Accelerator and Brake: Dynamic Persuasion with Dead Ends

We study optimal dynamic persuasion in a bandit experimentation model where a principal, unlike in standard settings, has a single-peaked preference over the agent's stopping time. This non-monotonic preference arises because maximizing the agent's effort is not always in the principal's best interest, as it may lead to a dead end. The principal privately observes the agent's payoff upon success and uses the information as the instrument of incentives. We show that the optimal dynamic information policy involves at most two one-shot disclosures: an accelerator before the principal's optimal stopping time, persuading the agent to be optimistic, and a brake after the principal's optimal stopping time, persuading the agent to be pessimistic. A key insight of our analysis is that the optimal disclosure pattern -- whether gradual or one-shot -- depends on how the principal resolves a trade-off between the mean of stopping times and its riskiness. We identify the Arrow-Pratt coefficient of absolute risk aversion as a sufficient statistic for determining the optimal disclosure structure.