  We study a dynamic model of Bayesian persuasion in sequential decision-making
settings. An informed principal observes an external parameter of the world and
advises an uninformed agent about actions to take over time. The agent takes
actions in each time step based on the current state, the principal's
advice/signal, and beliefs about the external parameter. The action of the
agent updates the state according to a stochastic process. The model arises
naturally in many applications, e.g., an app (the principal) can advice the
user (the agent) on possible choices between actions based on additional
real-time information the app has. We study the problem of designing a
signaling strategy from the principal's point of view. We show that the
principal has an optimal strategy against a myopic agent, who only optimizes
their rewards locally, and the optimal strategy can be computed in polynomial
time. In contrast, it is NP-hard to approximate an optimal policy against a
far-sighted agent. Further, if the principal has the power to threaten the
agent by not providing future signals, then we can efficiently compute a
threat-based strategy. This strategy guarantees the principal's payoff as if
playing against an agent who is far-sighted but myopic to future signals.
