Bayesian Learning in Social Networks

Abstract

We study the (perfect Bayesian) equilibrium of a model of learning over a general social network. Each individual receives a signal about the underlying state of the world, observes the past actions of a stochastically-generated neighborhood of individuals, and chooses one of two possible actions. The stochastic process generating the neighborhoods defines the network topology (social network). The special case where each individual observes all past actions has been widely studied in the literature. We characterize pure-strategy equilibria for arbitrary stochastic and deterministic social networks and characterize the conditions under which there will be asymptotic learning-that is, the conditions under which, as the social network becomes large, individuals converge (in probability) to taking the right action. We show that when private beliefs are unbounded (meaning that the implied likelihood ratios are unbounded), there will be asymptotic learning as long as there is some minimal amount of "expansion in observations". Our main theorem shows that when the probability that each individual observes some other individual from the recent past converges to one as the social network becomes large, unbounded private beliefs are sufficient to ensure asymptotic learning. This theorem therefore establishes that, with unbounded private beliefs, there will be asymptotic learning in almost all reasonable social networks. We also show that for most network topologies, when private beliefs are bounded, there will not be asymptotic learning. In addition, in contrast to the special case where all past actions are observed, asymptotic learning is possible even with bounded beliefs in certain stochastic network topologies.Keywords: information aggregation, learning, social networks, herding, information cascades.JEL Classification: C72, D83. * We thank Lones Smith and Peter Sorensen for useful comments and suggestions. We gratefully acknowledge financial support from the AFOSR and the NSF.

An Overview of Social Networks and Economic Applications

“…Allowing for heterogeneous preferences, and other idiosyncratic preferences that favor particular types of restaurants, can also change the results. These are the subjects of a set of papers such as Bikhchandani, Hirshleifer and Welch (1998), Smith and Sorensen (2000), Celen and Kariv (2004a,b), and Acemoglu et al (2008). For example, Acemoglu et al (2008) provide results on how social learning depends on how neighborhoods of which agents observe which other agents develop over time, and also on how accurate various agents' signals are.…”

“…These are the subjects of a set of papers such as Bikhchandani, Hirshleifer and Welch (1998), Smith and Sorensen (2000), Celen and Kariv (2004a,b), and Acemoglu et al (2008). For example, Acemoglu et al (2008) provide results on how social learning depends on how neighborhoods of which agents observe which other agents develop over time, and also on how accurate various agents' signals are. Although the results show that conclusions about social learning are somewhat case-specific, the results do have nice intuitions about things like the precision of information and who observes whom.…”

Herding with collective preferences

“…10 Acemoglu et al (2010) analyze a complementary set of issues about observational learning in networks by maintaining payoff independence and instead using a network to represent the structure of observability of actions.…”

Communication and Influence

“…Information would then come from direct communication and from learning of the past activity of some or all agents in the organization. This relates to a recent literature on social learning in networks (see, for example, Bala andGoyal 1998 andAcemoglu et al 2011) and we plan to pursue this analysis in future research.…”