Regret Minimization in Single-Dimensional Contract-Design with Binary Actions

We study principal-agent problems in which a principal commits to an outcome-dependent payment scheme (i.e., a contract) in order to induce an agent to take a costly action leading to a favorable outcome. We consider the online extension of the classical (one-shot) principal-agent problem, in which the principal repeatedly interacts with agents by proposing contracts over multiple rounds. The principal has no information about the agents and, crucially, does not observe their actions. As a result, the principal must learn an optimal contract using only the realized outcomes observed at each round. We focus on the setting with binary-actions and single-dimensional agent types, where the agent's private type represents their cost per unit-of-effort. For adversarial-type sequences, we provide tight $\widetilde\Theta(T^{2/3})$ regret guarantees. Remarkably, this rate is completely independent of the number of outcomes $m$. The upper bound is based on two key components: 1) a reduction to one-dimensional threshold optimization problem and 2) a non-uniform discretization to handle the non-Lipschitz nature of the problem. Moreover, in the case of a single (fixed) hidden type, we show that it is possible to improve the rates and provide a tight $\widetilde\Theta(\sqrt{T})$ regret bound. Our algorithm is based on an explore-then-commit strategy where we first approximately learn the hidden type via a stochastic binary search, and then we commit to a ``robustified'' near-optimal contract.