Robust Contracts with Exploration

We study a two-period moral hazard problem; there are two agents, with action sets that are unknown to the principal. The principal contracts with each agent sequentially, and seeks to maximize the worst-case discounted sum of payoffs, where the worst case is over the possible action sets. The principal observes the action chosen by the first agent, and then offers a new contract to the second agent based on this knowledge, thus having the opportunity to explore in the first period. We introduce and compare three different notions of dynamic worst-case considerations. Within each notion, we define a suitable rule of updating and characterize the principal's optimal payoff guarantee. We find that linear contracts are robustly optimal not only in static settings, but also in dynamic environments with exploration.