This work considers a repeated principal-agent bandit game, where the
principal can only interact with her environment through the agent. The
principal and the agent have misaligned objectives and the choice of action is
only left to the agent. However, the principal can influence the agent's
decisions by offering incentives which add up to his rewards. The principal
aims to iteratively learn an incentive policy to maximize her own total
utility. This framework extends usual bandit problems and is motivated by
several practical applications, such as healthcare or ecological taxation,
where traditionally used mechanism design theories often overlook the learning
aspect of the problem. We present nearly optimal (with respect to a horizon
$T$) learning algorithms for the principal's regret in both multi-armed and
linear contextual settings. Finally, we support our theoretical guarantees
through numerical experiments.

这项研究考虑了一个反复的委托人 - 代理人强盗游戏，委托人仅能通过代理人与环境进行交互。委托人和代理人的目标不一致，选择行动的权限仅留给代理人。然而，委托人可以通过提供激励来影响代理人的决策，这些激励为代理人的回报增加。委托人旨在迭代学习一种激励策略以最大化自己的总效用。该框架扩展了传统的强盗问题，并受到了诸多实际应用的启发，如医疗保健或生态税收，这些领域中传统机制设计理论经常忽视问题的学习方面。我们提出了几乎最优（就 $T$ 而言）的学习算法，用于委托人在多臂和线性背景环境中的遗憾，并通过数值实验验证了我们的理论保证。