Incentive Compatibility and Belief Restrictions

We study a framework for robust mechanism design that can accommodate various degrees of robustness with respect to agents’ beliefs, and which includes both the belief-free and Bayesian settings as special cases. For general belief restrictions, we characterize the set of incentive compatible direct mechanisms in general environments with interdependent values. The necessary conditions that we identify, based on a first-order approach, provide a unified view of several known results, as well as novel ones, including a robust version of the revenue equivalence theorem that holds under a notion of generalized independence that also applies to non-Bayesian settings. Our main characterizations informthe design of belief-based terms, in pursuit of various objectives in mechanism design, including attaining incentive compatibility in environments that violate standard single-crossing and monotonicity conditions. We discuss several implications of these results. For instance, we show that, under weak conditions on the belief restrictions, any allocation rule can be implemented, but full rent extraction need not follow. Information rents are generally possible, and they decrease monotonically as the robustness requirements are weakened.