highlights potential trade-offs between incentives in forming versus cooperating on ties. As we show, these trade-offs generate an endogenous source of tie heterogeneity, which in turn affects the role of the network in pooling incentive constraints. Second, we identify the range of equilibrium network outcomes in environments with local monitoring. Our PBE characterization provides arguably the weakest conditions that an equilibrium network should satisfy, while our results for BFE identify a very robust class of equilibrium network outcomes. The BFE result also generates new predictions on network structure in a local monitoring environment. The prior literature has often used normative criteria (i.e., efficiency) to analyze a subset of the PBE (e.g., Ali and Miller, 2013; Balmaceda and Escobar, 2017). Our result for BFE similarly highlights the importance of clustering and density for cooperative outcomes, but results from an entirely different mechanism that requires robust punishment strategies under local monitoring. This leads to a distinct structural prediction, which is more permissible than requiring completely-connected cliques or the complete network. Thus, our results provide a rationale for a network property (triadic closure) that has received long-standing attention in the broader networks literature (Simmel, 1908; Granovetter, 1973; Watts, 1999).
The remainder of the paper is organized as follows. In Section 2, we present our framework: a dynamic network game where players form heterogeneous ties and then decide whether to cooperate on ties in subsequent periods. Section 3 presents our main results. First, we consider the efficient outcomes that would emerge if players could perfectly commit to cooperate. We then assume that players are unable to commit to cooperate, but are able to partially monitor the behavior of other players. Finally, our main analysis considers network outcomes that can emerge in PBE and BFE under local monitoring. In Section 4, we illustrate some of the substantive restrictions of our results in the context of a risk-sharing and information-sharing examples. Section 5 concludes. Proofs are provided in a Supplementary Appendix.
Section snippets
Model
There is a finite set of players , time is indexed , and players have a common discount factor . The model has two phases. In period , players form bilateral ties in a formation phase. After ties are formed, in periods , players decide whether to cooperate on their ties in a continuation phase.
The ties between players can be viewed as a network, where nodes are players and ties are bilateral relationships of varying strength. Tie strength is determined by
Equilibrium network outcomes
In this section, we analyze equilibrium network outcomes. Our focus is on environments where players cannot commit to cooperative behavior, and efficient networks are then generally not incentive compatible. For partial monitoring we give a strong equivalence result for PBE and BFE. We then study environments with local monitoring, where PBE and BFE identify a range of equilibrium networks outcomes.
Examples
In our framework, the role of a network is most pronounced when no pair of players can form a tie that satisfies the BIC. Such environments are characterized by the following condition: Assumption 1 For all , either for all or for all .
Conclusion
We develop a model to study the formation of bilateral ties and enforcement of cooperative behavior on the resulting network. Our analysis primarily focuses on an environment where (i) players cannot pre-commit to cooperation, (ii) the formation of strategically isolated bilateral ties is not incentive compatible, and (iii) players cannot observe the bilateral interactions of others. As a result, cooperative behavior can emerge only when ties are embedded in a network that can pool incentive
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We are grateful to the editor, Xavier Vives, an associate editor and three referees for very helpful comments. We also thank Chris Barrett, Kaushik Basu, Larry Blume, Adam Brandenburger, David Easley, Ani Guerdjikova, Johannes Hoerner, Willemien Kets, Corey Lang, Samreen Malik, Debraj Ray and Fernando Vega-Redondo for useful discussions. Toth gratefully acknowledges support from the NSF Expeditions in Computing grant on Computational Sustainability, #0832782.