We analyze the problem of fully implementing a social choice set in ex post equilibrium. We identify an ex post monotonicity condition that is necessary and — in economic environments — suﬀicient for full implementation in ex post equilibrium. We also identify an ex post monotonicity no veto condition that is suﬀicient. Ex post monotonicity is satisﬁed in all single crossing environments with strict ex post incentive constraints. In many economically signiﬁcant environments, ex post implementation can be achieved in the direct mechanism. We show by means of two classic examples that ex post monotonicity does not imply nor is it implied by Maskin monotonicity (necessary and almost suﬀicient for complete information implementation). The single unit auction with interdependent valuations is shown to satisfy ex post monotonicity but not Maskin monotonicity. Ex post implementation in the direct mechanism is also possible in this case. We describe an example where the Pareto correspondence fails ex post monotonicity but Maskin monotonicity is satisﬁed.
Bergemann, Dirk and Morris, Stephen, "Ex Post Implementation" (2005). Cowles Foundation Discussion Papers. 1784.