Ex post implementation

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 su¢ cient for full implementation in ex post equilibrium. We also identify an ex post monotonicity no veto condition that is su¢ cient. Ex post monotonicity is satis…ed in all single crossing environments with strict ex post incentive constraints. In many economically signi…cant 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 su¢ cient 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 satis…ed. Keywords: Ex Post Equilibrium, Implementation, Single Crossing, Interdependent Values. JEL Classification: C79, D82 This research is supported by NSF Grant #SES-0095321. The …rst author gratefully acknowledges support through a DFG Mercator Research Professorship at the Center of Economic Studies at the University of Munich. We bene…ted from discussions with Matt Jackson, Andy Postlewaite, Phil Reny, Mike Riordan and Roberto Serrano. We would like to thank seminar audiences at California Institute of Technology, Columbia University, Cornell University, New York University, Nu¢ eld College, Princeton University, University of Michigan, the Paris Roy seminar and the Cowles Foundation Conference on "Robust Mechanism Design" for helpful comments. Parts of this paper were reported in early drafts of our work on Robust Mechanism Design (Bergemann and Morris (2001)). yDepartment of Economics, Yale University, 28 Hillhouse Avenue, New Haven, CT 06511, [email protected]. zDepartment of Economics, Yale University, 30 Hillhouse Avenue, New Haven, CT 06511, [email protected].