Bayesian learning: engineered versus social networks

We consider a classical model of distributed decision making, originally developed in engineering contexts [2,4,5], but which has also attracted recent attention in the social sciences [1]. Suppose that there are two hypotheses on the state of the world and that each of several nodes (or agents, decision makers, sensors, etc.) receives a conditionally independent measurement, with a different distribution under each hypothesis. The nodes act in sequence, and their interactions are modeled by a given directed acyclic graph. Each node computes a binary message, on the basis of its own observation and the messages it receives from its predecessors, and sends it to its successors. If we interpret the binary message as a decision in favor of one or the other hypothesis, a central question is whether the network will “learn, ” that is, whether the decision by the nth node is correct, with probability that approaches zero as n→∞. In the engineering version of the problem, the decision making rules of each node are chosen by a global designer, whose objective is to miminize the probability of an incorrect decision by the last node. In contrast, in the social science literature each node is assumed to act myopically, that is, to choose a message that minimizes its own probability of error. We overview positive and negative results under the two alternative formulations, for two simple network structures: a tandem, whereby a node only hears the message of its immediate predecessor, and an augmented tandem, where a node hears the message of its K immediat