Random Cap: Optimal Informationally Robust Delegation

Are simple delegation rules optimal under ambiguity? We study delegation when the principal knows the mean, but not the distribution, of the agent's private information. In a quadratic constant-bias environment, the robustly optimal randomized mechanism is a random cap: the principal draws and reveals an upper bound below which the agent chooses freely. Randomization strictly outperforms every deterministic cap by hedging against cap-specific worst-case distributions. We characterize random caps through nondecreasing and concave expected-action rules and construct the solution using a saddle-point approach. The worst-case distribution features an exponential survival function over its continuous region and an atom at the upper endpoint. Under regularity conditions, the result extends to convex-order ambiguity. When the mean is below the agent's bias, the optimal mechanism additionally requires an incentive-neutral outcome lottery.