Continuous Implementation

It is well-known that mechanism design literature makes many simplifying informational assumptions in particular in terms of common knowledge of the environment among players. In this paper, we introduce a notion of continuous implementation and characterize when a social choice function is continuously implementable. More specifically, we say that a social choice function is continuously (partially) implementable if it is (partially) implementable for types in the model under study and it continues to be (partially) implementable for types "close" to this initial model. We …rst show that if the model is of complete information a social choice function is continuously (partially) implementable only if it satis…es Maskin’s monotonicity. We then extend this result to general incomplete information settings and show that a social choice function is continuously (partially) implementable only if it is fully implementable in iterative dominance. For …nite mechanisms, this condition is also su¢ cient. We also discuss implications of this characterization for the virtual implementation approach. Journal of Economic Literature Classi…cation Numbers: C79, D82 Keywords: High order beliefs, Robust implementation First version: February 15, 2007. This version: May 11, 2009. We wish to thank Dirk Bergemann, Philippe Jehiel, Atsushi Kajii, Eric Maskin, Nicolas Vieille and seminar participants at Paris School of Economics, the EEA meeting in Budapest, Princeton University, the Institute for Advanced Study in Princeton, Rutgers University, Mc Guil University, the world congress of game theory in Northwestern, the workshop on new topics in mechanism design in Madrid, GRIPS workshop on Global Games in Tokyo and the PSE-Northwestern joint workshop. We are especially grateful to Stephen Morris for helpful remarks and discussions at various stages of this research. Olivier Tercieux thanks the Institute for Advanced Study at Princeton for …nancial support through the Deutsche Bank membership. He is also grateful to the Institute for its hospitality.