Quickest time detection and constrained optimal social learning with variance penalty

This paper considers Bayesian quickest time change detection with phase-type distributed change and a variance stopping penalty. Using lattice programming and stochastic orders, we prove that the optimal decision policy has a threshold switching curve structure on the space of posterior distributions. We then consider example in constrained optimal social learning. Each agent is benevolent and chooses its mode to reveal full information or herd to optimize a social welfare function to facilitate social learning. It is proved that the optimal decision for quickest time herding is characterized by a switching curve.