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 parsimonious 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 a nondecreasing and concave expected-action rule 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. Moreover, when the mean is below the agent's bias, an optimum can be implemented by supplementing the random cap with an incentive-neutral outcome lottery, while pure random caps are strictly suboptimal.