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, which encompasses 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. Our main results, which we obtain based on a first-order approach, inform the design of transfers via 'belief-based' terms to attain incentive compatibility. In environments that satisfy a property of generalized independence, our results imply a robust version of revenue equivalence in non-Bayesian settings. Instead, under a notion of comovement between types and beliefs, which extends the idea of correlated information to non-Bayesian settings, we show that any allocation rule can be implemented, even if standard single-crossing and monotonicity conditions do not hold. Yet, unless the environment is Bayesian, information rents typically remain, and they decrease monotonically as the robustness requirements are weakened.