Quickest Time Herding and Detection for Optimal Social Learning

This paper considers social learning amongst rational agents (for example, sensors in a network). We consider three models of social learning in increasing order of sophistication. In the first model, based on its private observation of a noisy underlying state process, each agent selfishly optimizes its local utility and broadcasts its action. This protocol leads to a herding behavior where the agents eventually choose the same action irrespective of their observations. We then formulate a second more general model where each agent is benevolent and chooses its sensor-mode to optimize a social welfare function to facilitate social learning. Using lattice programming and stochastic orders, it is shown that the optimal decision each agent makes is characterized by a switching curve on the space of Bayesian distributions. We then present a third more general model where social learning takes place to achieve quickest time change detection. Both geometric and phase-type change time distributions are considered. It is proved that the optimal decision is again characterized by a switching curve We present a stochastic approximation (adaptive filtering) algorithms to estimate this switching curve. Finally, we present extensions of the social learning model in a changing world (Markovian target) where agents learn in multiple iterations. By using Blackwell stochastic dominance, we give conditions under which myopic decisions are optimal. We also analyze the effect of target dynamics on the social welfare cost.