Robust mechanisms : the curvature case

This note considers the problem of a principal (she) who faces a privately informed agent (he) and only knows one moment of the distribution from which his types are drawn. Payoffs are non-linear in the allocation and the principal maximizes her worst-case expected profits. We recast the robust design problem as a zero-sum game played by the principal and an adversarial nature who seeks to minimize her expected payoffs. The robust mechanism and the worst case distribution are, then, the Nash equilibrium of such game. A robustness property of the optimal mechanism imposes restrictions on the principal's ex-post profit function. These restrictions then lead to the optimal mechanism. The robust mechanism entails exclusion of low types and distortions at the intensive margin that (in a precise sense) are larger than what those that prevail in standard Bayesian mechanism design problems.