Robust virtual implementation: Toward a reinterpretation of the Wilson doctrine

We study a mechanism design problem where arbitrary restrictions are placed on the sets of first-order beliefs of agents. Calling these restrictions ∆, we use ∆-rationalizability (Battigalli and Siniscalchi (2003)) as our solution concept, and require that a mechanism virtually implement a socially desirable outcome. We obtain two necessary conditions, ∆-incentive compatibility and ∆-measurability and show that the latter is satisfied as long as a particular zero-measure set of first order beliefs is ruled out. In environments allowing small transfers of utility among agents, these two conditions are also sufficient. JEL Classification: C72, D78, D82.