Harnessing the Continuous Structure: Utilizing the First-order Approach in Online Contract Design

This work studies the online contract design problem. The principal's goal is to learn the optimal contract that maximizes her utility through repeated interactions, without prior knowledge of the agent's type (i.e., the agent's cost and production functions). We leverage the structure provided by continuous action spaces, which allows the application of first-order conditions (FOC) to characterize the agent's behavior. In some cases, we utilize conditions from the first-order approach (FOA) in economics, but in certain settings, we are able to apply FOC without additional assumptions, leading to simpler and more principled algorithms. We illustrate this approach in three problem settings. Firstly, we study the problem of learning the optimal contract when there can be many outcomes. In contrast to prior works that design highly specialized algorithms, we show that the problem can be directly reduced to Lipschitz bandits. Secondly, we study the problem of learning linear contracts. While the contracting problem involves hidden action (moral hazard) and the pricing problem involves hidden value (adverse selection), the two problems share a similar optimization structure, which enables direct reduction between the problem of learning linear contracts and dynamic pricing. Thirdly, we study the problem of learning contracts with many outcomes when agents are identical and provide an algorithm with polynomial sample complexity.