  In practice, incentive providers (i.e., principals) often cannot observe the
reward realizations of incentivized agents, which is in contrast to many
principal-agent models that have been previously studied. This information
asymmetry challenges the principal to consistently estimate the agent's unknown
rewards by solely watching the agent's decisions, which becomes even more
challenging when the agent has to learn its own rewards. This complex setting
is observed in various real-life scenarios ranging from renewable energy
storage contracts to personalized healthcare incentives. Hence, it offers not
only interesting theoretical questions but also wide practical relevance. This
paper explores a repeated adverse selection game between a self-interested
learning agent and a learning principal. The agent tackles a multi-armed bandit
(MAB) problem to maximize their expected reward plus incentive. On top of the
agent's learning, the principal trains a parallel algorithm and faces a
trade-off between consistently estimating the agent's unknown rewards and
maximizing their own utility by offering adaptive incentives to lead the agent.
For a non-parametric model, we introduce an estimator whose only input is the
history of principal's incentives and agent's choices. We unite this estimator
with a proposed data-driven incentive policy within a MAB framework. Without
restricting the type of the agent's algorithm, we prove finite-sample
consistency of the estimator and a rigorous regret bound for the principal by
considering the sequential externality imposed by the agent. Lastly, our
theoretical results are reinforced by simulations justifying applicability of
our framework to green energy aggregator contracts.
