  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.
