  Randomized experiments can be susceptible to selection bias due to potential
non-compliance by the participants. While much of the existing work has studied
compliance as a static behavior, we propose a game-theoretic model to study
compliance as dynamic behavior that may change over time. In rounds, a social
planner interacts with a sequence of heterogeneous agents who arrive with their
unobserved private type that determines both their prior preferences across the
actions (e.g., control and treatment) and their baseline rewards without taking
any treatment. The planner provides each agent with a randomized recommendation
that may alter their beliefs and their action selection. We develop a novel
recommendation mechanism that views the planner's recommendation as a form of
instrumental variable (IV) that only affects an agents' action selection, but
not the observed rewards. We construct such IVs by carefully mapping the
history -- the interactions between the planner and the previous agents -- to a
random recommendation. Even though the initial agents may be completely
non-compliant, our mechanism can incentivize compliance over time, thereby
enabling the estimation of the treatment effect of each treatment, and
minimizing the cumulative regret of the planner whose goal is to identify the
optimal treatment.
